bluuebunny/crossref_metadata_2025 huggingface dataset:
Identity
Authors: Williamson Timothy
Published: September 2018

Anything whatsoever has the relation of identity to itself, and to nothing else. Things are identical if they are one thing, not two. We can refute the claim that they are identical if we can find a property of one that is not simultaneously a property of the other. The concept of identity is fundamental to logic. Without it, counting would be impossible, for we could not distinguish in principle between counting one thing twice and counting two different things. When we have acquired the concept, it can still be difficult to make this distinction in practice. Misjudgments of identity are possible because one thing can be presented in many guises.

Identity judgments often involve assumptions about the nature of things. The identity of the present mature tree with the past sapling implies persistence through change. The non-identity of the actual child of one couple with the hypothetical child of a different couple is implied by the claim that ancestry is an essential property. Knowledge of what directions are involves knowledge that parallel lines have identical directions. Many controversies over identity concern the nature of the things in question. Others concern challenges to the orthodox conception just sketched of identity itself.
Proper names
Authors: Forbes Graeme
Published: September 2018

The Roman general Julius Caesar was assassinated on 14 March 44 bc by conspirators led by Brutus and Cassius. It is a remarkable fact that, in so informing or reminding the reader, the proper names ‘Julius Caesar‘, ‘Brutus’ and ‘Cassius’ are used to refer to three people each of whom has been dead for about two thousand years. Our eyes could not be used to see any of them, nor our voices to talk to them, yet we can refer to them with our words.

The central philosophical issue about proper names is how this sort of thing is possible: what exactly is the mechanism by which the user of a name succeeds in referring with the name to its bearer? As the example indicates, whatever the mechanism is, it must be something that can relate the use of a name to its bearer even after the bearer has ceased to exist.

In modern philosophy of language there are two main views about the nature of the mechanism. On one account, which originated with Frege, a use of a name expresses a conception or way of thinking of an object, and the name refers to whatever object fits, or best fits, that conception or way of thinking. Thus with ‘Cassius‘, for example, I may associate the conception ‘the conspirator whom Caesar suspected because of his size’ (recalling a famous speech in Shakespeare’s Julius Caesar). Conception theories are usually called ‘sense’ theories, after Frege’s term ‘Sinn‘. The other account is the ‘historical chain’ theory, due to Kripke and Geach. In Geach’s words, ’ for the use of a word as a proper name there must in the first instance e someone acquainted with the object named…. But…the use of a given name for a given object…can be handed on from one generation to another…Plato knew Socrates, and Aristotle knew Plato, and Theophrastus knew Aristotle, and so on in apostolic succession down to our own times. That is why we can legitimately use "Socrates" as a name the way we do’ (1969–70: 288–9).
Quantifiers, generalized
Authors: Westerstahl Dag
Published: September 2018

Generalized quantifiers are logical tools with a wide range of uses. As the term indicates, they generalize the ordinary universal and existential quantifiers from first-order logic, ‘∀x’ and ‘∃x’, which apply to a formula A(x), binding its free occurrences of x. ∀xA(x) says that A(x) holds for all objects in the universe and ∃xA(x) says that A(x) holds for some objects in the universe, that is, in each case, that a certain condition on A(x) is satisfied. It is natural then to consider other conditions, such as ‘for at least five’, ‘at most ten’, ‘infinitely many’ and ‘most’. So a quantifier Q stands for a condition on A(x), or, more precisely, for a property of the set denoted by that formula, such as the property of being non-empty, being infinite, or containing more than half of the elements of the universe. The addition of such quantifiers to a logical language may increase its expressive power.

A further generalization allows Q to apply to more than one formula, so that, for example, Qx(A(x),B(x)) states that a relation holds between the sets denoted by A(x) and B(x), say, the relation of having the same number of elements, or of having a non-empty intersection. One also considers quantifiers binding more than one variable in a formula. Qxy,zu(R(x,y),S(z,u)) could express, for example, that the relation (denoted by) R(x,y) contains twice as many pairs as S(z,u), or that R(x,y) and S(z,u) are isomorphic graphs.

