The notion of universe allows one to examine from another angle the difference between the canonical elements of **Bool**, **yes** and **no**, and the expressions **true** and **false** as applied to a proposition. As mentioned above in the case of the empty set, the dialogical setting allows reading the statement

X!A

as expressing that player X states that there is at least one possible solution or answer to the enquiry A.

In the case of

X! Bool

the statement expresses that X is committed that at least one of the two possible answers to the enquiry associated with the set **Bool**holds. For example

X 0 : Bool

which is certainly different from establishing that there is no possible answer to the enquiry

X! ¬Bool

Now, one consequence of this distinction is that in general we cannot
demonstrate—that is, we cannot develop a winning strategy—in such a system that the
elements of **Bool** are different, or not identical: **¬Id(Bool, yes, no)**, unless we assume
that **yes** and **no** are associated to the codes of two disjoint sets, which are elements
of a *universe*. In fact it was shown by Jan Smith (1988, pp. 842-843) by means of a
metamathematical demonstration, that for any type A, the demonstration of an inequality
of the form **¬Id(A, a, b)** requires *universes* constituted by codes of sets.

**Demonstration that yes and no are not identical in Bool**

In order to develop a winnings strategy for **¬Id(Bool, yes, no)** —i.e.
**Id(Bool, no, yes)** ⊃⊥—we follow the basic ideas of Martin-Löf's (1984, p. 51) and
Nordström, Petersson & Smith's (1990, p. 86) demonstration of Peano's fourth axiom.

The main idea is introduce a predicate defined over **Bool**, more precisely the
function **G(x)** that evaluates in the universe **u**.<fnref n="136" /> Since it evaluates in **u**, the function
yield codes, namely, if **x** is **no**, then it yields **n**⁰, and it yields **n**¹, if **x** is **yes**. The codes
**n**⁰ and **n**¹ are codes for the empty set N⁰ and the unary set N¹ respectively. So, **yes** and
**no** are associated to two disjoint sets in **u**: thus, since the predicate **G(x)** applies to **yes**,
but yields the empty set when applied to **no**, then **yes** and **no** cannot be identical.
Moreover, the assumption that both of the canonical elements of **Bool** are identical
would lead to conclude that the empty set is inhabited, and this proves its negation.<fnref n="137" />

<footnote n="136">¹³⁶ Since it is a predicate over **Bool**, it follows the rules for the analysis of these kind of statements (in the CTT its definition stems from the elimination rules for **Bool**).</footnote>

<footnote n="137">¹³⁷ The CTT-demonstration in a nutshell is the following, with the arrow standing for functions, and the original notation of Martin-Löf where **n_0** is the notation for code and **t, f** are the canonical elements of **Bool** (see chapter II):</footnote>

<footnote>Define a family of sets $G : \mathbf{Bool} \to \mathcal{U}$. $G(x) :=_{\mathrm{dr}}$ if $x$ then $\mathbf{n}_1$, else $\mathbf{n}_0 : \mathcal{U}$ [$x : \mathbf{Bool}$].</footnote>

<footnote>$F : \mathbf{Bool} \rightarrow \mathit{set}$, by $F(x) :=_{\mathrm{dr}} \mathfrak{F}(G(x)) : \mathit{set}$ [$x : \mathbf{Bool}$].</footnote>

<footnote>tt : 𝕿(G(t)) (given tt : N¹, G(t) = n_1 : 𝕿, and 𝕿(G(t)) = N¹: set, thus tt : F(t).</footnote>
In the dialogical setting we formulate a Socratic rule specific to $G(x)$. We also provide the rule of synthesis specific to the unary set $\mathbb{N}^1$.

<table><caption>Table 48: Specific Socratic rules for $G(x)$ and synthesis of $\mathbb{N}^1$</caption><thead><tr><th></th><th>Move</th><th>Challenge</th><th>Defence</th></tr></thead><tbody><tr><td rowspan="3">Specific Socratic rule to G(x)</td><td>X G(x): U(x: Bool)</td><td>Y no: Bool<br>Y yes: Bool</td><td>X (G(no) = n<sup>0</sup>: U<br>X (G(yes) = n<sup>1</sup>: U<br>X (G(yes) = n<sup>1</sup>: U)</td></tr><tr><td>X G(no) = n<sup>0</sup>: U</td><td>Y ? T(G(no))</td><td>X T(G(no)) = N<sup>0</sup>: set</td></tr><tr><td>X G(yes) = n<sup>1</sup>: U</td><td>Y ? T(G(yes))</td><td>X T(G(yes)) = N<sup>1</sup>: set</td></tr><tr><td>Synthesis of N<sup>1</sup></td><td>X ! N<sup>1</sup></td><td>Y? N<sup>1</sup></td><td>X yesyes: N<sup>1</sup></td></tr></tbody></table>

**Note**

In the case of $Y$'s challenge of $X G(x): U(x: Bool)$, if $Y$ is $P$, then the challenge assumes that $O$ already conceded **yes, no : Bool**.

We will only display here the relevant play for the determination of the winning strategy for $\neg Id(Bool, yes, no)$. The thesis is stated under the condition that $O$ concedes the codes $n^0$ and $n^1$ are elements of $U$, the canonical answers (elements) of $Bool$ and the special predicate (function) $G(x)$ [$x : Bool$] defined by the specific Socratic rule given above.

<table><caption>Play 19: $Id(Bool, yes, no) \supset \perp [n^0, n^1: U; N^1; yes, no : Bool; G(x): U[x: Bool]]$</caption><thead><tr><th></th><th>O</th><th></th><th>P</th><th></th></tr></thead><tbody><tr><td>0.1</td><td>n<sup>0</sup>, n<sup>1</sup>: U</td><td></td><td>! Id(Bool, yes, no) &supset; &perp;</td><td>0</td></tr><tr><td>0.2</td><td>! N<sup>1</sup></td><td></td><td>[n<sup>0</sup>, n<sup>1</sup>: U; N<sup>1</sup>;<br>yes, no : Bool;<br>G(x): U[x: Bool]]</td><td></td></tr><tr><td>0.3</td><td>yes, no : Bool</td><td></td><td></td><td></td></tr><tr><td>0.4</td><td>G(x) : U[x: Bool]</td><td></td><td></td><td></td></tr><tr><td>1</td><td>m := 1</td><td></td><td>n := 2</td><td>2</td></tr><tr><td>3</td><td>p<sub>1</sub> : Id(Bool, yes, no)</td><td>0</td><td>you<sub>gave up</sub>(19): &perp;</td><td>20</td></tr><tr><td>5</td><td>G(yes) = n<sup>1</sup>: U</td><td>0.4</td><td>yes : Bool</td><td>4</td></tr><tr><td>7</td><td>T(G(yes)) = N<sup>1</sup>: set</td><td>5</td><td>? T(G(yes))</td><td>6</td></tr><tr><td>9</td><td>yesyes : N<sup>1</sup></td><td>4</td><td>? N<sup>1</sup></td><td>8</td></tr><tr><td>11</td><td>yesyes : T(G(yes))</td><td>9, 7</td><td>? T(G(yes)) / N<sup>1</sup></td><td>10</td></tr><tr><td>13</td><td>G(no) = n<sup>0</sup>: U</td><td>0.4</td><td>no : Bool</td><td>12</td></tr><tr><td>15</td><td>T(G(no)) = N<sup>0</sup>: set</td><td>13</td><td>? T(G(no))</td><td>14</td></tr><tr><td>17</td><td>Lbz-Id-subst(p<sub>1</sub>, yesyes): T(G(no))</td><td>11, 3</td><td>? Lbz-Id-subst no/yes</td><td>16</td></tr><tr><td>19</td><td>Lbz-Id-subst(p<sub>1</sub>, yesyes) : N<sup>0</sup></td><td>17, 15</td><td>T(G(no)) / N<sup>0</sup></td><td>18</td></tr></tbody></table>

P wins.

Assume $z : Id(Bool, t, f)$, then $subst(z, tt) : F(f)$.

Hence, $(\lambda.z)subst(z, tt) : \neg Id(Bool, t, f)$.
## Description of the play

* **Moves 0-5:** After O's challenge (move 3) on the thesis, P counter-attacks (4) the concession 0.4, following the prescription of the Socratic rules specific to $G(x)$. P can carry out this challenge because of concession 0.3. In fact it is justified in the Copy-cat rule—we skip here the further challenge of O asking for a justification and P's answer with the reflexivity **yes** = **yes**: **Bool**.

* **Moves 6-7** follow from implementing the decoding prescription for $G(x)$.

* **Moves 8-11:** After O provides the local reason **yes** for the unary set $N^1$, P asks O to substitute $N^1$ by $T(G(\textbf{yes}))$, given the equality conceded move 7.

* **Moves 12-15:** P repeats moves 4,6, but chooses this time **no**: **Bool**—we also skip here the moves leading to the reflexivity **no** = **no**: **Bool**.

* **Moves 16-20:** Move 16 is the crucial move and leads to the victory of P; P requests of O to replace **yes** with **no** within move 11, given the identity conceded in move 3 and given the *Leibniz-substitution rule* for *Id*. O's response (17) and her concession (15) that $T(G(no))$ and $N^0$ are equal sets leads her to state the giving up move 19. Indeed, in move 19 O is forced to admit that following her own moves the empty set (of answers) is not empty. In move 20, P can answer O's first challenge which was left unanswered by bringing forward the local reason *you gave up* (19) for defending his statement, which happens to be $\perp$ (see section VII.2.1, SR7: Winning rule for plays). He thus wins the play.

After a *recapitulation* of the possible moves, P can bring forward *you gave up* (19) as a strategic reason for grounding his thesis and thus assert:

$$you_{gave\ up}(19): \mathit{Id}(\mathit{N}, 0, s(x)) \supset \perp$$

We did not include this in the play, since we did not develop the whole strategy.

## The fourth axiom of Peano's arithmetic

The demonstration of the *fourth axiom of Peano's arithmetic*, “0 is identical to no successor of a natural number”:

$$(\forall x: \mathbb{N}) \neg \mathit{Id}(\mathbb{N}, 0, s(x))$$

is very close to the precedent one: Peano's fourth axiom was demonstrated first by Martin-Löf (1984, p. 51) using strong elimination rules for *Id*. Nordström, Petersson & Smith (1990, p. 86) provide a demonstration without those rules. Instead of a function defined over **Bool**, what is required is a function $H(x)$, defined over the natural numbers such that the value is the code for the unary set if the $x$ is 0, and it is the code for the empty set if $x$ is the successor of any natural number—thus there will be a predicate that applies to 0 but not to any other natural number, which contradicts that 0 and the successor of a natural number are identical. We leave to the diligent reader the development of both the dialogical rules for $H(x)$ and of the relevant play for building the winning strategy—notice that $H(x)$, will be defined following the rules of analysis for predicates defined over $\mathbb{N}$.

The conceptual background underlying these demonstrations is that in order to demonstrate that the canonical elements of **Bool** and $\mathbb{N}$ are different, we need to have a look from the outside of the respective sets and assume that there is a universe such that the Boolean **1** amounts to a code for the **truth**, the unary set; and **0** amounts to a code for
the **false**, namely the empty set. This elucidates George Boole's own use of **1** and **0**, both as selective functions and as the universal domain $\mathbb{T}$ and the empty set $\perp$.<fnref n="138" />

Let us now extend the set **Bool** and study some applications for truth-functional non-classical logics

**Integrating many-valued logics: Operations within larger sets**

Given some finite set $\mathbb{N}^n$ as defined above, we can define operations over it. For example the three-element set $\mathbb{N}^3$ can yield operations that correspond to a three-valued logic, based on the answers, **yes**, **?**, **no**. So $(a+^3b)$ is equal to **?**, if one of the elements is equal to **no** and the other to **?**, or both $a$ and $b$ are equal to **?**.

More generally, for any set $\mathbb{N}^n$ with elements **0**, **1**, ..., **n**, with minimum **0** and maximum **n**,<fnref n="139" /> and with the help of the following definition of $x \le y$ and its inverse $x \ge y$:

$$x \le y = (\exists z: \mathbb{N}) \mathit{Id}(\mathbb{N}, x + z, y) : \mathbf{prop} [x: \mathbb{N}, y: \mathbb{N}]$$

$$x \ge y = (\exists z: \mathbb{N}) \mathit{Id}(\mathbb{N}, x - z, y) : \mathbf{prop} [x: \mathbb{N}, y: \mathbb{N}]$$

we obtain the following:

* $(a \times^n b)$ is equal to $a = m$ if $m \le m' = b$, otherwise it is equal to $m' = b$;

* $(a+^n b)$ is equal to $a = m$ if $m \ge m' = b$, otherwise it is equal to $m' = b$;

* $\sim^n a$ is equal to $n - m$, where $n$ is the maximum and $m = a$.

What is more, $(a \to^n b)$ can be defined as $\sim^n a + \sim^n b$.

The dialogical formulation of this generalization is straightforward:

* the defender states that some operation is an element of $\mathbb{N}^n$;

* the challenger requests that the defender shows the operation is equal to some element of $\mathbb{N}^n$;

* once the defender chooses one of the elements, the challenger can request that he shows that this choice satisfies the $\le$ (or $\ge$) condition defining that operation.

For the sake of simplicity we will not display this last request. The following example should be enough. Assume that the defender stated

$$X (a \times^n b) = m = a: \mathbb{N}^n$$

The challenger can then ask to check if $m$ satisfies the $m \le m'$ condition required by the operator $\times^n$.<fnref n="140" /> Challenge and defence have the following form

$$Y? m \le m'$$

(Does $m$ satisfy the condition $m \le m'$ ?)

$$X m \le m': \mathbb{N}^n$$

<footnote n="138">¹³⁸ For a discussion on this ambiguity see (Prior, 1949).</footnote>

<footnote n="139">¹³⁹ Where **0** can be interpreted as corresponding to the lowest truth-value and **n** to the highest truth-value of some **n**-valued logic.</footnote>

<footnote n="140">¹⁴⁰ Again we assume the definition of "$\le$" and "$\ge$". The dialogical formulation of it deploys a Socratic rule specific to that relation, namely, if player $X$ states $n \le m$, given $x: \mathbb{N}, y: \mathbb{N}$, then $Y$ can ask $X$ to choose a $z: \mathbb{N}$ such that $n + z = m$. Similarly for "$\ge$".</footnote>
<table><caption>Table 49: dialogical rules for operations in many-valued logics</caption><thead><tr><th rowspan="2"></th><th rowspan="2">Move</th><th colspan="2">Synthesis of local reasons</th><th rowspan="2">Synthesis of strategic reasons</th></tr><tr><th>Challenge</th><th>Defence</th></tr></thead><tbody><tr><td>Product</td><td>X (a ×<sup>n</sup> b): N<sup>n</sup></td><td>Y ? = (a ×<sup>n</sup> b)</td><td>X (a ×<sup>n</sup> b) = m<br>= a: N<sup>n</sup><br>If m ≤ m', where<br>m' = b: N<sup>n</sup><br>or<br>X (a ×<sup>n</sup> b) = m'<br>= b: N<sup>n</sup><br>if m > m'</td><td>P (a ×<sup>n</sup> b) = m [⟨m, m'⟩]<sup>o</sup>: N<sup>n</sup><br>if m ≤ m'<br><br>P (a ×<sup>n</sup> b) = m' [⟨m, m'⟩]<sup>o</sup>: N<sup>n</sup><br>if m > m'<br><br>given<br>O m = a: N<sup>n</sup>, m' = b: N<sup>n</sup></td></tr><tr><td>Addition</td><td>X (a +<sup>n</sup> b): N<sup>n</sup></td><td>Y ? = (a +<sup>n</sup> b)</td><td>X (a +<sup>n</sup> b) = m<br>= a: N<sup>n</sup><br>If m ≥ m', where<br>m' = b: N<sup>n</sup><br><br>X (a +<sup>n</sup> b) = m'<br>= a: N<sup>n</sup><br>if m < m'</td><td>P (a +<sup>n</sup> b) = m [⟨m, m'⟩]<sup>o</sup>: N<sup>n</sup><br>if m ≥ m'<br><br>P (a +<sup>n</sup> b) = m' [⟨m, m'⟩]<sup>o</sup>: N<sup>n</sup><br>if m < m'<br><br>given<br>O m = a: N<sup>n</sup>, m' = b: N<sup>n</sup></td></tr><tr><td>Implication</td><td>X (a →<sup>n</sup> b): N<sup>n</sup></td><td colspan="3">Equivalent to the rules for addition.</td></tr><tr><td>Negation</td><td>X ~<sup>n</sup> a: N<sup>n</sup></td><td>Y ? = ~<sup>n</sup> a</td><td>X ~<sup>n</sup> a = n - m,<br>where<br>m = a: N<sup>n</sup></td><td>P ~<sup>n</sup> a = n - m [⟨m, n⟩]<sup>o</sup>: N<sup>n</sup><br>given<br>O m = a: N<sup>n</sup></td></tr></tbody></table>

Integrating many-valued logics: The Logics of Formal Inconsistency and the White Bullet Operator

In a recent paper, E. A. Barrio, N. Clerbout & S. Rahman (2017) developed a dialogical reconstruction of the so-called *Logics of Formal Inconsistency* (LFI)—see (Carnielli, Coniglio, & Marcos, 2007). The LFIs are logics tolerant to some amount of inconsistency but in which some versions of explosion (ex falso) still hold. Thus, the LFIs are a form of paraconsistent logics, that is, logics where ex falso sequitur quodlibet does not generally hold, and so inconsistencies are tolerated. However, the LFIs do not tolerate all forms of inconsistencies but only those considered to be relevant in a context.<fnref n="141" /> In fact LFI constitute a whole family of logics distinguished by the kind of inconsistency they allow.