In general, then, a quantifier (the attribute ‘generalized’ is often dropped) is syntactically a variable-binding operator, which stands semantically for a relation between relations (on individuals), that is, a second-order relation. Quantifiers are studied in mathematical logic, and have also been applied in other areas, notably in the semantics of natural languages. This entry first presents some of the main logical facts about generalized quantifiers, and then explains their application to semantics.
Quantification and inference
Authors: King Jeffrey C.
Published: September 2018

Quantificational reasoning in natural languages contains occurrences of expressions which, though name-like from a syntactic perspective, intuitively seem to be importantly different from ordinary names. For example, in attempting to argue that every F is G, we might consider an ‘arbitrary’ F, give ‘it’ the ‘name’ ‘n’, and go on to argue that n is G. That such an occurrence of ‘n’ is somehow different from occurrences of ordinary names can be seen by noting, for example, that arguing from the claim that Bertrand Russell is F to the claim that Bertrand Russell is G in general does not suffice for drawing the conclusion that every F is G. Similarly, given that some F is G, in general we do not think it legitimate to assert that Bertrand Russell is F and G, and continue reasoning from that claim. It seems that such occurrences of ‘n’ differ semantically from similar occurrences of ordinary names. Thus the question arises as to the semantics of occurrences of such expressions. An adequate semantic account must justify appropriate inferences to and from sentences containing these terms.
Quantifiers
Authors: Hintikka Jaakko , Sandu Gabriel
Published: September 2018

The quantifiers ‘some’ and ‘every’ were the object of the very first logical theory, Aristotelian syllogistic. An example of a syllogism is ‘Every Spartan is Greek, every Greek is European, therefore every Spartan is European’. In such inferences, no quantifier is governed by another one. Contrast this with ‘Everybody loves somebody’. Modern logic is often taken to have begun when Frege systematized for the first time the logic of quantifiers, including such dependent ones. In general, much of what has passed as logic over the centuries is in effect the study of quantifiers. This is especially clear with the area of logic variously known as quantification theory, lower predicate calculus or elementary logic. Some philosophers have even sought to limit the scope of logic to such a study of quantifiers. Yet the nature of quantifiers is a delicate matter which is captured incompletely by the logic initiated by Frege and Russell.
Fallacies
Authors: Walton Douglas
Published: September 2018

Fallacies are common types of arguments that have a strong tendency to go badly wrong, or to be used as deceptive tricks when two parties reason together. In some instances they are simply careless errors in thinking, while in other cases they are techniques of argumentation used by one arguer more or less deliberately to try to fool another into accepting a false conclusion. Fallacies have been described in logic textbooks since the time of Aristotle; their study, long neglected, has begun to be revived in recent years, as its practical importance in natural language reasoning has become apparent.
Type/token distinction
Authors: Wetzel Linda
Published: September 2018

The type/token distinction is related to that between universals and particulars. C.S. Peirce introduced the terms ‘type’ and ‘token’, and illustrated the distinction by pointing to two senses of ‘word’: in one, there is only one word ‘the’ in the English language; in the other, there are numerous words ‘the’ on the physical page you are now looking at. The latter are spatiotemporal objects composed of ink; they are said to be word tokens of the former, which is said to be the word type and is abstract. Phonemes, letters and sentences also come in types and tokens.
Necessary truth and convention
Authors: Sidelle Alan
Published: September 2018