The main result of (Barrio, Clerbout, & Rahman, 2017) is to provide a formal framework applicable to situations in which inconsistent information may appear during

<footnote n="141">¹⁴¹ Closely related are the important *adaptive logics* of (Batens, 1980) which are contextually sensitive to different inconsistent situations. They however seemed to have a more inferentialist background than the family of paraconsistent logics that came out of the work of Newton da Costa by 1970—see (d’Ottaviano & da Costa, 1970); for an overview of these logics and their origin see (Bonbenrieth Miserda, 1996); for a recent presentation of new developments see (Carnielli & Coniglio, Paraconsistent Logic: Consistency, Contradiction and Negation, 2016). (Beirlaen & Fontaine, 2016) develop a dialogical reconstruction of some adaptive logics.</footnote>
certain argumentative interactions, but always within some limits, in particular always
keeping some “safe” propositions for which inconsistency is not tolerated. This result has
been obtained from the dialogical inferentialist point of view: Barrio, Clerbout & Rahman
(2017) reconstructed in their section 5 the many-valued semantics of two of the LFIs into
structural rules.

This is therefore a nice example of how one can unify a family of logics in one of
the frameworks. As already suggested with the case of Boolean operators, we shall
embed the truth-functional semantics of one of the logics studied, namely the *Logic of
Pragmatic Truth* or *Quasi-Truth* (MPT) of (Coniglio & Silvestrini, 2014), within our
general framework. However, a generalization for all of the LFIs seem to be
straightforward.

The truth-functional semantics for MPT includes the operators of product, addition, and negation we described above for $\mathbb{N}^3$ (let us here use the standard three values, $0$, $\frac{1}{2}$, $1$, where $0$ is the minimum and $1$ the maximum), and adds a different negation and a new implication, that we indicate with the superscript MPT, as well as a consistency operator. The dialogical formulation of these operators in the lines proposed for $\mathbb{N}^n$ is straightforward.

• $(a \times^3 b): \mathbb{N}^{\text{MPT}}$;
(a \times^3 b)$ is equal to $0$ if $a = 0$, otherwise it is equal to $b$;

• $(a+^3b): \mathbb{N}^{\text{MPT}}$;
$(a+^3b)$ is equal to $\mathbf{1}$ if $a = \mathbf{1}$, otherwise it is equal to $b$;

• $\sim^3a: \mathbb{N}^{\text{MPT}}$,
$\sim^3a$ is equal to $\mathbf{1-m}$, where (where $m = a$).

• $\sim^{\text{MPT}}a: \mathbb{N}^{\text{MPT}}$,
$\sim^{\text{MPT}}a$ is equal to $\mathbf{1}$ if $a = \mathbf{0}$, otherwise it is equal to $\mathbf{0}$.

• $(a \rightarrow^{\text{MPT}} b): \mathbb{N}^{\text{MPT}}$;
$(a \rightarrow^{\text{MPT}} b)$ is equal to $\mathbf{0}$ if $\{b=\mathbf{0} \mid a=\mathbf{1}\}$ or if $\{b=\mathbf{0} \mid a=\frac{1}{2}\}$ otherwise it is equal to $\mathbf{1}$.

• $a^{\circ}: \mathbb{N}^{\text{MPT}}$,
$a^{\circ}$ is equal to $\mathbf{0}$ if $a = \frac{1}{2}$, otherwise it is equal to $\mathbf{1}$.

The idea of the *white-bullet* operator “o”, called *consistency operator* is to create a
fragment in which some of the truth-functional objects behave like in classical logic, that
is like in our framework for **Bool**. The dialogical reconstruction of this operator deployed
the operator "V°" which triggers the opening of a subplay in which the rules of the game
are classical (Barrio, Clerbout, & Rahman, 2017).

In the present framework we will study the white-bullet operator as another truth-
functional operator, that is as a non-canonical element of the three-elements set $\mathbb{N}^{\text{MPT}}$, but
also as a function that evaluates the elements of some fixed subset $C$ of non-canonical
elements of $\mathbb{N}^{\text{MPT}}$ as the codes of the universe $\mathbf{u}$ (described above) and those codes,
whose decoding yields the empty set falsum ($\mathbb{N}^0$, or $\perp$) and the unary set verum ($\mathbb{N}^1$, or
$\top$). This provides the insight that "o" triggers a transfer from $\mathbb{N}^3$ to **Bool**.

Thus the insight we gain here is that $V^\circ$ should be understood as the following
function, with $x: C^{\text{MPT}}$ as an abbreviation for $\{x : \mathbb{N}^{\text{MPT}} | C(x)\}$:
<table><caption>Table 50: dialogical rules from $\mathbb{N}^3$ to **Bool**</caption><thead><tr><th></th><th>Move</th><th>Challenge</th><th>Defence</th></tr></thead><tbody><tr><td rowspan="3">White bullet</td><td rowspan="3">X V°(x): u [x: C<sup>MPT</sup>]</td><td>Y a = 0: C<sup>MPT</sup></td><td>X V°(a) = n<sup>1</sup>: u</td></tr><tr><td>Y a = 1: C<sup>MPT</sup></td><td>X V°(a) = n<sup>1</sup>: u</td></tr><tr><td>Y a = 1/2: C<sup>MPT</sup></td><td>X V°(a) = n<sup>0</sup>: u</td></tr><tr><td></td><td rowspan="3">X V°(a): u</td><td rowspan="3">Y ? T(V°(a))</td><td>X T(V°(a)) = T: set<br>if a = 0: C<sup>MPT</sup></td></tr><tr><td></td><td>X T(V°(a)) = T: set<br>if a = 1: C<sup>MPT</sup></td></tr><tr><td></td><td>X T(V°(a)) = T: set<br>if a = 1/2: C<sup>MPT</sup></td></tr></tbody></table>

We can also deploy **Id** within **Bool** for rendering empirical propositions. Let us discuss this issue now.

# X.4 Empirical Quantities and Material Dialogues

## X.4.1 Empirical Quantities as Finite Sets of Answers

As already mentioned in the opening of the chapter, non-canonical elements of the set **Bool** can be used to study the meaning of empirical propositions, though what we need in particular is the notion of *empirical quantity*. This notion has been introduced by Martin-Löf in applying CTT to the empirical realm (Martin-Löf, 2014): whereas quantities of mathematics and logic are determined by computation, empirical quantities are determined by experiments and observation. An example of a mathematical quantity is 2 + 2; it is determined by a computation yielding the number 4. An example of an empirical quantity is the colour of some object. This is not determined by computation; rather, one must look at the object under normal conditions.

In the dialogical framework, we can consider empirical quantities as answers to a question. For example, give the question

*Are Cheryl's eyes blue?*

The yes or no answer, obtained through some kind of empirical procedure received in a given context, can be defined over the set **Bool**, namely as being equal to **yes** or **no**. The following question however might involve many different answers:

*What is the colour of Cheryl's eyes?*

If **X** stands for the empirical quantity *Colour of Cheryl's eyes*, we might define the possible answers over some finite set $\mathbb{N}^n$ of natural numbers:

*X* = 1: $\mathbb{N}^n$ if Cheryl's eyes are brown

*X* = 2: $\mathbb{N}^n$ if Cheryl's eyes are green

*X* = 3: $\mathbb{N}^n$ if Cheryl's eyes are blue
...
$\mathbf{X} = n: \mathbb{N}^n$ if Cheryl's eyes are...

Certainly the question *Are Cheryl's eyes blue?* can also be defined over a larger set, if several degrees of colour are to be included as an answer, or alternatively degrees of certainty (definitely blue, quite blue, slightly blue...). Let us assume then another set $\mathbb{N}^j$ for the degree of colour:

* $\mathbf{Y} = 0_1: \mathbb{N}^j$ if Cheryl's eyes are dark blue

* $\mathbf{Y} = 0_2: \mathbb{N}^j$ if Cheryl's eyes are light blue.

* $\mathbf{Y} = 0_3: \mathbb{N}^j$ if Cheryl's eyes are green-blue.

...

* $\mathbf{Y} = 0_j: \mathbb{N}^j$ if Cheryl's eyes are...

The general dialogical rule for an empirical quantity can thus be rendered:

<table><caption>**Table 51: general dialogical rule for an empirical quantity**</caption><thead><tr><th></th><th>Move</th><th>Challenge</th><th>Defence</th></tr></thead><tbody><tr><th>Empirical quantity</th><td>$\mathbf{X} \mathbf{X} : \mathbb{N}^n$</td><td>$\mathbf{Y} ? = \mathbf{X}$</td><td>$\mathbf{X} m_1 = \mathbf{X} : \mathbb{N}^n$<br>$\cdots$<br>$\mathbf{X} m_n = \mathbf{X} : \mathbb{N}^n$<br>(the defender chooses)</td></tr></tbody></table>

Notice that determining the value of the empirical quantity is an empirical procedure, specific to that quantity; the result of carrying out such a procedure is determined by the rules for that quantity. Moreover, the value of two different empirical quantities might be the same: the quantities only indicate that *the way of determining* the answer to the question might be the same. Take for example these two enquiries

(1) *Did Jorge Luis Borges compose the poem “Ajedrez”?*

(2) *Is Ibn al-Haytham the author of Al-Shukūk ‘alā Batlamyūs (Doubts Concerning Ptolemy)?*

These two enquiries involve determining the value of the empirical quantity $\mathbf{X}$ for (1) and $\mathbf{Y}$ for (2), which can be the same: they can both be **yes** for instance if the underlying set is **Bool**.

This leads to a Socratic rule specific to statements of the form $\mathbf{X}, \mathbf{Y}, \mathbf{Z} : \mathbb{N}^n$. For example, given the set $\mathbb{N}^n$, $\mathbf{P}$ can defend the challenges

* $\mathbf{O} ? = \mathbf{X}$ with the statement $\mathbf{P} m_1 = \mathbf{X} : \mathbb{N}^n$

* $\mathbf{O} ? = \mathbf{Y}$ with the statement $\mathbf{P} m_2 = \mathbf{Y} : \mathbb{N}^n$

* $\mathbf{O} ? = \mathbf{Z}$ with the statement $\mathbf{P} m_3 = \mathbf{Z} : \mathbb{N}^n$

## Incompatibility

A system of rules that targets the development of a more complex meaning network might include incompatibility rules formulated as challenges. Thus, instead of establishing the simple use of Copy-cat, the game might include more sophisticated rules specific to a particular empirical quantity. For example, if a player responded **yes** to the enquiry associated with $\mathbf{X}$

(3) *Did the Greek won in 480 BC the sea-battle take of Salamis?*
that is, if he stated **yes** = **X** : **Bool**; this player might not be allowed to respond **yes** to the enquiry associated with **Z**

(4) Did Xerxes won in 480 BC the sea-battle of Salamis?

that is, he might not be entitled to further state **yes** = **Z** : **Bool**. That is, the other player
may challenge the right to answer both (3) and (4) with **yes**:

(5) Both answers cannot be yes.

that is, she can challenge his two statements by stating that $\neg(\mathbf{Id}(\mathbf{Bool}, \mathbf{yes}, \mathbf{X}) \wedge \mathbf{Id}(\mathbf{Bool}, \mathbf{yes}, \mathbf{Z}))$. The first player would then have to give up. This challenge would be calling upon some *formal incompatibility* between two statements.

<table><caption>Table 52: Formal incompatibility</caption><thead><tr><th></th><th>Moves</th><th>Challenge</th><th>Defence</th></tr></thead><tbody><tr><th>Formal incompatibility</th><td><p><b>P yes</b> = <b>X</b> : <b>Bool</b></p><p>and</p><p><b>P yes</b> = <b>Z</b> : <b>Bool</b></p></td><td><p><b>O</b> ¬(<b>Id</b>(<b>Bool</b>, <b>yes</b>, <b>X</b>)</p><p>∧ <b>Id</b>(<b>Bool</b>, <b>yes</b>, <b>Z</b>))</p></td><td><p><b>P</b> gives up</p></td></tr></tbody></table>

But there is another kind of incompatibility challenge, calling upon *contentual incompatibility*. Consider for instance (4): if a player answers **yes**, Xerxes won in 480 BC the sea-battle of Salamis, then the other player can challenge this through contentual incompatibility: the challenger simply states the formally incompatible answer to the challenged statement: *Id Bool yes X*, The Greek won in 480 BC the sea-battle of Salamis. The challenged player must then give up.

<table><caption>Table 53: Contentual incompatibility</caption><thead><tr><th></th><th>Moves</th><th>Challenge</th><th>Defence</th></tr></thead><tbody><tr><th>Contentual incompatibility</th><td><b>P yes</b> = <b>Z</b> : <b>Bool</b></td><td><b>O</b> <b>Id</b>(<b>Bool</b>, <b>yes</b>, <b>X</b>)</td><td><b>P</b> gives up</td></tr></tbody></table>

### X.4.2 Dependent Empirical Quantities

Another more sophisticated form of dealing with empirical quantities is to implement a structure where one empirical quantity might depend on another one. For example let us define the empirical quantity **Y** as the function $b(\mathbf{X})$: $\mathbb{N}_j^n [\mathbf{X}: \mathbb{N}^n]$ such that

$$ 
\begin{align*}
\mathbf{Y} &:= \text{df } b(\mathbf{X}) : \mathbb{N}_j^n [\mathbf{X}: \mathbb{N}^n] \\
b(\mathbf{X}) &= j_i : \mathbb{N}^j, \text{ given } \mathbf{X} = n_m : \mathbb{N}^m \\
\dots \\
b(\mathbf{X}) &= j_k : \mathbb{N}^j, \text{ given } \mathbf{X} = n_n : \mathbb{N}^n, \text{ if } \dots
\end{align*}
$$

Suppose we are interested in determining the meaning of some empirical propositions; this can involve for instance establishing that stating that something has a determinate colour (say, *red*) would presuppose that the player already answered the question whether the object at stake is *coloured or not*.
In this case also the rules of the game might include rules for challenging empirical quantities on the basis of a certain evaluation of another empirical quantity on which the first is dependent; this would be like challenging that something is red by denying that the empirical quantity that yields the evaluation **X** has a positive response to the question if the object at stake has a colour.

## Dependent Empirical Quantities and Futures Contingents

Among empirical quantities are the quantities of future events, the indicator then being whether or not the event occurs, following an analogous practice in mathematics, defining such a quantity **X** by making it equal to *yes* if the event occurs and to *no* if the event does not occur. We are dealing here with empirical quantities because an empirical method is required for determining their value.

Martin-Löf used such empirical quantities of future events for dealing with Aristotle's sea-battle puzzle: the question

*Will a sea-battle take place tomorrow?*

can have two answers, **X** = *yes* tomorrow a sea-battle will take place (the event will occur), or **X** = *no* tomorrow a sea-battle will not take place (the event will not occur). In a dialogical setting, if we replace the empirical quantity **X** by a variable *x*, we would obtain the following thesis (in the form of a hypothetical):

P! (*Id(Bool, x, yes) ∨ Id(Bool, x, no))* [*x* : Bool]

Since we do have a winning strategy for (*∀x: Bool*)(*Id(Bool, x, yes) ∨ *Id(Bool, x, no)*) (see section X.2.3), it is possible to assert this thesis even though for practical reasons we cannot yet determine the value of *x*.

## Application in law

This piece of logical analysis finds a nice application<fnref n="142" /> in what Leibniz has called *suspensive conditions* or also *moral conditions*,<fnref n="143" /> determining some *conditional right* such as:

*Primus must pay 100 dinar to Secundus if a ship arrives from Asia*
(within a set time frame)

According to Leibniz, this problem should be coupled with both a logical and an epistemic analysis: the contracting parties must not have any information yet whether the antecedent (that a ship arrives from Asia) is true or false (if they know, then the contract is not conditional). The right established by the contract should be considered to be legally binding, despite the fact that the condition has not yet been satisfied.

Klev (2014b) uses in this context Martin-Löf's notion of empirical quantity: let **X** be an empirical quantity equal to *yes* if a ship arrives and equal to *no* if no ship has arrived within a certain time span. The empirical method here that can be used to determine whether **X** = *yes* or **X** = *no* can for instance consist in standing on the dock for the specified time span and recording the incoming ships. We can now define a function *b* on the set **Bool** := {*yes*, *no*} by setting

<footnote n="142">¹⁴² Traditional legal approaches to conditional right considered suspensive conditions through the notions of existence or legal fiction (Magnier, 2015, p. 72). New recent logical reconstructions of conditional right have been triggered by the work of (Armgardt, 2001; 2008; 2010), such as the studies of (Thiercelin, 2009; 2010), (Magnier, 2013; 2015), (Rahman, 2015).</footnote>

<footnote n="143">¹⁴³ *Doctrina conditionum* in Leibniz (1964), see also (Armgardt, 2001).</footnote>
$$b(\mathbf{no}) = 0 \text{ and}$$

$$b(\mathbf{yes}) = 100$$

where 0 and 100 are to be understood as the amount of money to be paid.

Since $\mathbf{X}$, being a (non-canonical) element of $\mathbf{Bool}$, is equal to either **yes** or **no**, $b(\mathbf{X})$ is well defined, it's evaluation being either 0 or 100. So $b(\mathbf{X})$ is understood as the amount to be payed by Primus to Secundus [$\mathbf{X} : \mathbf{Bool}$].

Thus, in the dialogical framework using empirical quantities, the thesis stated by $\mathbf{P}$ would be the following:

$\mathbf{P}! \textit{Primus must pay } b(\mathbf{X}) \textit{ dinar to Secundus.}$

Notice that this thesis is not hypothetical but has a categorical form: the condition *If a ship arrives is not given in a hypothesis, it is built straight into the empirical quantity $\mathbf{X}$; one should therefore probably speak of dependent obligation rather than conditional right.* The ruling dependents on the value of $\mathbf{X}$, though leaving the possibility open of not yet being in a position to determine $\mathbf{X}$, but as soon as it is determined, so is $b(\mathbf{X})$, and thereby Primus's debt to Secundus:

* if we can determine that $\mathbf{X}$ is **no**, then we can assert the debt to be $b(\mathbf{no}) = 0$;

* if we can determine that $\mathbf{X}$ is **yes**, then we can assert the debt to be $b(\mathbf{yes}) = 100$.

What one is obliged to do depends on the value of such a determined empirical quantity.

What is more, Islamic jurists also have intensive discussions on the issue, and have been precursors of Leibniz's rejection of the roman notion of *retroactivity*. As pointed out by Yvon Linant de Bellefonds (1965, pp. 425-430), the Islamic jurists considered that only a restricted set of suspensive (mu'allaq) conditions (ta'liq) yield legally binding contracts. It might be argued that, from the logical point of view, their rejection was based on a hypothetical analysis of conditional right. An indication of this is that transfers of goods are excluded from contracts with suspensive conditions. A suspensive condition—unless a clear time frame was defined—might introduce a too hazardous parameter for the establishment of the juridical act. In fact, if the time frame is clearly defined and the condition not absolutely contingent, then it was considered not falling under suspensive conditions. Thus, contracts stipulating *too vague conditions* such as *If next year I will have a profitable harvest, then B*, where not considered to be legally binding. However, if the condition is set in a clear time frame, then it is not considered as falling under what they understood to be suspensive. In fact, only a reduced set of cases were allowed, including those juridic acts that can in principle be revoked, such as a will: since it can be revoked, the fulfilment of the will might be formulated as including an *explicit suspensive condition—tacit conditions* have another structure, see (Linant de Bellefonds, 1965, pp. 429-430).

This might perhaps lead to distinguish between *dependent obligations* (rather than *sus suspensive conditions*) and *conditional right* (dependent upon suspensive conditions). In relation to the latter, a possible reconstruction that stresses the hypothetical character of the conditional right and deploys empirical quantities is the following:

$$ \mathbf{P!} (\mathbf{Id}(\mathbf{Bool}, x, \mathbf{yes}) \supset \mathbf{Id}(\mathbb{N}, y, \mathbf{yes})) \\ \wedge (\mathbf{Id}(\mathbf{Bool}, x, \mathbf{no}) \supset \mathbf{Id}(\mathbb{N}, y, \mathbf{no})) [x, y: \mathbf{Bool}] $$
Where **x** is stands for a variable for the empirical quantity **X**, *Ashraf fulfils condition C* [explicitly established as a condition in Zayd's will], and where **y** stands for a variable for the empirical quantity **Y**, *Ashraf receives 100 dinar after Zayd's death* (according to Zayd's will).

The procedure determining the value of **y** is eminently empirical: it amounts to decide if the contract is or not legally binding (this amounts to verifying if it the condition meets the requirements settled for *mu'allaq ta'liq* (suspensive condition)). Similar applies to the determination of **x**.<fnref n="144" />

### X.4.3 Conclusion on Empirical Quantities

Local reasons in general, and empirical quantities in particular, take care of some old and new criticisms raised against the standard dialogical approach to meaning formulated by (Lorenzen & Lorenz, 1978)—see sections I.1, p. 14 and XI.4 for further details on these criticisms and our take on the issue.

It is fair to say that the notion of *material dialogues* seem to be underdeveloped in relation to the formal dialogues that gathered much more attention. However, let us stress that the fathers of dialogical logic where aware of the need of a contentual (*material* was the chose term) basis right from the beginning; they tackled the issue with different devices. Lorenz (1970) in particular dedicated to this issue very thorough and deep studies, most of them collected in (Lorenz, 2010a; 2010b). Moreover, the rules for integrating empirical quantities within the dialogical framework described above are directly inspired by the *predicator rules* already discussed in (Lorenz & Mittelstrass, 1967). Predicator rules, the dialogical counterparts of semantic definitions, are part of the play level; the formalistic interpretation of the dialogical framework pointed out in the opening of this chapter comes for a large part from neglecting this level of meaning. Chapter XI, our concluding chapter, will deal more at large on this neglect of the play level.

*Predicator rules* are part of the *Orthosprache* project of the Erlangen Constructivism proposed by 1970,<fnref n="145" /> which challenged at that time also the approach of mainstream analytic theory of meaning. The underlying idea is the explicit and constructive development, *by example* (*exemplarisch*), of a contentual language in order to build a specific scientific terminology (Kamlah & Lorenzen, 1972, pp. 70-111).

The qualification “by example” refers to one of the major tenets of the overall philosophy of language of the Erlangen School, namely the idea that we grasp an individual as exemplifying something—type theoreticians will say, as exemplifying a type (see below):

> Yet even science cannot avoid the fact that things do not proffer themselves everywhere as different of their own accord, more often in important areas (e.g. in the social or historical sciences) science must decide for itself what it wants to regard as of the same kind and what is of different kind, and address them accordingly. [...]

> As we have said already, the world does not “consist of objects” (of “things in themselves”) which are subsequently named by men [...].

> In the world being disclosed to us all along through language we tend to grasp the

<footnote n="144">¹⁴⁴ See (Rahman & Iqbal, 2018) for a general dialogical approach to legal reasoning in the context of Islamic jurisprudence.</footnote>

<footnote n="145">¹⁴⁵ The term “Orthosprache” was introduced by Lorenzen in 1972, quoted in a footnote in the second edition of the *Logische Propädeutik* (Kamlah & Lorenzen, 1972, p. 73 footnote 1) and discussed in the *bible of the Erlangen School: Konstruktive Logik, Ethik und Wissenschaftstheorie* (Lorenzen & Schwemmer, 1975).</footnote>
individual object as individual at the same time that we grasp it as specimen of... Further, when we say "This is a bassoon" we mean thereby "This instrument is a bassoon" [...] or when we say "This is a blackbird", we presuppose that our discussion partner already knows "what kind of an object is meant", that we are talking about birds. (Kamlah & Lorenzen, 1984, p. 37)

Accordingly, the predicates of the *Orthosprache* are introduced via the study of exemplification instances. Now, a scientific terminology does not only consist in a set of predicates or even of sentences expressing propositions: an adequate scientific language constitutes a system of conceptual interrelations. The main logical device of the *Orthosprache* project is to establish the corresponding transitions by *predicator rules* that normalize the passage from one predicator to the other. Moreover, these transition rules are formulated within a dialogical frame, so that given the predicator rule

$$x \in A \Rightarrow x \in B$$

(where x is a free variable and "A" and "B" are predicates)

we have: if a player brings forward an object of which predicator *A* is said to apply, then he is also committed to ascribe the predicator *B* to the same object. The idea is that if, for example, someone claims *k* is a bassoon, then he is committed to the further claim *k* is a musical instrument (where *k* is an individual constant: in the *Logische Propädeutik* the application of these norms proceeds by substituting individual constants for free variables). The Constructivists of Erlangen called *material-analytic norms* such transition rules which structure a (fully interpreted) scientific language by setting the *boundaries of a predicator*. *Material-analytical propositions* (or, more literally, *material-analytical truths*) are defined as the universally quantified propositions based on such material-analytic norms (Lorenzen & Schwemmer, 1975, p. 215).

# X.5 Some General Epistemological Consequences

## X.5.1 Play Level and Material Dialogues

The philosophical background to our dialogical approach of Martin-Löf's notion of empirical quantity is set in describing how to *internalize* empirical data into the rules of the play (Peregrin, 2014, pp. 34-36, 100-104); or, to put it in Sellars' words, describing how to place empirical data *within the space of reasons*. As it is well known, Sellars introduces the notion of space of reasons in the context of observational reports such as "This is green". According to Sellars (1991, pp. 129-194), such a report expresses a state of knowledge, if the one who brings forward the reports is able to justify his assertion by calling on some further, more general, knowledge about the reliability of such reports:

> The essential point is that in characterizing an episode or a state as that of knowing, we are not giving an empirical description of that episode or state; we are placing it in the logical space of reasons, of justifying and being able to justify what one says. (Sellars, 1991, p. 169)

Brandom's (1994; 2000; 2008) interpretation of the *space of reasons* intends to provide an inferentialist reading to both the internalization of empirical data, and the broader knowledge that is required for the reliability of such reports: the relations in the space of reasons are constituted by possibilities of reaching positions of entitlement or commitment *by inference* from prior positions of entitlement or commitment; thus what language needs to become the games of giving and asking for reasons' vehicle are inferential rules. To be able to *give* reasons we must be able to make claims that can serve as reasons for other claims; hence our language must provide for sentences that *entail* other sentences. To be able to *ask for* reasons, we must be able to make claims that count as a *challenge* to other claims; hence our language must provide for sentences that are
*incompatible* with other sentences. Our language must therefore be structured by these entailment and incompatibility relations.

One must also add on the one hand the relation from which commitments and entitlements are inherited: by committing myself to *This is a dog* I also and thereby commit myself to *This is an animal*, and being entitled to *It is raining* I am also and thereby entitled also to *The streets are wet*; and on the other hand the relation of co-inheritance of incompatibilities (*A* is in this relation to *B* iff whatever is incompatible with *B* is incompatible with *A*).

The relation between dialogical logic and the games of asking and giving reasons has already been pointed out by (Keiff, 2007) and (Marion, 2006; 2009; 2010):

> *My suggestion is simply that dialogical logic is perfectly suited for a precisification of these ‘assertion games’. This opens the way to a ‘game-semantical’ treatment of the ‘game of giving and asking for reasons’: ‘asking for reasons’ corresponds to ‘attacks’ in dialogical logic, while ‘giving reasons’ corresponds to ‘defences’. In the Erlangen School, attacks were indeed described as ‘rights’ and defences as ‘duties’, so we have the following equivalences:*

*Right to attack* ⇔ *asking for reasons*

*Duty to defend* ⇔ *giving reasons*

> *The point of winning ‘assertion games’, i.e., successfully defending one’s assertion against an opponent, is that one has thus provided a justification or reason for one’s assertion. Referring to the title of the book [Making it Explicit], one could say that playing games of ‘giving and asking for reasons’ implicitly presupposes abilities that are made explicit through the introduction of logical vocabulary. (Marion, 2010, p. 490)*

An important component for linking Brandom’s interpretation of Sellar’s space of reasons with the dialogical framework, namely the strategic level, is stressed in section 1.2 of (Keiff, 2007):

> *Traditionnellement, la logique est présentée comme la science des arguments (ou du raisonnement) préservant la vérité, et les objets de cette théorie sont déterminés par rapport à cette propriété : les constantes logiques sont les unités syntaxiques dans les énoncés qui constituent un argument que l’on ne peut altérer tout en garantissant la préservation de la vérité. Ce que l’on peut reformuler en termes brandomiens : les constantes logiques sont définies comme les unités syntaxiques qu’on ne peut altérer tout en préservant l’identité des conditions d’assertabilité. Mais l’approche dialogique détermine son objet de façon plus précise : elle définit les conditions d’assertabilité en termes de stratégies de justification.<fnref n="146" />*

Yet, despite the dialogical framework’s close links with Brandom’s inferentialism, there is also—as argued by (Clerbout & Rahman, 2015, pp. ix-xi)—an important difference: the play-level. Indeed from dialogical point of view, strategies are constituted by plays: if we are prepared to determine meaning from the point of view of dialogical games, the constitution of the strategy is a process that cannot be left by side. To put it other words, *not every sequence of moves in games of asking for reasons and providing them is necessarily inferential*, only those plays leading to winning strategies are.

<footnote n="146">¹⁴⁶ Logic is traditionally presented as the science of arguments (or of reasoning) preserving truth, and the objects of this theory are determined according to this property: the logical constants are syntactical units in the expressions constituting an argument that cannot be changed while still warranting the truth preservation. To put it in Brandomian terms: logical constants are defined as syntactical units that cannot be changed while preserving the identity of the assertability conditions. But the dialogical approach determines its object in a more precise fashion: this approach defines the assertability conditions in terms of strategies of justification.</footnote>
To put in the nice of words of Peregrin (2014, pp. 228-229), the prescription for the interaction of questions and answers at the play level provides *the material by the means the which we reason*, not the material that prescribes *how to reason*.<fnref n="147" />

> This is a crucial point, because it is often taken for granted that the rules of logic tell us how to reason precisely in the tactical sense of the word. But what I maintain is that this is wrong, the rules do not tell us how to reason, they provide us with things with which, or in terms of which, to reason. (Peregrin, 2014, pp. 228-229)

This idea that *not every move in the space of reasons is inferential* might be related to McDowell's (2004; 2009) worry concerning Brandom's interpretation of Sellars:

> Someone can know what colour something is by looking at it only if she knows enough about the effects of different sorts of illumination on colour appearances. The essential thing for our purposes is that the relation of this presupposed knowledge to the knowledge that presupposes it—support in Sellars’s second dimension—is not that the presupposing knowledge is inferentially grounded on the presupposed knowledge. (McDowell, 2004)

> We only need to register that it is experience that yields the knowledge expressed in observation reports. Recognizing the second dimension puts us in a position to understand observation reports properly. The knowledge they express is not inferentially grounded on other knowledge of matters of fact, but—in the crucial departure from traditional empiricism—it presupposes other knowledge of fact. (McDowell, 2009, p. 223)

Our reconstruction of the controversy between Brandom and McDowell is based on a double articulation:

* the difference between the play and the strategy level, and

* the difference between dependences upon empirical quantities and dependences as structured by premises-conclusion.

For short, while the dialogical framework allows for interaction through questions and answers that cannot be reduced to the strategy level—though it may well have the general intent of constituting them, see (Keiff, 2004, pp. 41-42)—, the richer language of immanent reasoning can analyze empirical reports as constituted by empirical quantities and the propositions that bear them, that is, as statements involving local reasons adduced in favour of certain propositions. In this regard, we can analyze this kind of report

*(1) *This apple looks green to me*

as the play level statement of some concrete player, say, *Eloise*:

$$Eloise X = 3: \mathbb{N}^5$$

<footnote n="147">¹⁴⁷ In fact Peregrin (2014) uses the dialogical framework to develop a new approach of the issue on the normativity of logic: he understands the normativity of logic not in the sense of prescriptions on how to reason, but rather as providing the material by the means of which we reason. If we link this proposal with the distinction between the play level and strategy level, we can distinguish prescriptions that aim the development of a play and provide the material for reasoning, from those proper to the tactics, considering the optimal means on how to win. These last prescriptions dictate the design of feasible strategies; Peregrin's suggestion leads to dividing the strategy level with tactics singling out the subset of feasible strategies from the whole set of strategies. While tactical considerations try to find the optimal way to achieve victory, normativity in a more general and fundamental level involves the play level, that is, the level where instruments of reasoning and meaning are forged. Moreover, Peregrin links the normativity of logic with another main conceptual tenet of the dialogical framework, namely, the public feature of the speech-acts underlying an argumentative approach to reasoning. See in particular (Peregrin, 2014, pp. 228-229).</footnote>
(where **X** is the empirical quantity that encodes the response to the enquiry on *the apple being green*).

Moreover, determining the response to such an empirical quantity **X** may well be
dependent upon another empirical quantity **Y**:

$$
Eloise \ X :=_{df} b(Y) = 3: \mathbb{N}^5 [Y : \mathbf{Bool}]
$$

(where **Y** is the empirical quantity that encodes the response to the enquiry on *the apple being coloured*).

Notice that we are here like McDowell making an empirical quantity dependent upon another one by means of a function between those quantities rather that expressing the dependence by means of inferences.<fnref n="148" />

The rules of the play level *internalize* the empirical features by prescribing the rules specific to the empirical quantity at stake. However this does not mean that we cannot move from the statement *it looks green to me* to the assertion *it is green*: a winning strategy is required for this, strategy that can be totally rendered by inferential moves. In a winning strategy, it is sufficient for Eloise to show that she can defend her statement, given the material rules set by the game, against any challenge of her antagonist Abelard playing according to these rules.

X.5.2 The Dialogical Internalization and the Myth of the Given

Let us stress the point that, if our reconstruction of Sellars' observational reports by
means of empirical quantities is correct, acknowledging the legitimacy of such reports
*does not* fall into the so-called *Myth of the Given* (Sellars, 1956): in our approach,
empirical quantities are non-canonical elements of some set in the context of CTT; in
such a context there is no way to approach some object without apprehending it as
determining what it is. Indeed two main tenets of CTT are

1. No entity without type

2. No type without semantical equality

If we recall the Curry-Howard isomorphism between types and propositions we have

*Every entity is bearer of a proposition.*

This is what the *internalization* of empirical content within a dialogical stance is about: bringing forward local reasons for a proposition. <fnref n="149" /> Moreover, the dialogical approach understands “no type without semantical equality” as making semantical equality result from the interaction of giving and asking for reasons, which would take care of Brandom's (1994; 1997; 2000; The Centrality of Sellars's Two-Ply Account of Observation to the Arguments of 'Empiricism and the Philosophy of Mind', 2002) worries in interpreting observational reports the way McDowell suggested.

The discussion between McDowell and Brandom has interesting parallels with the
opposition between Hintikka's (1973) notion of *outdoor-games* and Lorenzen & Lorenz's

<footnote n="148">¹⁴⁸ In the early stages of the development of the dialogical framework, meaning dependences where normed by means of transition rules between *predicators*, at the play level. See section X.4.3.</footnote>

<footnote n="149">¹⁴⁹ This is the sense of *internalization* discussed by (Peregrin, 2014, pp. 34-36, 100-104). However, since he does not use the CTT language, he does not have the means for distinguishing the empirical quantity from the set (proposition) it instantiates.</footnote>
(1978) *indoor-games*. Indeed Hintikka (1973, pp. 77-82), who acknowledges the close links between dialogical logic and game-theoretical semantics, launched an attack against the philosophical foundations of dialogical logic because of its *indoor*—or purely formal— approach to meaning as use. He argues that *formal proof games* are not much help in accomplishing the task of linking the linguistic rules of meaning with the real world:

> In contrast to our games of seeking and finding, the games of Lorenzen and Stegmüller are ‘dialogical games’ which are played ‘indoors’ by means of verbal ‘challenges’ and ‘responses’. [...]