Necessary truths have always seemed problematic, particularly to empiricists and other naturalistically-minded philosophers. Our knowledge here is a priori - grounded in appeals to what we can imagine or conceive (or can prove on that basis) - which seems hard to reconcile with such truths being factual, short of appealing to some peculiar faculty of a priori intuition. And what mysterious extra feature do necessary truths possess which makes their falsity impossible? Conventionalism about necessity claims that necessary truths obtain by virtue of rules of language, such as that ‘vixen’ means the same as ‘female fox’. Because such rules govern our descriptions of all cases - including counterfactual or imagined ones - they generate necessary truths (‘All vixens are foxes’), and our a priori knowledge is just knowledge of word meaning. Opponents of conventionalism argue that conventions cannot ground necessary truths, particularly in logic, and have also challenged the notion of analyticity (truth by virtue of meaning). More recent claims that some necessary truths are a posteriori have also fuelled opposition to conventionalism.
Computability theory
Authors: Mundici Daniele , Sieg Wilfried
Published: September 2018

The effective calculability of number-theoretic functions such as addition and multiplication has always been recognized, and for that judgment a rigorous notion of ‘computable function’ is not required. A sharp mathematical concept was defined only in the twentieth century, when issues including the decision problem for predicate logic required a precise delimitation of functions that can be viewed as effectively calculable. Predicate logic emerged from Frege’s fundamental ‘Begriffsschrift’ (1879) as an expressive formal language and was described with mathematical precision by Hilbert in lectures given during the winter of 1917–18. The logical calculus Frege had also developed allowed proofs to proceed as computations in accordance with a fixed set of rules; in principle, according to Gödel, the rules could be applied ‘by someone who knew nothing about mathematics, or by a machine’.

Hilbert grasped the potential of this mechanical aspect and formulated the decision problem for predicate logic as follows: ‘The Entscheidungsproblem [decision problem] is solved if one knows a procedure that permits the decision concerning the validity, respectively, satisfiability of a given logical expression by a finite number of operations.’ Some, for example, von Neumann (1927), believed that the inherent freedom of mathematical thought provided a sufficient reason to expect a negative solution to the problem. But how could a proof of undecidability be given? The unsolvability results of other mathematical problems had always been established relative to a determinate class of admissible operations, for example, the impossibility of doubling the cube relative to ruler and compass constructions. A negative solution to the decision problem obviously required the characterization of ‘effectively calculable functions’.

For two other important issues a characterization of that informal notion was also needed, namely, the general formulation of the incompleteness theorems and the effective unsolvability of mathematical problems (for example, of Hilbert’s tenth problem). The first task of computability theory was thus to answer the question ‘What is a precise notion of "effectively calculable function"?’. Many different answers invariably characterized the same class of number-theoretic functions: the partial recursive ones. Today recursiveness or, equivalently, Turing computability is considered to be the precise mathematical counterpart to ‘effective calculability’. Relative to these notions undecidability results have been established, in particular, the undecidability of the decision problem for predicate logic. The notions are idealized in the sense that no time or space limitations are imposed on the calculations; the concept of ‘feasibility’ is crucial in computer science when trying to capture the subclass of recursive functions whose values can actually be determined.
Church’s theorem and the decision problem
Authors: Parikh Rohit
Published: September 2018

Church’s theorem, published in 1936, states that the set of valid formulas of first-order logic is not effectively decidable: there is no method or algorithm for deciding which formulas of first-order logic are valid. Church’s paper exhibited an undecidable combinatorial problem P and showed that P was representable in first-order logic. If first-order logic were decidable, P would also be decidable. Since P is undecidable, first-order logic must also be undecidable.