> If one is merely interested in suitable technical problems in logic, there may not be much to choose between the two types of games. However, from a philosophical point of view, the difference seems to be absolutely crucial. Only considerations which pertain to ‘games of exploring the world’ can be hoped to throw any light on the role of our logical concepts in the meaningful use of language. (Hintikka, 1973, p. 81)

This kind of worries have been dealt away by integrating Socratic rules specific to a given predicate and by incorporating empirical quantities.
# XI. CONCLUDING REMARKS: A PLAIDOYER FOR THE PLAY LEVEL

To some extent, the criticisms the dialogical approach to logic has been subject to have provided an opportunity for clarifying its basic tenets. Moreover, our responses to the objections have highlighted crucial distinctions constituting the originality and flexibility of this logical framework. We will therefore in this concluding chapter consider some recent objections raised against the dialogical framework in order to pinpoint some of its fundamental features, whose importance may not have appeared clearly enough through the main body of the book; namely, dialogue-definiteness, player-independence, and the dialogical conception of propositions. Showing how and why these features have been developed, and specifying their point and the level they operate on, will enable us to vindicate the play level and thus disarm the objections that have been raised against the dialogical framework for having neglected this crucial level.

We shall first come back on the central notion of dialogue-definiteness and on the dialogical conception of propositions, which are essential for properly understanding the specific role and importance of the play level. We shall then be able to address three objections to the dialogical framework, due to a misunderstanding of the notion of *Built-in Opponent*, of the principles of dialogue-definiteness and of player-independence, and of the reflexion on normativity that constitutes the philosophical foundation of the framework; all of these misunderstandings can be reduced to a misappraisal of the play level. We shall then go somewhat deeper in the normative aspects of the dialogical framework, according to the principle that logic has its roots in ethics. A last section before our final words will come back to the origins of immanent reasoning, sketching some important aspects of Dialogical Constructivism which rest on the learning process constituting the core of intersubjectivity.

## XI.1 Dialogue-Definiteness and Propositions

The dialogical theory of meaning is structured in three levels, that of the local meaning (determined by the particle rules for the logical constants), of the global meaning (determined by the structural rules), and the strategic level of meaning (determined by what is required for having a winning strategy). The material level of consideration is part of the global meaning, but with particular rules so precise that they determine only one specific expression (through a modified Socratic rule). A characteristic of the local meaning is that the rules are player independent: the meaning is thus defined in the same fashion for each player; they are bound by the same sets of duties and rights when they start a dialogue. This normative aspect is thus constitutive of the play level (which encompasses both the local meaning and the global meaning): it is even what allows one to judge that a dialogue is taking place. In this regard, meaning is immanent to the dialogue: what constitutes the meaning of the statements in a particular dialogue solely rests on rules determining interaction (the local and the global levels of meaning). The strategy level on the other hand is built on the play level, and the notion of demonstration operates on the strategy level (it amounts to having a winning strategy).

Two main tenets of the dialogical theory of meaning can be traced back to Wittgenstein, and ground in particular the pivotal notion of dialogue-definiteness:
1. the internal feature of meaning (the *Unhintergehbarkeit der Sprache*<fnref n="150" />), and

2. the meaning as mediated by language-games.

As for the first Wittgensteinian tenet, the internal feature of meaning, we already mentioned in the introduction that if we relate the notion of internalization of meaning with both language-games and fully-interpreted languages of CTT, then a salient feature of the dialogical approach to meaning can come to fore: the expressive power of CTT allows all these actions involved in the dialogical constitution of meaning to be incorporated as an explicit part of the object-language of the dialogical framework.<fnref n="151" />

In relation to the second tenet, the inceptors of the dialogical framework observed that if language-games are to be conceived as mediators of meaning carried out by social interaction, these language-games must be games actually playable by human beings: it must be the case that *we can actually perform them*,<fnref n="152" /> which is captured in the notion of dialogue-definiteness.<fnref n="153" /> Dialogue-definiteness is essential for dialogues to be mediators of meaning, but it is also constitutive of what propositions are, as Lorenz clearly puts it:

> [...] for an entity to be a proposition there must exist an individual play, such that this entity occupies the initial position, and the play reaches a final position with either win or loss after a finite number of moves according to definite rules. (Lorenz, 2001, p. 258)

A proposition is thus defined in the standard presentation of dialogical logic (see chapters III-V) as a dialogue-definite expression, that is, an expression *A* such that there is an individual play about *A*, that can be said to be lost or won after a finite number of steps, following given rules of dialogical interaction.<fnref n="154" />

The notion of *dialogue-definiteness* is in this sense the backbone of the dialogical theory of meaning: it provides the basis for implementing the human-playability requirement and the notion of proposition.

Dialogue-definiteness sets apart rather decisively the level of strategies from the level of plays, as Lorenz's notion of dialogue-definite proposition does not amount to a set of winning strategies, but rather to an individual play. Indeed, a winning strategy for a player **X** is a sequences of moves such that **X** wins *independently of the moves of the antagonist* (see section III.5 and chapter V). It is crucial to understand that the

<footnote n="150">¹⁵⁰ See *Tractatus Logico-Philosophicus*, 5.6.</footnote>

<footnote n="151">¹⁵¹ Moreover as discussed in section X.5, the dialogical conception of CTT internalization enables subjective reports such “this looks green *to me*” to be rendered.</footnote>

<footnote n="152">¹⁵² As observed by (Marion, 2006, p. 245), a lucid formulation of this point is the following remark of (Hintikka, 1996a, p. 158) who shared this tenet (among others) with the dialogical framework:</footnote>

<footnote>> [Finitism] was for Wittgenstein merely one way of defending the need of language-games as the sense that [sic] they had to be actually playable by human beings. [...] Wittgenstein shunned infinity because it presupposed constructions that we human beings cannot actually carry out and which therefore cannot be incorporated in any realistic language-game. [...] What was important for Wittgenstein was not just the finitude of the operations we perform in our calculi and other language-games, but the fact that we can actually perform them. Otherwise the entire idea of language-games as meaning mediators will lose its meaning. The language-games have to be humanly playable. And that is not possible if they involve infinitary elements. Thus it is the possibility of actually playing the meaning-conferring language-games that is the crucial issue for Wittgenstein, not finitism as such.</footnote>

<footnote n="153">¹⁵³ The fact that these language-games must be finite does not rule out the possibility of a (potentially) infinite number of them.</footnote>

<footnote n="154">¹⁵⁴ While establishing particle rules the development rules have not been fixed yet, so we might call those expressions *propositional schemata*.</footnote>
qualification *independently of the moves of the antagonist* amounts to the fact that the one claiming *A* has to play under the restriction of the Copy-cat rule: if possessing a winning strategy for player **X** involves being in possession of a method (leading to the win of **X**) allowing to choose a move for any move the antagonist might play, then we must assume that the propositions brought forward by the antagonist are justified. There is a winning strategy if **X** can base his moves leading to a win by endorsing himself those propositions whose justification is rooted on **Y**'s authority (see Martin-Löf's discussion of this point in our Preface). For short, the act of endorsing is what lies behind the so-called Copy-cat rule and structures dialogues for immanent reasoning: it ensures that **X** can win whatever the contender might bring forward in order to contest *A* (within the limits set by the game).

Furthermore, *refuting*, that is bringing up a strategy *against A*, amounts to the dual requirement: that the antagonist **Y** possess a method that leads to the loss of **X**! *A*, whatever **X** is can bring forward, and that she can do it under the Copy-cat restriction:

*X*! *A* is *refuted*, if the antagonist *Y* can bring up a sequence of moves such that she (*Y*) can win playing under the Copy-cat restriction.

Refuting is thus different and stronger than contesting: while contesting only requires that the antagonist **Y** brings forward at least one counterexample in a kind of play where **Y** does not need to justify her own propositions, *refuting* means that **Y** must be able to lead to the loss of **X**! *A*, whatever *X*'s justification of his propositions might be.

In this sense, the assumption that every play is a finitary open two-person zero-sum game does not mean that either there is a winning strategy for *A* or a winning strategy *against A*: the play level cannot be reduced to the strategy level.

For instance, if we play with the Last-duty first development rule (see section IV.4.2), **P** will lose the individual plays relevant for the constitution of a strategy for $\forall \neg A$. So $A \forall \neg A$ is *dialogue-definite*, though there is no winning strategy *against A*.

The distinction between the play level and the strategy level thus emerges from the combination of dialogue-definiteness and the Copy-cat rule.

The classical reduction of strategies *against A* to the falsity of *A* (by means of the saddle-point theorem) assumes that the win and the loss of a *play* reduce to the truth or the falsity of the thesis. But we claim that the existence of the play level and a loss in one of the plays introduces a qualification that is not usually present in the purely proof-theoretic approach; to use the previous example, we know that **P** does not have a winning strategy for ! *A* $\forall \neg A$ (playing under the intuitionistic development rule SR1i), but neither will **O** have one against it if she has to play under the Copy-cat rule herself (notice the switch in the burden of the restriction of the Copy-cat rule when *refuting* a thesis). Let us identify the player who has to play under the Copy-cat restriction by highlighting her moves:

Play 20 *against P* ! *A* $\forall \neg A$

<table><thead><tr><th colspan="2">O</th><th colspan="2">P</th></tr></thead><tbody><tr><td></td><td></td><td>! A $\forall \neg A$</td><td>0</td></tr><tr><td>1</td><td>n := 1</td><td>m := 2</td><td>2</td></tr><tr><td>3</td><td>?v</td><td>! A</td><td>4</td></tr><tr><td colspan="2"></td><td colspan="2">P wins</td></tr></tbody></table>
The same obviously happens with the negation of the thesis (! ¬(A ∨ ¬A)), albeit there is a winning strategy against a contradiction (! A ∧ ¬A):

Play 21 *against P* ! A ∧ ¬A

<table><thead><tr><th colspan="2">O</th><th colspan="2">P</th></tr></thead><tbody><tr><td></td><td></td><td>! A ∧ ¬A</td><td>0</td></tr><tr><td>1</td><td>n := 2</td><td>m := 3</td><td>2</td></tr><tr><td>3</td><td>? L<sup>^</sup></td><td>0</td><td>4</td></tr><tr><td>5</td><td>? R<sup>^</sup></td><td>0</td><td>6</td></tr><tr><td>7</td><td>! A</td><td></td><td>—</td></tr><tr><td colspan="2">O wins</td><td></td><td></td></tr></tbody></table>

The distinction between the play and the strategy level can be understood as a consequence of introducing the notion of dialogue-definiteness which amounts to a win or a loss at the play level, though strategically seen, the proposition at stake may be (proof-theoretically) undecidable. Hence, some criticisms to the purported lack of dynamics to dialogical logic are off the mark if they are based on the point that "games" of dialogical logic are deterministic:<fnref n="155" /> plays are deterministic in the sense that they are dialogue-definite, but strategies are not deterministic in the sense that for every proposition there would either be a winning strategy for it or a winning strategy *against* it.

Before ending this section let us quote quite extensively (Lorenz, 2001), who provides a synopsis of the historical background that lead to the introduction of the notion of dialogue-definiteness and the distinction of the deterministic conception of plays—which obviously operates at the level of plays—from the proof-theoretical undecidable propositions—which operate at the level of strategies:

> [...] It was Alfred Tarski who, in discussions with Lorenzen in 1957/58, when Lorenzen had been invited to the Institute for Advanced Study at Princeton, convinced him of the impossibility to characterize arbitrary (logically compound) propositions by some decidable generalization of having a decidable proof-predicate or a decidable refutation-predicate.

> [...] It became necessary to search for some decidable predicate which may be used to qualify a linguistic entity as a proposition about any domain of objects, be it elementary or logically compound. Decidability is essential here, because the classical characterization of a proposition as an entity which may be true or false, has the awkward consequence that of an undecided proposition it is impossible to know that it is in fact a proposition. This observation gains further weight by L. E. J. Brouwer's discovery that even on the basis of a set of "value-definite", i.e., decidably true or false, elementary propositions, logical composition does not in general preserve value-definiteness. And since neither the property of being proof-definite nor the one of being refutation-definite nor properties which may be defined using these two, are general enough to cover the case of an arbitrary proposition, some other procedure had to be invented which is both characteristic of a proposition and satisfies a decidable concept. The concept looked for and at first erroneously held to be synonymous with argumentation[¹⁵⁶] turned out to be the concept of dialogue about a

<footnote n="155">¹⁵⁵ For such criticisms see (Trafford, 2017, pp. 86-88).</footnote>

<footnote n="156">¹⁵⁶ Lorenz identifies argumentation rules with rules at the strategy level and he would like to isolate the interaction displayed by the moves constituting the play level—see (Lorenz, 2010a, p. 79). We deploy the term *argumentation-rule* for request-answer interaction as defined by the local and structural rules. It is true that nowadays argumentation-rules has even a broader scope including several kinds of communicative interaction and this might produce some confusion on the main goal of the dialogical framework which is in principle, to provide an argumentative understanding of logic rather than the logic of argumentation.</footnote>
proposition A (which had to replace the concept of truth of a proposition A as well as the concepts of proof or of refutation of a proposition A, because neither of them can be made decidable). Fully spelled out it means that for an entity to be a proposition there must exist a dialogue game associated with this entity, i.e., the proposition A, such that an individual play of the game where A occupies the initial position, i.e., a dialogue D(A) about A, reaches a final position with either win or loss after a finite number of moves according to definite rules: the dialogue game is defined as a finitary open two-person zero-sum game. Thus, propositions will in general be dialogue-definite, and only in special cases be either proof-definite or refutation-definite or even both which implies their being value-definite. Within this game-theoretic framework where win or loss of a dialogue D(A) about A is in general not a function of A alone, but is dependent on the moves of the particular play D(A), truth of A is defined as existence of a winning strategy for A in a dialogue game about A; falsehood of A respectively as existence of a winning strategy against A. Winning strategies for A count as proofs of A, and winning strategies against A as refutations of A. The meta-truth of "either 'A is true' or 'A is false'" which is provable only classically by means of the saddlepoint theorem for games of this kind may constructively be reduced to the decidability of win or loss for individual plays about A. The concept of truth of dialogue-definite propositions remains finitary, and it will, as it is to be expected of any adequate definition of truth, in general not be recursively enumerable. The same holds for the concept of falsehood which is conspicuously defined independently of negation. (Lorenz, 2001, pp. 257-258)

XI.2 The Built-in Opponent and the Neglect of the Play Level

In recent literature Catarina Duthil Novaes (2015) and James Trafford (2017, pp. 102-105) deploy the term *internalization* for the proposal that natural deduction can be seen as having an internalized Opponent, thereby motivating the inferential steps. This form of internalization is called the *built-in Opponent*. The origin of this concept is linked to Göran Sundholm who, by 2000, in order to characterize the fundamental links between natural deduction and dialogical logic, introduced in his lectures and talks the term *implicit interlocutor*. Yet, since the notion of *implicit interlocutor* was meant to link the strategy level with natural deduction, the concept of *built-in Opponent*—being the *implicit interlocutor’s* offspring—inherited the same strategic perspective on *logical truth*. Thus, logical truth can be seen as the encoding of a process through which the Proponent succeeds in defending his assertion against a stubborn *ideal interlocutor*.<fnref n="157" />

From the dialogical point of view however, the ideal interlocutor of the strategy
level is the result of a process of selecting the relevant moves from the play level.
Rahman, Clerbout & Keiff (2009), in a paper dedicated to the Festschrift for Sundholm,
designate the process as *incarnation*, using Jean-Yves Girard's term. Their thorough
description of the incarnation process already displays those aspects of the *cooperative
endeavour*, which was formulated by Duthil Novaes (2015) and quoted by Trafford
(2017, p. 102) as a criticism of the dialogical framework. Their criticism seems to rest on
the idea that the dialogues of the dialogical framework are not truly cooperative, since
they are reduced to constituting logical truth. If this is really the point of their criticism, it
is simply wrong, for the play level would then be completely neglected: the
intersubjective in-built and implicit cooperation of the *strategy* level (which takes care of
inferences) grows out of the *explicit* interaction of players at the *play* level in relation to
the formation-rules; accepting or contesting a local reason is a process by the means of

<footnote>However, once this distinction has been drawn nothing prevents to develop the interface dialogical-
understanding of logic/logical structure of a dialogue. In fact, it is our claim that in order to study the
logical structure of a dialogue, the dialogical conception of logic provides the right venue.</footnote>

<footnote n="157">¹⁵⁷ With "ideal" we mean an interlocutor that always make the optimal choices in order to collaborate in the task of testing the thesis.</footnote>
which players cooperate in order to determine the meaning associated to the action-schema at stake.<fnref n="158" />

It is fair to say that the standard dialogical framework, not enriched with the language of CTT, did not have the means to fully develop the so-called material dialogues, that is dialogues that deal with content. Duthil Novaes (2015, p. 602)—but not Trafford (2017, p. 102)—seems to be aware that dialogues are a complex interplay of adversarial and cooperative moves,<fnref n="159" /> even in Lorenzen and Lorenz' standard formulation. However, since she understands this interplay as triggered by the built-in Opponent at the strategy level, her suggestions or corrections motivated by reflections on the Opponent's role cannot be made explicit in the framework, and the way this role contributes in finally constituting a winning strategy cannot be traced back.<fnref n="160" />

Duthil Novaes' (2015, pp. 602-604) approach leads her to suggest that monotonicity is a consequence of the role of the Opponent as a stubborn adversary, which takes care of the non-defeasibility of the demonstration at stake; from this perspective, she contends that the standard presentations of dialogical logic, being mostly adversarial or competitive, are blind to defeasible forms of reasons and are thus "[...] rather contrived forms of dialogical interaction, and essentially restricted to specific circles of specialists" (Duthil Novaes, 2015, p. 602). But this argument is not compelling when considering the strategy level as being built from the play level: setting aside the point on content mentioned above, if we conceive the constitution of a strategy as the end-result of the complementary role of competition and cooperation taking place at the play level, we do not seem to need—at least in many cases—to endow the notion of inference with non-monotonic features. The play level is the level were cooperative interaction, either constructive or destructive, can take place until the definitive answer—given the

<footnote n="158">¹⁵⁸ In fact, when Trafford (2017) criticizes dialogical logic in his chapter 4, he surprisingly claims that this form of dialogical interaction does not include the case in which the plays would be open-ended in relation to the logical rules at stake, though it has already been suggested—see for instance in (Rahman & Keiff, 2005, pp. 394-403)—how to develop what we called *Structure Seeking Dialogues* (SSD). Moreover, Keiff's (2007) PhD-dissertation is mainly about SSD. The idea behind SSD is roughly the following; let us take some inferential practice we would like to formulate as an action-schema, mainly in a teaching-learning situation; we then search for the rules allowing us to make these inferential practices to be put into a schema. For example: we take the third excluded to be in a given context a sound inferential practice; we then might ask what kind of moves *P* should be allowed to make if he states the third excluded as thesis. It is nonetheless true to say that SSD were studied only in the case of modal logic.</footnote>

<footnote n="159">¹⁵⁹ To put it in her own words: "*the majority of dialogical interactions involving humans appear to be essentially cooperative, i.e., the different speakers share common goals, including mutual understanding and possibly a given practical outcome to be achieved.*" (Duthil Novaes, 2015, p. 602)</footnote>

<footnote n="160">¹⁶⁰ See for instance her discussion of countermoves (Duthil Novaes, 2015, p. 602): indefeasability means that the Opponent has no available countermove: "A countermove in this case is the presentation of one single situation, no matter how far-fetched it is, where the premises are the case and the conclusion is not—a counterexample." The question then would be to know how to show that the Opponent has no countermove available. The whole point of building winning strategies from plays is to *actually construct* the evidence that there is no possible move for the Opponent that will lead her to win: that is a winning strategy. But when the play level is neglected, the question remains: how does one know the Opponent has no countermove available? It can actually be argued that the mere notion of countermove tends to blurr the distinction between the level of plays and of strategies: a *countermove* makes sense if it is 'counter' to a winning strategy, as if the players were playing at the strategy level, but that is something we explicitly reject, see for instance section III.5; see also our discussion of strategic reasons in section VII.7 which inserts strategy level considerations within the *play* level, not the other way round. At the play level, on the other hand, there are only simple *moves*: these can be challenges, defences, counterattacks, but *countermoves* do not make any sense.</footnote>
structural and material conditions of the rules of the game—has been reached.<fnref n="161" /> The strategy level is a recapitulation that retains the end result.

These considerations should also provide an end to Trafford's (2017, pp. 86-88) search for *open-ended* dialogical settings: open-ended dialogical interaction, to put it bluntly, is a property of the play level. Certainly the point of the objection may be to point out either that this level is underdeveloped in the literature—a fact that we acknowledge with the provisos formulated above—, or that the dialogical approach to meaning does not manage to draw a clean distinction between local and strategic meaning—the section on *tonk* below intends to make this distinction as clear as possible.

At this point of the discussion we can say that the role of the (built-in) Opponent in Lorenzen and Lorenz' dialogical logic has been fully misunderstood. Indeed, the role of *both interlocutors* (implicit or not) is not about assuring logical truth by checking the non-defeasibility of the demonstration at stake, but their role is about implementing both the dialogical definiteness of the expressions involved and the internalization of meaning.<fnref n="162" />

## XI.3 Pathological cases and the Neglect of the Play Level

The notorious case of *tonk* has been several times addressed as a counterargument to inferentialism and also to the “indoor-perspective” of the dialogical framework. This also seems to constitute the background of how Trafford (2017, p. 86) for instance reproduces the circularity objection against the dialogical approach to logical constants. At this point of the discussion, Trafford (2017, pp. 86-88) is clearly aware of the distinction between the rules for local meaning and the rules of the strategy level, though he points out that the local meaning is vitiated by the strategic notion of justification. This is rather surprising as (Rahman & Keiff, 2005), (Rahman, 2012), (Rahman, Clerbout, & Keiff, 2009), and (Rahman & Redmond, 2016) have shown it is precisely the case of *tonk* that provides a definitive answer to the issue.

In this respect, three well distinguished levels of meaning are respectively determined by specific rules:

* the local meaning of an expression establishes how a statement involving such an expression is to be attacked and defended (through the particle rules);

* the global meaning of an expression results from structural rules prescribing how to develop a play having this expression for thesis;

* the strategy rules (for **P**) determine what options **P** must consider in order to show that he does have a method for winning whatever **O** may do—in accordance with the local and structural rules.

It can in a quite straightforward fashion be shown (see below) that an inferential formulation of rules for *tonk* correspond to *strategic rules* that *cannot be constituted* by the formulation of *particle rules*. The player-independence of the particle rules— responsible for the branches at the strategy level—do not yield the strategic rules that the inferential rules for *tonk* are purported to prescribe.

For short, the dialogical take on *tonk* shows precisely how distinguishing rules of local meaning from strategic rules makes the dialogical framework immune to *tonk*. As this distinction is central to the dialogical framework and illustrates the key feature of

<footnote n="161">¹⁶¹ See (Rahman, 2015) and (Rahman & Iqbal, 2018).</footnote>

<footnote n="162">¹⁶² Notice that if the role of the Opponent in adversial dialogues is reduced to checking the achievement of logical truth, one would wonder what the role of the Opponent might be in more coperation-featured dialogues: A *soft* interlocutor ready to accept weak arguments?</footnote>
player-independence of particle rules, we will now develop the argument; we will then be able to contrast this pathological *tonk* case to another case, that of the black-bullet operator.

### XI.3.1 The *tonk* challenge and player-independence of local meaning

To show how the dialogical framework is immune to *tonk* through the importance and priority it gives to the play level, winning strategies are linked to semantic tableaux. According to the dialogical perspective, if tableaux rules (or any other inference system for that matter) are conceived as describing the core of strategic rules for **P** (see section V.2 for the core of a strategy), then the tableaux rules should be justified by the play level, and not the other way round: the *tonk* case clearly shows that contravening this order yields pathological situations. We will here only need conjunction and disjunction for dealing with *tonk*.<fnref n="163" />

A systematic description of the winning strategies available for **P** in the context of the possible choices of **O** can be obtained from the following considerations: if **P** is to win against any choice of **O**, we will have to consider two main different dialogical situations, namely those

(a) in which **O** has uttered a complex formula, and those

(b) in which **P** has uttered a complex formula.

We call these main situations the **O**-cases and the **P**-cases, respectively. In both of these situations another distinction has to be examined:

(i) **P** wins by choosing

i.1. between two possible challenges in the **O**-cases (a), or

i.2. between two possible defences in the **P**-cases (b),

iff he can win with *at least one* of his choices.

(ii) When **O** can choose

ii.1. between two possible defences in the **O**-cases (a), or

ii.2. between two possible challenges in the **P**-cases (b),

**P** wins iff he can win *irrespective* of **O**'s choices.

The description of the available strategies will yield a version of the semantic tableaux of Beth that became popular after the landmark work on semantic-trees by Raymond Smullyan (1968), where **O** stands for **T** (left-side) and **P** for **F** (right-side), and where situations of type **ii** (and not of type **i**) will lead to a branching-rule.

<table><caption>Table 54: tableaux and **P**-winning strategies for conjunction and disjunction</caption><thead><tr><th>(P)-Chooses</th><th>(O)-Chooses</th></tr></thead><tbody><tr><td>(P) A ∨ B<br>⟨O?⟩ (P)A<br>⟨O?⟩ (P)B<br><br>The expressions of the form ⟨X ...⟩ constitute interrogative utterances.<br>(O)A ∧ B</td><td>(P)A ∧ B<br>⟨O?∧<sub>1</sub>⟩(P)A | ⟨P?∧<sub>2</sub>⟩(P)B<br><br>The expressions of the form ⟨X ...⟩ constitute interrogative utterances.<br>(O)A ∨ B</td></tr></tbody></table>

<footnote n="163">¹⁶³ (Clerbout, 2014a; 2014b) worked out the most thorough method for linking winning strategies and tableaux.</footnote>
<figure><img src="image_45.png" /></figure>

<figure><img src="image_46.png" /></figure>

However, as mentioned above, tableaux are not dialogues. The main point is that dialogues are built bottom up, from local to global meaning, and from global meaning to validity. This establishes the priority of the play level over the winning strategy level. From the dialogical point of view, Prior's original *tonk* contravenes this priority.

Let us indeed temporarily assume that we can start not by laying down the local meaning of *tonk*, but by specifying how a winning strategy for *tonk* would look like with the help of **T**(left)-side and **F**(right)-side tableaux-rules (or sequent-calculus) for logical constants; in other words, let us assume that the tableaux-rules are necessary and sufficient to set the meaning of *tonk*.

Prior's *tonk* rules are built for half on the disjunction rules (taking up only its introduction rule), and for half on the conjunction rules (taking up only its elimination rule). This renders the following tableaux version for the undesirable *tonk*:<sup>164</sup>

$$ \frac{\textbf{(O)} \text{ [or (T)] } AtonkB}{\textbf{(O)} \text{ [(T)] } B} $$

$$ \frac{\textbf{(P)} \text{ [or (F)] } AtonkB}{\textbf{(P)} \text{ [(F)] } A} $$

*Tonk* is certainly a nuisance: if we apply the cut-rule, it is possible to obtain a closed tableau for **T**A, **F**B, for any **A** and **B**. Moreover, there are closed tableaux for both {**T**A, *Atonk*¬**A**} and {**T**A, ¬(*Atonk*¬**A**)}.

From the dialogical point of view, the rejection of *tonk* is linked to the fact there is no way to formulate rules for its local meaning that meet the condition of being player-independent: if we try to formulate rules for local meaning matching the ones of the tableaux, the defence yields a different response, namely the tail of *tonk* if the defender is **O**, and the head of *tonk* if the defender is **P**:

<table><caption>Table 55: **O**-tonk rule for challenge and defence</caption><thead><tr><th><b>O</b>-move</th><th><b>Challenge</b></th><th><b>Defence</b></th></tr></thead><tbody><tr><td><b>O</b> ! AtonkB</td><td><b>P</b> ?<sub>tonk</sub></td><td><b>O</b> ! B</td></tr></tbody></table>

<table><caption>Table 56: **P**-tonk rule for challenge and defence</caption><thead><tr><th><b>P</b>-move</th><th><b>Challenge</b></th><th><b>Defence</b></th></tr></thead><tbody><tr><td><b>P</b> ! AtonkB</td><td><b>O</b> ?<sub>tonk</sub></td><td><b>P</b> ! A</td></tr></tbody></table>

The fact that we need two sets of rules for the challenge and the defence of a *tonk* move means that the rule that should provide the local meaning of *tonk* is *player-dependent*, which should not be the case.

Summing up, within the dialogical framework *tonk*-like operators are rejected because there is no way to formulate player-independent rules for its local meaning that justify the tableaux rules designed for these operators. The mere possibility of writing

<footnote n="164">164 Cf. (Rahman, 2012, pp. 222-224).</footnote>
tableaux rules that cannot be linked to the play level rules shows that the play level rules are not vitiated by strategic rules.

This brief reflection on *tonk* should state our case for both, the importance of distinguishing the rules of the play level from those of the strategy level, and the importance of including in the rules for the local meaning the feature of *player-independence*: it is the player-independence that provides the meaning explanation of the strategic rules, not the other way round.

### XI.3.2 The black-bullet challenge and dialogue-definiteness

Now, Trafford (2017, pp. 37-41) contests the standard inferentialist approach to the meaning of logical constants by recalling the counterexample of Stephen Read, the *black-bullet* operator. Indeed, Read (2008; 2010) introduces a different kind of pathological operator, the black-bullet •, a zero-adic operator that says of itself that it is false. Trafford (2017, p. 39 footnote 35) suggests that the objection also extends to CTT; this claim however is patently wrong, since those counterexamples would not meet the conditions for the constitution of a type.<fnref n="165" /> Within the dialogical framework, though player-independent rules for black-bullet can be formulated (as opposed to *tonk*), they do not satisfy dialogue-definiteness.

Let us have the following tableaux rules for the black-bullet, showing that it certainly is pathological: they deliver closed tableaux for both • and ⊢:

$$ \frac{\frac{\begin{array}{c} (P) \bullet \\ \langle O? \rangle \\ (P) \cdot \supset \bot \end{array}}{\begin{array}{c} (O) \bullet \\ \langle P? \rangle \\ (O) \cdot \supset \bot \end{array}} $$

$$ \frac{\begin{array}{c} (O) \bullet \\ \langle P? \rangle \\ (O) \cdot \supset \bot \end{array}}{\begin{array}{c} (P) \bullet \\ \langle O? \rangle \\ (P) \cdot \supset \bot \end{array}} $$

We can in this case formulate the following player-independent rules:

<table><caption>**Table 57: black-bullet player-independent particle rules**</caption><thead><tr><th>Move</th><th>Challenge</th><th>Defence</th></tr></thead><tbody><tr><td>X ! •</td><td>Y ?.</td><td>X ! • ⊢⊥</td></tr></tbody></table>

The black-bullet operator seems therefore to meet the dialogical requirement of player-independent rules, and would thus have local meaning. But if it does indeed have player-independent rules, the further play on the defence (which is a negation) would require that the challenger concedes the antecedent, that is black-bullet itself:

<table><caption>**Table 58: deploying the black-bullet challenges**</caption><thead><tr><th colspan="2">Y</th><th colspan="2">X</th></tr></thead><tbody><tr><td></td><td>...</td><td></td><td>...</td></tr><tr><td></td><td></td><td>! •</td><td>i</td></tr><tr><td>i + 1</td><td>?.</td><td>! • ⊢⊥</td><td>i + 2</td></tr></tbody></table>

<footnote n="165">¹⁶⁵ Klev (2017, p. 12 footnote 7) points out that the introduction rule of such kind of operator fails to be meaning-giving because the postulated canonical set $\Lambda(A)$ occurs negatively in its premiss, and that the restriction avoiding such kind of operators have been already formulated by (Martin-Löf, 1971, pp. 182-183), and by (Dybjer, 1994).</footnote>
<table><tr><td>i + 3</td><td>! •</td><td>i + 2</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td>i + 3</td><td>?</td><td>i + 4</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>

Obviously, this play sequence can be carried out indefinitely, regardless of which player initially states black-bullet. So the apparently acceptable player-independent rules for playing black-bullet would contravene dialogue-definiteness; and the only way of keeping dialogue-definiteness would be to give up player-independence!<fnref n="166" />

## Conclusion: the meaning of expressions comes from the play level

The two pathological cases we have discussed, the *tonk* and the black-bullet operators, stress the difference between the play level and the strategy level and how the meaning provided by rules at the strategy level does not carry to the local meaning. Thus, from the dialogical point of view, the rules determining the meaning of any expression are to be rooted at the play level, and at this level *what is to be admitted and rejected as a meaningful expression amounts to the formulation of a player-independent rule, that prescribe the constitution of a dialogue-definite proposition (where that expression occurs as a main operator)*.

Notice that if we include material dialogues (see chapter X), the distinction between logical operators and non-logical operators is not important any more. If we enrich the dialogical framework with the CTT-language, this feature comes more prominently to the fore. What the dialogical framework adds to the CTT framework is, as pointed out by Martin-Löf (2017a; 2017b), to set a pragmatic layer where normativity finds its natural place.

## XI.4 Other than logical constants

In his recent book, Jaroslav Peregrin (2014) marshals the distinction between the play level and the strategy level (that he calls *tactics*) in order to offer another insight, more general, into the issue of normativity mentioned at that start of our volume (see section I.2). Indeed, Peregrin understands the normativity of logic not in the sense of a prescription on *how to reason*, but rather as *providing the material by the means of which we reason*.

> It follows from the conclusion of the previous section that the rules of logic cannot be seen as tactical rules dictating feasible strategies of a game; they are the rules constitutive of the game as such. (MP does not tell us how to handle implication efficiently, but rather what implication is.) This is a crucial point, because it is often taken for granted that the rules of logic tell us how to reason precisely in the tactical sense of the word. But what I maintain is that this is wrong, the rules do not tell us how to reason, they provide us with things with which, or in terms of which, to reason. (Peregrin, 2014, pp. 228-229)

Peregrin endorses at this point the dialogical distinction between rules for plays and rules for strategies. In this regard, the prescriptions for developing a *play* provide the *material* for reasoning, that is, the material allowing a play to be developed, and without which there would not even be a play; whereas the prescriptions of the *tactical* level (to use his terminology) prescribe how to win, or how to develop a winning-strategy:

<footnote n="166">¹⁶⁶ We could provide at the local level of meaning a set of player-independent rules, and add some special structural rule in order to force dialogue-definiteness—see (Rahman, 2012, p. 225); however, such kinds of rules would produce a mismatch in the formation of black-bullet: the formulation of the particle rule would have to assume that black-bullet is an operator, but the structural rule would have to assume it is an elementary proposition.</footnote>
> This brings us back to our frequently invoked analogy between language and chess. There are two kinds of rules of chess: first, there are rules of the kind that a bishop can move only diagonally and that the king and a rook can castle only when neither of the pieces have previously been moved. These are the rules constitutive of chess; were we not to follow them, we have seen (Section 5.5) we would not be playing chess. In contrast to these, there are tactical rules telling us what to do to increase our chance of winning, rules advising us, e.g., not to exchange a rook for a bishop or to embattle the king by castling. Were we not to follow them, we would still be playing chess, but with little likelihood of winning. (Peregrin, 2014, pp. 228-229)

This observation of Peregrin plus his criticism on the standard approach to the dialogical framework, according to which this framework would only focus on *logical constants* (Peregrin, 2014, pp. 100, 106)—a criticism shared by many others since (Hintikka, 1973, pp. 77-82)<fnref n="167" />—naturally leads to the main subject of our book, namely immanent reasoning, or linking CTT with the dialogical framework.