Church’s theorem is a negative solution to the decision problem (Entscheidungsproblem), the problem of finding a method for deciding whether a given formula of first-order logic is valid, or satisfiable, or neither. The great contribution of Church (and, independently, Turing) was not merely to prove that there is no method but also to propose a mathematical definition of the notion of ‘effectively solvable problem’, that is, a problem solvable by means of a method or algorithm.
Herbrand’s theorem
Authors: Ungar A.M.
Published: September 2018

According to Herbrand’s theorem, each formula F of quantification theory can be associated with a sequence F1, F2, F3,… of quantifier-free formulas such that F is provable just in case Fn is truth-functionally valid for some n. This theorem was the centrepiece of Herbrand’s dissertation, written in 1929 as a contribution to Hilbert’s programme. It provides a finitistically meaningful interpretation of quantification over an infinite domain. Furthermore, it can be applied to yield various consistency and decidability results for formal systems. Herbrand was the first to exploit it in this way, and his work has influenced subsequent research in these areas. While Herbrand’s approach to proof theory has perhaps been overshadowed by the tradition which derives from Gentzen, recent work on automated reasoning continues to draw on his ideas.
Complexity, computational
Authors: Urquhart Alasdair
Published: September 2018

The theory of computational complexity is concerned with estimating the resources a computer needs to solve a given problem. The basic resources are time (number of steps executed) and space (amount of memory used). There are problems in logic, algebra and combinatorial games that are solvable in principle by a computer, but computationally intractable because the resources required by relatively small instances are practically infeasible.

The theory of NP-completeness concerns a common type of problem in which a solution is easy to check but may be hard to find. Such problems belong to the class NP; the hardest ones of this type are the NP-complete problems. The problem of determining whether a formula of propositional logic is satisfiable or not is NP-complete. The class of problems with feasible solutions is commonly identified with the class P of problems solvable in polynomial time. Assuming this identification, the conjecture that some NP problems require infeasibly long times for their solution is equivalent to the conjecture that P≠NP. Although the conjecture remains open, it is widely believed that NP-complete problems are computationally intractable.
Non-constructive rules of inference
Authors: Hazen A.P.
Published: September 2018

For some theoretical purposes, generalized deductive systems (or, ‘semi-formal’ systems) are considered, having rules with an infinite number of premises. The best-known of these rules is the ‘ω-rule’, or rule of infinite induction. This rule allows the inference of ∀nΦ(n) from the infinitely many premises Φ(0), Φ(1),… that result from replacing the numerical variable n in Φ(n) with the numeral for each natural number. About 1930, in part as a response to Gödel’s demonstration that no formal deductive system had as theorems all and only the true formulas of arithmetic, several writers (most notably, Carnap) suggested considering the semi-formal systems obtained, from some formulation of arithmetic, by adding this rule. Since no finite notation can provide terms for all sets of natural numbers, no comparable rule can be formulated for higher-order arithmetic. In effect, the ω-rule is valid just in case the relevant quantifier can be interpreted substitutionally; looked at from the other side, the validity of some analogue of the ω-rule is the essential mathematical characteristic of substitutional quantification.
Model theory
Authors: Hodges Wilfrid
Published: September 2018

Model theory studies the relations between sentences of a formal language and the interpretations (or ‘structures’) which make these sentences true or false. It offers precise definitions of truth, logical truth and consequence, meanings and modalities. These definitions and their consequences have revolutionized the teaching of elementary logic.

Model theory also forms a branch of mathematics concerned with the ways in which mathematical structures can be classified. This technical work has led to philosophically interesting results in at least two areas: it has thrown light on the nature of the set-theoretic universe, and in nonstandard analysis it has suggested new forms of argument (where we prove something different from what we intended, but then use a general model-theoretic argument to change the result into what we wanted).

The word ‘model’ has many other uses. For example, model theory is not about scientific theories as models of the world. It is also a controversial question – not considered here – how model theory is connected with the ‘mental models’ which appear in the psychology of reasoning.
Semantic paradoxes and theories of truth
Authors: McGee Vann
Published: September 2018

The Cretan philosopher Epimenides said that Cretans always lie. Assuming, for the sake of argument, the mendacity of all other statements by Cretans, we get a paradox: if what Epimenides said was true, it must have been a lie, whereas if what he said was a lie, it would have made his statement true. The citizens of Crete have long since forgiven the insult, but semantics has never recovered.