The criticism according to which the focus would be on logical constants and not on the meaning of other expressions does indeed fall to some extent on the standard dialogical framework, as little studies have been carried out on material dialogues in this basic framework;<fnref n="168" /> but the enriched CTT language in material dialogues deals with this shortcoming.

Yet this criticism seems to dovetail this other criticism, summoned by Martin-Löf as starting point in his Oslo lecture:

> I shall take up criticism of logic from another direction, namely the criticism that you may phrase by saying that traditional logic doesn't pay sufficient attention to the social character of language. (Martin-Löf, 2017a, p. 1)

The focus on the social character of language not only takes logical constants into account, of course, but it also considers other expressions such as elementary propositions or questions, as well as the acts bringing these expressions forward in a dialogical interaction, like statements, requests, challenges, or defences—to take examples from the dialogical framework—and how these acts made by persons intertwine and call for—or put out of order—other specific responses by that person or by others. In this regard, the social character of language is put at the core of immanent reasoning through the normativity present in dialogues: normativity involves, within immanent reasoning, rules of interaction which allow us to consider assertions as the result of having intertwined rights and duties (or permissions and obligations). This central normative dimension of the dialogical framework at large, which stems from questionning what is actually being done when implementing the rules of this very framework, entails that objections according to which the focus would be only on logical constants will always be, from the dialogical perspective, slightly off the mark.

## XI.5 Normativity and the Dialogical Framework: A New Venue for the Interface Pragmatics-Semantics

In his Oslo and Stockholm lectures, Martin-Löf's (2017a; 2017b) delves in the structure of the deontic and epistemic layers of statements within his view on dialogical

<footnote n="167">¹⁶⁷ For a response to this see our chapter X on material dialogues.</footnote>

<footnote n="168">¹⁶⁸ This kind of criticism does not seem to have been aware of (Lorenz, 1970; 2009; 2010a; 2010b), carrying out a thorough discussion on predication from a dialogical perspective, which discusses the interaction between perceptual and conceptual knowledge. However, perhaps it is fair to say that this philosophical work has not been integrated into the dialogical logic—we will come back to this subject below.</footnote>
logic. In order to approach this normative aspect which pervades logic up to its technical parts, let us discuss the following extracts of “Assertion and Request”:<fnref n="169" />

[...] we have this distinction, which I just mentioned, between, on the one hand, the social character of language, and on the other side, the non-social [...] view of language. But there is a pair of words that fits very well here, namely to speak of the monological conception of logic, or language in general, versus a dialogical one. And here I am showing some special respect for Lorenzen, who is the one who introduced the very term dialogical logic.

The first time I was confronted with something of this sort was when reading Aarne Ranta's book Type-Theoretical Grammar in (1994). Ranta there gave two examples, which I will show immediately. The first example is in propositional logic, and moreover, we take it to be constructive propositional logic, because that does matter here, since the rule that I am going to show is valid constructively, but not valid classically. Suppose that someone claims a disjunction to be true, asserts, or judges, a disjunction to be true. Then someone else has the right to come and ask him, Is it the left disjunct or is it the right disjunct that is true? There comes an opponent here, who questions the original assertion, and I could write that in this way:

$$ ? \vdash A \lor B \ \mathbf{true} $$

And by doing that, he obliges the original assertor to answer either that A is true that is, to assert either that A is true or that B is true, so he has a choice, and we need to have some symbol for the choice here.

$$ (\mathbf{Dis}) \quad \frac{\vdash A \lor B \ \mathbf{true} \qquad ? \vdash A \lor B \ \mathbf{true}}{\vdash A \ \mathbf{true} | \vdash B \ \mathbf{true}} $$

Ranta's second example is from predicate logic, but it is of the same kind. Someone asserts an existence statement,

$$ \vdash (\exists x : A)B(x) \ \mathbf{true} $$
and then someone else comes and questions that

$$ ? \vdash (\exists x : A)B(x) \ \mathbf{true} $$

And in that case the original assertor is forced, which is to say, he must come up with an individual from the individual domain and also assert that the predicate B is true of that instance.

[...] So, what are the new things that we are faced with here? Well, first of all, we have a new kind of speech act, which is performed by the oh, I haven't said that, of course I will use the standard terminology here, either speaker and hearer, or else respondent and opponent, or proponent and opponent, as Lorenzen usually says, so that's terminology but the novelty is that we have a new kind of speech act in addition to assertion.

[...] So, let's call them rules of interaction, in addition to inference rules in the usual sense, which of course remain in place as we are used to them.

[...] Now let's turn to the request mood. And then it's simplest to begin directly with the rules, because the explanation is visible directly from the rules. So, the rules that involve request are these, that if someone has made an assertion, then you may question his assertion, the opponent may question his assertion.

$$ (\mathbf{Req1}) \quad \frac{\vdash C}{? \vdash_{may} C} $$

Now we have an example of a rule where we have a may. The other rule says that if we have the assertion $\vdash C$, and it has been challenged, then the assertor must execute his knowledge how to do C. And we saw what that amounted too in the two Ranta examples, so I will write this schematically that he will continue by asserting zero, one, or more we have two in the existential case so I will call that schematically by C0.

$$ (\mathbf{Req2}) \quad \frac{\vdash C \qquad ? \vdash C}{\phantom{? \vdash} \phantom{C}} $$

<footnote n="169">¹⁶⁹ Transcription of (Martin-Löf, 2017a, pp. 1-3; 7).</footnote>
$$ \vdash_{must} C' $$

The Oslo and the Stockholm lectures of Martin-Löf (2017a; 2017b) contain challenging and deep insights in dialogical logic, and the understanding of *defences as duties* and *challenges as rights* is indeed at the core of the deontics underlying the dialogical framework.<fnref n="170" /> More precisely, the rules Req1 and Req2 do both, they condense the local rules of meaning, and they bring to the fore the normative feature of those rules, which additionally provides a new understanding for Sunholm's notion of *implicit interlocutor*: once we make explicit the role of the interlocutor, the deontic nature of logic comes out.<fnref n="171" /> Moreover, as Martin-Löf points out, and rightly so, they should not be called *rules of inference* but *rules of interaction*.

Accordingly, a dialogician might wish to add players **X** and **Y** to Req2, in order to stress both that the dialogical rules do not involve inference but *interaction*, and that they constitute a new approach to the action-based background underlying Lorenzen's (1955) *Operative Logik*. This would yield the following, where we substitute the horizontal bar for an arrow:<fnref n="172" />

$$ \begin{array}{ccc} \vdash_X C & & ? \vdash_{may} C \\ \text{(Req2)} & \searrow & \\ & & ? \vdash_{must} C' \end{array} $$

Such a rule does indeed condense the rules of local meaning, but it still does not express the choices while defending or challenging; yet it is the distribution of these choices that determines for example that the meaning of a disjunction is different from that of a conjunction: while in the former case (disjunction) the defender *must choose* a component, the latter (conjunction) requires of the challenger that, *her right to challenge is bounded to her duty to choose* the side to be requested (though she might further on request the other side). Hence, the rules for disjunction and conjunction (if we adapt them to Martin-Löf's rules) would be the following:

$$ \begin{array}{ccc} \vdash_X D & & ? \vdash_{may} D \\ \text{(Dis)} & \searrow & \\ & & ? \vdash_{must} D' \\ & & \begin{array}{l} \text{choose} \\ \text{one of the components} \\ \text{of the disjunction } D \end{array} \end{array} $$

$$ \begin{array}{ccc} \vdash_X C & ?_{left} \vdash_Y C & ?_{right} \vdash_Y C \\ \text{(Conj)} & \searrow & \\ & ? \vdash_{must} C' & \\ & \begin{array}{l} \text{assert} \\ \text{the left of } C \end{array} & \end{array} $$

$$ \begin{array}{ccc} \vdash_X C & ?_{right} \vdash_Y C & \\ \text{(Conj)} & \searrow & \\ & ? \vdash_{must} C'' & \\ & \begin{array}{l} \text{assert} \\ \text{the right of } C \end{array} & \end{array} $$

<footnote n="170">170 See (Lorenz, 1981, p. 120), who uses the expressions *right to attack* and *duty to defend*.</footnote>

<footnote n="171">171 This crucial insight of Martin-Löf on dialogical logic and on the deontic nature of logic seems to underlie recent studies on the dialogical framework which are based on Sundholm's notion of the *implicit interlocutor*, such as (Duthil Novaes, 2015) and (Trafford, 2017).</footnote>

<footnote n="172">172 In the context of Operative Logik operations are expressed by means of arrows of the form “⇒”.</footnote>
These rules can be considered as inserting in the rules the back and forth movement described by Martin-Löf (2017a, p. 8) with the following diagram:

<figure><img src="image_47.png" /></figure>

Notice however that these rules only determine the local meaning of disjunction and conjunction, not their global meaning. For example, while classical and constructive disjunction share the same rules of local meaning, they differ at the global level of meaning: in a classical disjunction the defender may come back on the choice he made for defending his disjunction, though in a constructive disjunction this is not allowed, once a player has made a choice he must live with it.

What is more, these rules are not rules of inferences (for example rules of introduction and elimination): they become rules of inference only when we focus on the choices **P** must take into consideration in order to claim that he has a winning strategy for the thesis. Indeed, as mentioned at the start of the present chapter (XI.3.1) strategy rules (for **P**) determine what options **P** must consider in order to show that he has a method for winning whatever **O** does, in accordance with the rules of local and global meaning.

The introduction rules on the one hand establish what **P** has to bring forward in order to assert it, when **O** challenges it. Thus in the case of a disjunction, **P** must *choose* and *assert* one of the two components. So, **P**'s obligation lies in the fact that he must *choose*, and so **P**'s *duty to choose* yields the introduction rule. Compare this with the conjunction where it is the *challenger* who has the *right to choose* (and who does not assert but request his choice). But in both cases, defending a disjunction and defending a conjunction, only one conclusion will be produced, not two: in the case of a conjunction, the challenger will ask one after the other (recall that it is an interaction taking place within a dialogue where each step alternates between moves of each of the players).

The elimination rules on the other hand prescribe what moves **O** must consider when she asserted the proposition at stake. So if **O** asserted a disjunction, **P** must be able to win whatever the choices of **O** be.

The case of the universal quantifier adds the *interdependence of choices* triggered by the *may-moves* and the *must-moves*: if the thesis is a universal quantifier of the form $(\forall x : A) B(x)$, **P** must assert $B(a)$, for whatever $a$ **O** may choose from the domain $A$: this is what correspond to the introduction rule. If it is **O** who asserted the universal quantifier, and if she also conceded that, $a : A$, then **P** may challenge the quantifier by choosing $a : A$, *and request of O* that she asserts $B(a)$; this is how the elimination rule for the universal quantifier are introduced in the dialogical framework (for details see chapter VII).

These distinctions can be made explicit if we enrich the first-order language of standard dialogical logic with expressions inspired by CTT. The first task is to introduce statements of the form “*p* : A”. On the right-hand side of the colon is the proposition *A*,
on the left-hand side is the *local reason* *p* brought forward to back the proposition *during a play*. The local reason is therefore *local* if the force of the assertion is limited to the level of plays. But when the assertion “*p* : *A*” is backed by a *winning strategy*, the judgement asserted draws its justification precisely from that strategy, thus endowing *p* with the status of a strategic reason that, in the most general cases, encodes an arbitrary choice of *O*.

The rock bottom of the dialogical approach is still the play level notion of dialogue-definiteness of the proposition, namely

> For an expression to count as a proposition *A* there must exist an individual play about the statement *X*! *A*, in the course of which *X* is committed to bring forward a local reason to back that proposition, and the play reaches a final position with either win or loss after a finite number of moves according to definite local and structural rules.<fnref n="173" />

The deontic feature of logic is here built directly within the dialogical concept of statements about a proposition. More generally, the point is that, as observed by Martin-Löf (2017a, p. 9), according to the dialogical conception, logic belongs to the area of ethics.

One way of explaining how this important aspect has been overseen or misunderstood might be that the usual approaches to the layers underlying logic got the order of priority between the deontic notions and the epistemic notions the wrong way round.<fnref n="174" />

The Oslo and Stockholm lectures of Martin-Löf propose a fine analysis of the inner and outer structure of the statements of logic from the point of view of speech-act theory, that put the order of priority mentioned above right; in doing so it pushes forward one of the most cherished tenets of the dialogical framework, namely that *logic has its roots in ethics*.

In fact, Martin-Löf's insights on dialogical logic as re-establishing the historical links of ethics and logic provides a clear answer to Wilfried Hodge's (2008)<fnref n="175" /> sceptical view in his section 2 as to what the dialogical framework's contribution is. Hodge's criticism seems to target the *mathematical* interest of a dialogical conception of logic, rather than a philosophical interest which does not seem to attract much of his interest.

In lieu of a general plaidoyer for the dialogical framework's philosophical contribution to the foundations of logic and mathematics, which would bring us too far, let us highlight these three points which result from the above discussions:<fnref n="176" />

1) the dialogical interpretation of epistemic assumptions offers a sound venue for the development of inference-based foundations of logic;

2) the dialogical take on the interaction of epistemic and deontic notions in logic, as well as the specification of the play level's role, display new ways of implementing the interface pragmatics-semantics within logic.

3) the introduction of *knowing how* into the realm of logic is of great import (Martin-Löf, 2017a; 2017b).

Obviously, formal semantics in the Tarski-style is blind to the first point, misunderstands the nature of the interface involved in the second, and ignores the third.

<footnote n="173">¹⁷³ See above, section XI.1, for a discussion on dialogue-definiteness.</footnote>

<footnote n="174">¹⁷⁴ See (Martin-Löf, 2017b, p. 9).</footnote>

<footnote n="175">¹⁷⁵ See also (Hodges, 2001) and (Trafford, 2017, pp. 87-88).</footnote>

<footnote n="176">¹⁷⁶ See also our Preface and Introduction (chapter I).</footnote>
XI.6 A brief historical study on material dialogues

Allow us now a brief historic interlude on the distinction between statements of the form *a:* *A* and of the form *A(a)* *true* (i.e. *b(a)* : *A(a)*), where *a:* *B* ), *B:* *set* and *A(x): prop [x : B]*. As discussed in (Rahman & Clerbout, 2015, pp. 145-146), this distinction is close to the (Lorenz & Mittelstrass, 1967) reconstruction of Plato’s notion of *correct naming* in the *Cratylus* (Plato, 1997),<fnref n="177" /> and has some links with the dialogical notion of *predicator rule*, which lies at the very basis of material dialogues.

Lorenz & Mittelstrass point out two fundamental speech-acts: *naming* ( Angeòmuàzèiv) and *stating* (lèyèeiv). The first speech-act, *naming*, amounts to the act of subsuming an entity under a concept, while the second, *stating*, establishes a proposition about a previously named entity. If the naming has been correctly carried out, the (named) entity reveals the concept it instantiates (*names reveal objects for what they are*). *Stating truly* concerns the truth of a proposition that has been constituted by instantiating a propositional function with a suitable element of a genus (a correctly named instantiation of a genus). Thus both acts, naming and stating, involve judgements: while on the one hand *naming* corresponds to the assertion that an entity instantiates a given genus, it says *what the entity is*, and has therefore the following form

a: A [A: genus]

on the other hand, *stating* corresponds to the act of building a proposition, such as *A(a)*,
out of the propositional function *A(x)* and the *genus B*.

In other words, the correct form of the result of an act of stating, or *saying how something is*, amounts to the judgment

$A(a): prop [a: B]$

that presupposes *A(x): prop [x : B]*, *B: genus*. In this regard,

* the act of naming *a*: *A* is said to be true iff *a* instantiates *A*; and

* the proposition *A(a)* is *true* iff *a* is one of the entities to which the propositional function *A(x)* applies (i.e, if *a* is of the genus *B*), and it is the case that *A(x)* can be said of *a*.

So the resulting proposition (in the context of our example) *A(a)* is true if *A(x)* can be
said of the entity *a*. In such a case *A(a)* would be *stated truly*.

The specialized literature harshly criticized Plato's claim, not that the results of acts of predication could be qualified as true or false, but that the results of acts of naming could be also. <fnref n="178" /> According to this criticism, while truth applies to propositions, it does

<footnote n="177">¹⁷⁷ For an endorsement of this interpretation see (Luce, 1969).</footnote>

<footnote n="178">¹⁷⁸ Viktor Ilievski (2013, pp. 12-13) provides a condensed formulation of this kind of criticisms:

*Socrates next proceeds briefly to discuss true and false speech, with an intention to point out to Hermogenes that there is a possibility of false, incorrect speech. It is a matter of very basic knowledge of logic that truth-value is to be attributed to propositions, or more precisely utterances, specific uses of sentences. Plato's Socrates acknowledges that, but he, somewhat surprisingly, ascribes truth-value to the constituents, or parts of the statements as well, on the assumption that whatever is true of the unit, has to be true of its parts as well. This seems to be an example of flagrant error in reasoning, known as the fallacy of division. Why would Plato's Socrates commit such a fallacy in the course of what seems to be a valid and*</footnote>
not apply to entities. (Lorenz & Mittelstrass, 1967, pp. 6-12) defended the old master by suggesting to read these passages as presupposing that in both cases we have the *same kind* of acts of predication,<fnref n="179" /> that is, both acts of predication can be qualified as true even if they do not involve the same form of predicator rule.

Resorting to the CTT-setting in order to develop the (Lorenz & Mittelstrass, 1967) interpretation, it follows that claiming both acts of predication can be qualified as true does not necessarily entail that *both* involve propositional functions: Plato's claim can be defended by carefully distinguishing both constituents of a judgement involving a specific propositional function, namely

i. the act of asserting that a given entity exemplifies the genus presupposed by the formation of that propositional function, and

ii. the act of asserting a proposition that results from substituting the variable of the relevant propositional function by a suitable instantiation.

According to this analysis, it is possible to endorse at the same time the following claims of Plato:

1. acts of *naming* and *stating* involve different acts of judgement;

2. both naming and stating can be qualified as *true*.

3. Neither *1 nor 2* assume<fnref n="180" /> that the truth of the result of an act of predication always involves a prescription on how to constitute a propositional function out of another one; rather, *stating*, that is saying *how something is*, presupposes *what it is*, that is *naming*.<fnref n="181" />

Thus, on or view, whereas the act of predication *tɛS* (*naming*) can be reconstructed as *t: S*; the act of predication *tɛP* (*stating*), can be reconstructed as *P(t) [t : S] true*.

This reconstruction makes explicit the (Lorenz & Mittelstrass, 1967, p. 6) point that *stating* presupposes *naming*. Indeed, let us take the expression *man*, and use it ambiguously again to express both the assertion *man true* (where *man* : *genus*), and the assertion *Man(a) true* (where *Man(a)*: *prop [a : living being]*). From what we presented earlier on CTT, both make perfect sense:

* *man true* iff *man* can be instantiated, and thus asserting that *a* exemplifies *man* amounts to the truth of *man*—provided *a* is indeed such an element;<fnref n="182" />

* *Man(a) true* if *a* is an instantiation of the genus *living being*, presupposed by the formation of the propositional function *Man(x)*; there is a method that takes

<footnote>stable argument? One obvious answer would be that the very theory he is about to expound presupposes the notion of names as independent bearers of meaning and truth, linguistic microcosms encapsulating within themselves both truth-value and reference. In other words, the theory of true and false names has to presuppose that names do not only refer or designate, or even do not only refer and sometimes suggest descriptions, but that they always necessarily represent descriptions of some kind.</footnote>

<footnote n="179">¹⁷⁹ It follows that a true sentence SP really does consist of the ' true parts ' S and P, i.e. *t ε S* and *t ε P*. In case of a false sentence SP, however, the second part *t ε P* is false, while the first part *t ε S* should ex definitione be considered as true, because any sentence is necessarily a sentence about something (Soph. 262e), namely the subject of it. The subject has to be effectively determined, i.e. it must be a thing correctly named, before one is going to state something about it. (Lorenz & Mittelstrass, 1967, p. 6)</footnote>

<footnote n="180">¹⁸⁰ As (Lorenz & Mittelstrass, 1967, p. 13) seem to:</footnote>

<footnote>Names, i.e. predicates, are tools with which we distinguish objects from each other. To name objects or to let an individual fall under some concept is on the other hand the means to state something about objects, i.e. to teach and to learn about objects, as Plato prefers to say.</footnote>

<footnote n="181">¹⁸¹ See section II.1.1.</footnote>

<footnote n="182">¹⁸² In fact (Lorenz & Mittelstrass, 1967, p. 6) pointed out, and rightly so, that both acts presuppose a contextually given entity.</footnote>
us from *a*: *living being* to *Man(a)*. Moreover, the falsity of *Man(a)* also presupposes that *a* is of the suitable genus presupposed by the propositional function *Man(x)*.<fnref n="183" />

If we follow this interpretation the fact that the judgement *Man(a)* true presupposes *Man(x)*: *prop [x: living being]* makes explicit the relation between naming and stating.<fnref n="184" />

In the dialogical framework we might say that the formation presupposed is the formation leading to specifying the Socratic rule for material dialogues. These considerations call for further generalization: the difference between the two *Cratylus* speech-acts dovetails the difference between categorical assertions, that involve independent types, and hypotheticals, that involve dependent ones. Let us summarize our suggestions in the following table:<fnref n="185" />

<table><caption>Table 59: Comparing Categorical and Hypothetical judgements in CTT and Plato</caption><thead><tr><th colspan="2">Categorical Judgements</th><th colspan="2">Hypothetical Judgements</th></tr><tr><th colspan="2">tεS</th><th colspan="2">tεP</th></tr><tr><th>CTT</th><th>Cratylus</th><th>CTT</th><th>Cratylus</th></tr></thead><tbody><tr><td>c : B</td><td>naming (όνομάζειν)</td><td>c(a): B(a) [a: A]</td><td>stating (λέγειν)</td></tr><tr><td rowspan="3">c is of type B</td><td>B names c</td><td rowspan="3">Presupposes the formation rule:<br>The propositional function B(x) yields a proposition provided x is an element of the set A</td><td>B is predicated of a under the condition that a exemplifies A<br>Presupposes the predicator rule:<br>B(x) yields a proposition provided x exemplifies A</td></tr><tr><td>c is B</td><td></td></tr><tr><td>c exemplifies (the genus) B</td><td></td></tr><tr><td>c:B true iff:</td><td>c is B true iff:</td><td>B(a) true iff:</td><td>B(a) true iff:</td></tr><tr><td>c is a canonical element of B, or<br>c is generated from a canonical one</td><td>c correctly exemplifies B</td><td>B applies to a</td><td>B(x) is correctly said of a</td></tr></tbody></table>

<footnote n="183">¹⁸³ Cf. Lorenz & Mittelstrass, 1967, p. 6).</footnote>

<footnote n="184">¹⁸⁴ (Lorenz & Mittelstrass, 1967, pp. 6-7) claim that *being correct* and *being true* are to be considered as synonymous.</footnote>

<footnote n="185">¹⁸⁵ The table is based on preliminary results of an ongoing research project by S. Rahman and Fachrur Rozie. Let us point out that we do not claim herewith that the CTT-notion of *type* is the same as Plato's notion of *genus*, but rather that they play the same role in judgements involving *type/genus*. The claim is that we can establish a kind of parallelism between the CTT use of judgements involving independent and dependent types on the one hand, and Plato's distinction on the other hand between acts of naming and acts of asserting a proposition. For a detailed comparison between the CTT notion of *type* and Plato's notion of *genus*, a detailed study is due.</footnote>
<table><tr><td>presupposes the formation rule of<br>the independent type B</td><td>presupposes the formation rule of<br>the genus B (and not of B(x))</td><td>presupposes the formation rule of<br>the set A<br>and the propositional function B(x) over the set A:<br>B(x) : prop [x : A]<br>where B(x) is a dependent type upon A.</td><td>presupposes the formation rule of<br>(the genus) A<br>and the predicate rule<sup>186</sup> for B(x):<br>x exemplifies A &harr; B(x)<br>where B(x) is defined over the genus A</td></tr></table>

Identifying versus Individuating

As pointed out in section II.1, the categorical predication of the form $a : B$ does not have the function-argument form that modern logic inherited from Frege, and responds to the question on *what something is*. The modern view conflating both forms of predication into a functional form can perhaps be put in the following terms: the act of predicating amounts to a process by the means of which some objects are selected from a given domain. If the process is set to select exactly one, then we have an *individuating predication*. According to this view, Frege-Russell's claim that individual constants can be substituted *salva veritate* by definite descriptions can be seen as pushing forward the idea that at the very end there is only one fundamental form of predication that amounts to a process of selecting by description, either exactly one or a whole class of objects of the given domain. This certainly is very different from Plato and Aristotle: according to Plato's approach the process associated to the act of naming cannot be reduced to relating some kind of sign with an object but establishes what that object is; in other words, the focus in Plato's naming is on *identifying*, not *individuating*. Moreover, it is not only the case that in the tradition of Plato the acts are different but, according to the wise insight of our forefathers, the source of knowledge is not a given indiscriminate domain.

This might provide some further insight on the division of waters between rigid designation and definite descriptions brought forward by Saul Kripke and Hilary Putnam (with some variation). Indeed, the theory of rigid designation assumes an indiscriminate domain of objects within which by some kind of individuation act of baptism exactly one individual is selected. According to this approach the selection happens in such a way that despite the fact that the initial act of selection might have been carried out by an individuating description (but not necessarily so), the further identification occurs by some kind of causally transmitted indexation.

If we push our parallelism forward, we could say that in Plato there is also a *chain of propagation* of some (perhaps initial) act of *identification*. However, the chain might propagate a mistake, and the role of the philosopher is to stop the further dissemination of the error and investigate about its *correctness* by the means of dialectical interaction.

<footnote n="186">¹⁸⁶ For the notion of *predicate rule* see (Lorenz & Mittelstrass, 1967). We use here a slightly modified version: the original idea is that those kinds of rules establish how to constitute a predicate from a different one. We propose a more basic predicate rule, namely, a rule that establishes how a predicate is ascribed to a certain kind of objects (the genus underlying the predicate).</footnote>
Thus the rigid-designation theory is a remarkable change from Plato's distinction, as it considers after all that both definite descriptions and rigid designators have the same role: they are individuating operations. Plato's theory of naming and stating establishes two different acts, one identifying while the other establishing how an object is.

As pointed out by (Lorenz & Mittelstrass, 1967, p. 6), the act of establishing presupposes a domain already set by the act of identification.<fnref n="187" /> Thus, acts of individuation, that did not attract much neither Plato's nor Aristotle's attention, might find their place as special cases of acts of establishing how things are (if our aim is to insert them in their framework). If that is so, then acts of individuation presuppose a given domain.

This certainly does not meet the perspective of those that strictly distinguish between individuation by flexible designation (or description) and individuation by rigid designation (or indexation). According to this view, both individuation acts are to be understood as two different irreducible ways of selecting objects within a domain, one by selecting an individual and the other as a unary set. The first form of individuation cannot be imported as such in the framework of the *Cratylus*,<fnref n="188" /> since it would assume that we could *individuate an object without any knowledge of what it is*.<fnref n="189" />

XI.7 Intersubjectivity, Dialogues, and Learning

The book *Logic, Language and Method. On Polarities in Human Experience*, published in 2010, includes papers written by Kuno Lorenz in a period extending over more than thirty years. These papers have planted the seeds for his further penetrating work (Lorenz, 2009; 2010a; 2010b), which can be considered as philosophical variations on *Das Dialogische Prinzip* (2010b, pp. 509-520) underlying what is often known as *Dialogical Constructivism*.

In the framework of Dialogical Constructivism, the analysis of the notion of intersubjectivity starts by the study of a situation where two persons are engaged in the process of acquiring a common action-competence in a situation of teaching and learning;<fnref n="190" /> what is at stake then is not simply mirroring an individual competence in another individual, but rather it is a *procedure* which incorporates from the very beginning this dialogical situation.<fnref n="191" /> Immanent reasoning, being an offspring of Dialogical Constructivism, inherits its philosophical background and sensitivity. In this regard, the rules of the play level are not actualizations by themselves, but are rather

<footnote n="187">¹⁸⁷ It can be seen as an early formulation of what is nowadays known as the *principle of comprehension* or *set-specification*.</footnote>

<footnote n="188">¹⁸⁸ The proviso “as such” leaves the door open to the possibility that acts of individuation could be carried out over an already separated subset. However, it is not clear if this will work out in Plato’s framework, and it seems that it will not work at all in CTT.</footnote>

<footnote n="189">¹⁸⁹ In contemporary modal logic we have also a more sophisticated theory of dealing with individuation, namely the theory of individuals as *World-Lines* of the late Jaakko Hintikka (1969), but the discussion of it will go far beyond the aims of this short excursus. (Tulenheimo, 2017) provides both a thorough critical discussion of Hintikka’s formulations and a new approach inspired by the notion of world-lines. Tulenheimo’s distinction between *world-bounded local objects*, and individuals, might perhaps be associated to the distinction between identification (in a world) and Individuating (establishing world links).</footnote>

<footnote n="190">¹⁹⁰ The bibliographic background of this section is based mainly on (Lorenz, 2010a, pp. 2017-2018) chapter *Procedural Principles of the Erlangen School. On the Interrelation between the principles of method, of dialogue, and of reason*.</footnote>

<footnote n="191">¹⁹¹ The act of executing must be distinguished from taking the action as an object: while executing an action, actor and execution are said to be indistinguishable.</footnote>
procedures for actualizing some action consisting in dealing with an object or appropriating that object, be it in a situation of teaching and learning, or any dialogical situation.

A consequence of Lorenz's (2017b, pp. 509-520) general dialogical principle is that the interface semantics-pragmatics should be understood

1. neither as the result of the *semantization of pragmatics*—where deontic, epistemic, ontological, and temporal modalities become truth-functional operators;

2. nor as the result of the *pragmatization of semantics*—where a propositional kernel, when put into use, is complemented by moods yielding assertions, questions, commands and so on.

Lorenz's view (2010a, pp. 71-79) is that the differentiation of semantic and pragmatic layers is the result of the articulation within one and the same utterance: each utterance displays in principle both features: it *signifies* (semantic layer) and it *communicates* (pragmatic layer).

Take for example one-word sentences such as:

*Rabbit!*

*Water!*

With these utterances the speaker is conveying at the same time *what* the object is and *how* the object is. But while the first aspect (*what*) is related to *object-constitution*, the second (*how*) is related to *object-description*; or, if we use the terminology of Wittgenstein's *Tractatus*, the first aspect relates to the *act of showing* and the second to the *act of saying*. Object-description is carried out by the use of predicates on an already constituted domain of objects. Lorenz recalls here Plato's *Cratylus* (388b), in which these two acts and their interdependence are distinguished as *naming* (which has the role of *indicating*) and *establishing* (with the role of *communicating*).<fnref n="192" /> Lorenz's view is that each utterance of a sentence has this double nature, not only one-word sentences. Thus:

(a) *Sam is smoking.*

has both roles, indicating as well as communicating; though according to this analysis, uttering such a sentence does not yield any ambiguity: uttering it simply displays within one movement *object-constitution* (or *construction*) and *object-description* (or *attribution*).

While the first, *object-constitution*, involves differentiating parts of a whole (including the processes of partitioning a whole by synthesis and analysis), the second, *object-description*, involves stating that a certain relation holds. In this regard, *attribution* is not a relation, but a means for stating that relations hold of objects. The usual procedure for representing attribution by using extensional class-membership relations thus blurs this distinction.

According to the language of immanent reasoning (borrowed from CTT),

(a) *Sam is smoking*

can be read as expressing either

(b) !Sam : Smoking

or

(c) !d(Sam) : Smoking(Sam) (Sam : Human)

From the point of view of Lorenz's Dialogical Constructivism, me might say that the colon in both claims separates, using his words, the significative, particular, part of

<footnote n="192">¹⁹² In the previous section (XI.6) we briefly present the discussion of the *Cratylus* found in (Lorenz & Mittelstrass, 1967).</footnote>
the expression from the communicative, universal, side of it, placed at the right side of the colon.

These considerations deserve further investigation, though this conclusion is not their place. But the point here is to stress that, according to the dialogical principle, pragmatization and semantization are two different aspects: *a*: *B* and *B(a)* are not the result of an ambiguity of some sort, but are simply two aspects, the semantic aspect and the pragmatic aspect.

## The dialogical nature of meaning and the I-You-perspective

From the point of view of Dialogical Constructivism however, it is not sufficient to simply observe utterances and to conceptually distinguish their two functions. Beyond descriptive observation, what it is needed is to be aware of the necessarily social character of the acquisition process by the means of which these expressions are used and some distinctions are drawn. Meaning is dialogical by nature and it emerges from participation and collective interaction.

In other words, Dialogical Constructivism—and immanent reasoning after it—is based on the idea that the acquisition of meaning is intimately linked to both, the accomplishment of acting, and living together and participating in the actions the utterances express:

> [...] the use of predicates as passed from one generation to the another is not usually accomplished by mere deictic activities, that is, in distantiated pointing, but rather "empractically" (Bühler), i.e. in very accomplishment of acting and living together. What "walking" or "eating" is [...] we learn these things linguistically only along with the activities themselves, at the same time. In living with one another over a long period of time we acquired the use of such predicates as "father", "brother" [...]. (Kamlah & Lorenzen, 1984, p. 36)

Lorenz (2010a, pp. 142-146) develops further this *empractical* approach, based on the work of Johan Gottfried Herder (1960 [1772]) Part II, who uses a dialogue model of teaching and learning to identify items of cultural process, inasmuch as both “doing” and “suffering”, terms that can be used to characterize the two roles which always occur together: that of teaching and that of learning. Indeed, from the point of view of the teacher, there is a predominant *active* feature of “doing”, while the learner, in its *passive* perspective, is taking the doing of the teacher as the way to actualize the action-schema in question. This duality also applies to performances by the means of which singulars are brought forward, and holds likewise of actualizations by the means of which particulars are brought forward.

In other words, within the model of an elementary dialogue situation in which two agents are engaged in the process of acquiring an action competence, the activities of actualizing and schematizing should not be understood as carrying out two separate actions; rather one acquires the competence of one and the same action by learning to play both the active and the passive role. Active actualization makes the action appear in *I*-perspective, passive schematization lets it appear in *You*-perspective. Performing an action-schema by *myself*, that has been experienced as a schema through *your* actualization of the schema, amounts to learning by means of an *I-You*-perspective.

In order to switch from an *I-You*-perspective to a *He-She*-perspective, the constitution of a *third-man perspective* is required. The perspective of the "third-man" results from the nesting of *I-You*-perspectives— (Lorenz, 2010a, p. 48). In such a framework,<fnref n="193" />

<footnote n="193">¹⁹³ See (Lorenz, 2010a, pp. 144-146).</footnote>
* *individuality*, i.e. a difference between individuals on the level of reflection, will be recognized only within some common activity where *I*- and *You*-perspectives are put into action.

* *sociality*, i.e. an equality of individuals on the level of reflection, can be exercised only by being conscious of the different approaches within the same common activity. This amounts to cooperation by means of individual contributions.

XI.8 Final Words

The play level is the level where meaning is forged: it provides the material with which we reason.<fnref n="194" /> It reduces neither to the (singular) performances that actualize the interaction-types of the play level, nor to the “tactics” for the constitution of the schema that yields a winning strategy.

We call our dialogues involving rational argumentation *dialogues for immanent reasoning* precisely because *reasons* backing a statement, that are now *explicit* denizens of the object-language of plays, are *internal* to the development of the dialogical interaction itself.

More generally, the emergence of concepts, so we claim, are not only games of giving and asking for reasons (games involving *Why*-questions) they are also games that include moves establishing *how is it that the reason brought forward accomplishes the explicative task*. Dialogues for immanent reasoning are dialogical games of *Why* and *How*.

More generally, Lorenz (2010a, pp. 140-147) associates *competition* with the process of *individualization* and cooperation with the process of *socialization*, so that both are different ways to display the *I* and the *You* perspectives. In this sense, the dialogical teaching-learning situation is where competition and cooperation interact: both intertwine in collective forms of dialogical interaction that take place at the play level.

The point is that within the dialogical framework actualizing and schematizing should not be understood as performing two separate actions: through these actions we acquire the competence that is associated to the meaning of an expression by *learning* to play both, *the active* and *the passive* role. This feature of Dialogical Constructivism stems from Herder’s view<fnref n="195" /> that the cultural process is a process of education, in which teaching and learning always occur together: dialogues display this double nature of the cultural process in which concepts emerge from a complex interplay of *why* and *how* questions.

If the reader allows us to condense our proposal once more, we might say that the perspective we are trying to bring to the fore is rooted in the intimate conviction that meaning and knowledge are something we do together; our perspective is thus an invitation to participate in the open-ended dialogue that is the human pursuit of knowledge and collective understanding, since philosophy’s endeavour is immanent to the kind of dialogical interaction that makes reason happen.

<footnote n="194">¹⁹⁴ To use Peregrin’s (2014, pp. 228-229) words.</footnote>

<footnote n="195">¹⁹⁵ See (Herder, 1960 [1772]), Part II.</footnote>
# APPENDIX: MAIN NOTATION FOR CTT

## Equality and Identity

* Judgemental equality

* $a = b : B$

* $A = B : \mathbf{set}$

* Identity-Predicate: $\mathbf{Id}(A, x, y)$, alternatively $x =_A y$

## Judgement

* Categorical

* $a : A$

* $a = b : A$

* $A : \mathbf{set}$, alternatively $A : \mathbf{prop}$

* $A = B : \mathbf{set}$, alternatively $A = B : \mathbf{prop}$

* Hypothetical

* $x : A \vdash b : B$, alternatively $b : B (x : A)$

* $x : A \vdash b = c : B$, alternatively $b = c : B (x : A)$

* $x : A \vdash B : \mathbf{set}$, alternatively $B : \mathbf{set} (x : A)$

* $x : A \vdash B = C : \mathbf{set}$, alternatively $B = C : \mathbf{set} (x : A)$

## Types

* proposition: **prop**

* sets: **set**

* natural numbers: $\mathbb{N}$

* propositional-function of prime numbers: **Pr**

## Operators over a family of sets

* $\Sigma$-operator $(\Sigma x : A)B$

* Proof object: $\langle a, b \rangle : (\Sigma x : A)B$

* Projectors for $c : (\Sigma x : A)B$ $\mathbf{fst}(c) : (\Sigma x : A)B$ $\mathbf{snd}(c) : (\Sigma x : A)B[\mathbf{fst}(c)]$

* $\Pi$-operator $(\Pi x : A)B$

* Proof object for $\Pi \lambda x.b : (\Pi x : A)B$ :

* Application for $c : (\Pi x : A)B, a : A$ $\mathbf{ap}(\lambda x.b, a) : B[a]$

* Disjoint union $A + B$

* Proof-object given $a : A$ $\mathbf{i}(a) : A + B$

* Proof-object given $b : B$ $\mathbf{j}(b) : A + B$

* Selector given $c : A + B, x : A \vdash d : C[i(x)], y : B \vdash e : C[j(y)]$ $\mathbf{D}(c, x.d, y.e) : C[c]$
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INDEX OF TABLES

<table><tr><td>Table 1: particle rules for conjunction</td><td>53</td></tr><tr><td>Table 2: particle rules for disjunction</td><td>54</td></tr><tr><td>Table 3: particle rules for implication</td><td>54</td></tr><tr><td>Table 4: particle rules for negation</td><td>54</td></tr><tr><td>Table 5: particle rules</td><td>55</td></tr><tr><td>Table 6: Particle rules for dialogical games: propositional connectives</td><td>66</td></tr><tr><td>Table 7: Particle rules for dialogical games: quantifiers</td><td>66</td></tr><tr><td>Table 8: Summing up the properties of the particle rules</td><td>67</td></tr><tr><td>Table 9: Summing up the particle rules</td><td>67</td></tr><tr><td>Table 10: SR2 special Copy-cat rule</td><td>70</td></tr><tr><td>Table 11: O-decisions in a play</td><td>78</td></tr><tr><td>Table 12: Formation rules, condensed presentation</td><td>97</td></tr><tr><td>Table 13: Synthesis of a local reason for implication</td><td>97</td></tr><tr><td>Table 14: synthesis of a local reason for conjunction</td><td>98</td></tr><tr><td>Table 15: Synthesis of a local reason for disjunction</td><td>98</td></tr><tr><td>Table 16: general structure for the synthesis of a local reason for a constant</td><td>98</td></tr><tr><td>Table 17: Formation rules</td><td>105</td></tr><tr><td>Table 18: Substitution rule within dependent statements (subst-D)</td><td>106</td></tr><tr><td>Table 19: synthesis rules for local reasons</td><td>108</td></tr><tr><td>Table 20: analysis rules for local reasons</td><td>109</td></tr><tr><td>Table 21: Non-reflexive cases of the Socratic rule</td><td>114</td></tr><tr><td>Table 22: Transmission of definitional equality I: Substitution within dependent or independent statements</td><td>115</td></tr><tr><td>Table 23: Transmission of definitional equality II</td><td>115</td></tr><tr><td>Table 24: Synthesis of strategic reasons for P:</td><td>138</td></tr><tr><td>Table 25: synthesis of the strategic reason for negation</td><td>138</td></tr><tr><td>Table 26: correspondence between synthesis of strategic reasons and introduction rules</td><td>139</td></tr><tr><td>Table 27: analysis rules for P-strategic reasons</td><td>142</td></tr><tr><td>Table 28: recalling the synthesis of local reasons</td><td>152</td></tr><tr><td>Table 29: correspondence between dialogical rules on P-moves and introduction rules</td><td>156</td></tr><tr><td>Table 30: correspondence between dialogical rules on O-moves and elimination rules</td><td>156</td></tr><tr><td>Table 31: Global rules for Bool and Boolean operators in immanent reasoning</td><td>191</td></tr><tr><td>Table 32: Specific Socratic rule for Bool</td><td>192</td></tr><tr><td>Table 33: Global rules for classical truth-functional operators</td><td>192</td></tr><tr><td>Table 34: Formation rule for Id</td><td>194</td></tr><tr><td>Table 35: Specific Socratic rule for the global synthesis of the local reason for P!Id(A, a, a) and the rules of equality</td><td>195</td></tr><tr><td>Table 36: Specific Socratic rule for triggering the reflexivity move O!Id(A, a, a)</td><td>195</td></tr><tr><td>Table 37: Analysis I: The Global Analysis of O p: Id(A, a, a)</td><td>196</td></tr><tr><td>Table 38: Leibniz-substitution for Id</td><td>196</td></tr></table>
<table><tr><td>Table 39: Lbz-Id-subst and Definitional Equality</td><td>197</td></tr><tr><td>Table 40: Formation rules for Eq</td><td>200</td></tr><tr><td>Table 41: Specific Socratic rule for the Global Synthesis of the local reason for X! Eq(A, a, a)</td><td>200</td></tr><tr><td>Table 42: Synthesis of N</td><td>201</td></tr><tr><td>Table 43: Analysis rules of N</td><td>201</td></tr><tr><td>Table 44: Specific Socratic rules for N</td><td>201</td></tr><tr><td>Table 45: Formation rule of Nn</td><td>203</td></tr><tr><td>Table 46: rules for Nn</td><td>203</td></tr><tr><td>Table 47: Formation rules for u</td><td>204</td></tr><tr><td>Table 48: Specific Socratic rules for Gx and synthesis of N1</td><td>206</td></tr><tr><td>Table 49: dialogical rules for operations in many-valued logics</td><td>209</td></tr><tr><td>Table 50: dialogical rules from N3 to Bool</td><td>211</td></tr><tr><td>Table 51: general dialogical rule for an empirical quantity</td><td>212</td></tr><tr><td>Table 52: Formal incompatibility</td><td>213</td></tr><tr><td>Table 53: Contentual incompatibility</td><td>213</td></tr><tr><td>Table 54: tableaux and P-winning strategies for conjunction and disjunction</td><td>229</td></tr><tr><td>Table 55: O-tonk rule for challenge and defence</td><td>230</td></tr><tr><td>Table 56: P-tonk rule for challenge and defence</td><td>230</td></tr><tr><td>Table 57: black-bullet player-independent particle rules</td><td>231</td></tr><tr><td>Table 58: deploying the black-bullet challenges</td><td>231</td></tr><tr><td>Table 59: Comparing Categorical and Hypothetical judgements in CTT and Plato</td><td>240</td></tr></table>
## List of Names

Aczel

Almog

Aristotle

Armagardt

Austin

Baire

Bar-Hillel

Barrio

Barnes

Batens

Beirlaen

Bell

Bishop

Bobenrieth

Borel

Brandom

Breckenridge

Brouwer

Cardascia

Carnap

Carnielli

Church

Clerbout

Conigilio

Cooper

Corcoran

Cratylus

Crubellier

Curry

Dango

Diaconescu

D'Ottaviano

Dummett

Dutilh Novaes

Dybjer

Ebbinghaus

Eckoubili

Felscher

Fine

Fischer

Fontaine

Fraenkel

Frege

Garner

Gentzen

Ginzburg

Girard

Goodman
Gorisse

Granström

Hale

Herder

Hintikka

Hodges

Hofmann

Howard

Ilievski

Jovanovic

Kamlah

Keiff

Klev

Kolmogorov

Krabbe

Lebesgue

Leibniz

Levy

Lorenz

Lorenzen

Luce

Lukasiewicz

Magidor

Magnier

Marion

Martin-Löf

McConaughey

McDowell

Mitelstrass

Myhle

Netz

Nordström

Nzokou

Parmenides

Paseau

Peregrin

Petersson

Piecha

Plato

Popek

Primiero

Quine

Rahman

Ranta

Read

Redmond

Rückert

Schröder-Heister

Schwemmer

Sellars
Smith

Socrates

Sundholm

Tasistro

Theaetetus

Thiercelin

Thompson

Trafford

Troelstra

Tulenheimo

van Daalen

van Heijenoort

van der Schaar

Wittgenstein

Zermelo
List of Subjects

abstract

abstraction

absurdum

application

arbitrary object

arbitrary reference

assertion

assumption

Bool

Boolean

axiom of choice

canonical

Cartesian

case-dependent

category

categorical

choice

computation

computational rule

concession

conjunction

constructive

content

context

copy-cat

core

core of the strategy

correct naming

course of values

critical

Curry-Howard isomorphism

decision

definitional

definitional equality

demonstration

dependent

dialectical

dialogical

dialogical roots of equality

dialogue

disjoint union

disjunction

dispense

double negation

ethesis

elimination

emergence of equality

empty set
equality

extensive form of a dialogical game

extensive form of a strategy

function

function type

game

game-tree

Geltung

global meaning

harmony

hypothesis

identity

instruction

judgement

knowledge

local meaning

material

m-dependent resolution

meaning

metalanguage

metalevel

metalogic

natural number

natural deduction

negation

nominal

object

object language

ontological

Opponent

pensée aveugle

play

local reason

posit

posit-substitution

predicate

predication

predicator

premiss

presupposition

projection

**prop**

Proponent

proposition

propositional equality

quantifier

range-course

resolution

resolution of functions

resolution of instructions
selection

sequence of moves

**set**

Socratic

Socratic Rule

starting rule

strategic

strategic object

strategy

strategy-level

structural rule

subset-separation

substitution

substitution of instructions

syntactic

terminal

transmission

tree

type

universal

validity

Wertverlauf

winn

winning strategy

**yes**

**yes-no**
LIST OF EXERCISES AND EXAMPLES

<table><thead><tr><th></th><th>Negation ¬</th><th>Negation ⊃⊥</th><th>Sections</th></tr></thead><tbody><tr><td>1</td><td>(A ∨ B) ⊃ (B ∨ A)</td><td></td><td>III.4</td></tr><tr><td>2</td><td>(A ∧ B) ⊃ A</td><td></td><td>III.4</td></tr><tr><td>3</td><td>A ∨ ¬A</td><td>A ∨ (A ⊃⊥)</td><td>III.4; IV.5; XI.1</td></tr><tr><td>4</td><td>¬¬A ⊃ A</td><td>((A ⊃⊥) ⊃⊥) ⊃ A</td><td>III.4; IV.5</td></tr><tr><td>5</td><td>((A ∨ B) ∧ ¬A) ⊃ B</td><td>((A ∨ B) ∧ (A ⊃⊥)) ⊃ B</td><td>V.3</td></tr><tr><td>6</td><td>((((A ∨ (B ∧ C)) ∧ ¬A) ⊃ (B ∧ C)) ∧ ((D ∨ E) ⊃ (D ∨ E)))</td><td>((((A ∨ (B ∧ C)) ∧ (A ⊃⊥)) ⊃ (B ∧ C)) ∧ ((D ∨ E) ⊃ (D ∨ E)))</td><td>V.3</td></tr><tr><td>7</td><td>(A ∧ B) ⊃ (B ∧ A)</td><td></td><td>VI.4</td></tr><tr><td>8</td><td>(∀x: A)(B(x) ⊃ C(x)): prop[A: set; B(x): prop; C(x): prop[x: A]]</td><td></td><td>VII.1</td></tr><tr><td>9</td><td>(∀x: D)Q(x) ⊃ Q(x)</td><td></td><td>VII.3; VII.5</td></tr><tr><td>10</td><td>B ∨ A [c: A ∨ B]</td><td></td><td>VII.6.1</td></tr><tr><td>11</td><td>¬¬(A ∨ ¬A)</td><td>((A ∨ (A ⊃⊥)) ⊃⊥) ⊃⊥</td><td>III.4; IV.5; VII.6.2</td></tr><tr><td>12</td><td>¬(A ∧ ¬A)</td><td>(A ∧ (A ⊃⊥)) ⊃⊥</td><td>VII.6.3</td></tr><tr><td>13</td><td>((A ⊃ B) ⊃ A) ⊃ A</td><td></td><td>V.2.1; VII.6.4</td></tr><tr><td>14</td><td>(A ∧ ¬B) ⊃ ¬(A ⊃ B)</td><td>(A ∧ (B ⊃⊥)) ⊃ ((A ⊃ B) ⊃⊥)</td><td>VII.6.5</td></tr><tr><td>15</td><td>(A ∧ B) ∧ C [c: A ∧ (B ∧ C)]</td><td></td><td>VII.6.6; VII.7.4; IX.5</td></tr><tr><td>16</td><td>(B ∧ A) ⊃ C [c: (A ∧ B) ⊃ C]</td><td></td><td>VII.6.7; VII.7.4; IX.5</td></tr><tr><td>17</td><td>¬(∀x: D)A(x)[((∃x: D)¬A(x))</td><td>(∀x: D)A(x) ⊃⊥ [(∃x: D)(A(x) ⊃⊥)]</td><td>VII.6.8</td></tr><tr><td>18</td><td>((∃x: D)A(x) ∧ (∃x: D)B(x)[((∃x: D)(A(x) ∧ B(x))])</td><td></td><td>VII.6.9</td></tr><tr><td>19</td><td>((∃x: D)(∃y: D)(A(x) ⊃ B(y)) [A(a) ⊃ B(b); a: D; b: D]</td><td></td><td>VII.6.10</td></tr><tr><td>20</td><td>((∃x: D)B(x) ∧ (∃x: D)P(x) [B(a) ∧ P(b); a: D; b: D]</td><td></td><td>VII.6.11</td></tr><tr><td>21</td><td>((∃x: D)(∃y: D)A(x, y)[((∃x: D)A(x, x)])</td><td></td><td>VII.6.12</td></tr><tr><td>22</td><td>((∀x: D)(∀y: D)(A(x, y) ∧ A(y, x)) [((∀x: D)(∀y: D)A(x, y)])</td><td></td><td>VII.6.13V<br>II.4</td></tr><tr><td>23</td><td>((∃x: D)(A(x) ⊃ (�forall x: D)A(x))<br>[( (�exists x: D)¬A(x)) ∨ ((forall x: D)A(x))]</td><td>((∃x: D)(A(x) ⊃ (�forall x: D)A(x))<br>[( (�exists x: D)(A(x) ⊃⊥)) ∨ ((forall x: D)A(x))]</td><td>VII.6.14</td></tr><tr><td>24</td><td>((∀x: A)(∃y: B(x))C(x, y) ⊃ (�exists f: (�forall x: A)B(x))C(x, f(x))</td><td></td><td>VIII.2</td></tr><tr><td>25</td><td>((∀x: Bool)(Id(Bool, x, yes) ∨ Id(Bool, x, no))</td><td></td><td>X.2.3</td></tr><tr><td>26</td><td>Id(Bool, yes, no) ⊃⊥ [n<sup>0</sup>, n<sup>1</sup>: u; N<sup>1</sup>; yes, no : Bool; G(x): u [x : Bool]]</td><td></td><td>X.3.3</td></tr><tr><td>27</td><td>A ∧ ¬A</td><td>A ∧ (A ⊃⊥)</td><td>XI.1</td></tr></tbody></table>