Alfred Tarski perceived the consequences of Epimenides’ paradox with particular clarity. Our common-sense intuitions about truth follow the paradigm: ‘Snow is white’ is true if and only if snow is white. As Tarski rigorously shows, if the language we are describing (the object language) is the same as the language in which we are formulating our theory (the metalanguage), this paradigm will be inconsistent with the rudimentary laws of syntax. The conclusion Tarski drew was that, if we are to develop a satisfactory theory of truth, our metalanguage must be essentially richer in expressive power than the object language. Since there is no human language essentially richer than English (or any other natural language), there can be no satisfactory theory of truth for English.

One earnestly hopes that this is not the end of the matter. Tarski’s analysis leaves open the prospect that we can develop a fully satisfactory theory of truth for a substantial fragment of English; also the prospect that we can develop a theory of truth for English as a whole which, while not fully satisfying our intuitions, is none the less useful and illuminating.

Both prospects have been substantially advanced by Saul Kripke’s ‘Outline of a Theory of Truth’, which exploits the idea that there are truth-value gaps, statements which are neither true nor false, and that Epimenides’ insult was one of them.

Invocation of truth-value gaps does not resolve the paradox in any straightforward way. If we let the phrase ‘the simple liar sentence’ refer to the sentence ‘The simple liar sentence is false’, we see that we can readily account for the paradoxical features of the sentence by declaring the sentence neither true nor false; but if we let the strengthened liar sentence be ‘The strengthened liar sentence is not true’, we get a sentence we cannot dispose of so tidily. If the strengthened liar is neither true nor false, then it is not true; but that it is not true is precisely what the sentence says.

Truth-value gaps have not vanquished the liar paradox. Nor have any of the alternatives, the most prominent of which are a contextualist account, which sees the English word ‘true’ as radically ambiguous, and so-called ‘revision theory’, which investigates the cyclic reasoning that occurs when we try to evaluate the simple liar sentence: if the sentence is true, then it must be false; but if, then, it is false, it must be true; and so on. While these approaches have not eliminated the paradox, they have opened new approaches that have greatly improved our prospects for finding a comfortable way to live with it.
Beth’s theorem and Craig’s theorem
Authors: Swijtink Zeno
Published: September 2018

Beth’s theorem is a central result about definability of non-logical symbols in classical first-order theories. It states that a symbol P is implicitly defined by a theory T if and only if an explicit definition of P in terms of some other expressions of the theory T can be deduced from the theory T. Intuitively, the symbol P is implicitly defined by T if, given the extension of these other symbols, T fixes the extension of the symbol P uniquely. In a precise statement of Beth’s theorem this will be replaced by a condition on the models of T. An explicit definition of a predicate symbol states necessary and sufficient conditions: for example, if P is a one-place predicate symbol, an explicit definition is a sentence of the form (x) (Px ≡φ(x)), where φ(x) is a formula with free variable x in which P does not occur. Thus, Beth’s theorem says something about the expressive power of first-order logic: there is a balance between the syntax (the deducibility of an explicit definition) and the semantics (across models of T the extension of P is uniquely determined by the extension of other symbols).

Beth’s definability theorem follows immediately from Craig’s interpolation theorem. For first-order logic with identity, Craig’s theorem says that if φ is deducible from ψ, there is an interpolant θ, a sentence whose non-logical symbols are common to φ and ψ, such that θ is deducible from ψ, while φ is deducible from θ. Craig’s theorem and Beth’s theorem also hold for a number of non-classical logics, such as intuitionistic first-order logic and classical second-order logic, but fail for other logics, such as logics with expressions of infinite length.
Paradoxes of set and property
Authors: Moore Gregory H.
Published: September 2018

Emerging around 1900, the paradoxes of set and property have greatly influenced logic and generated a vast literature. A distinction due to Ramsey in 1926 separates them into two categories: the logical paradoxes and the semantic paradoxes. The logical paradoxes use notions such as set or cardinal number, while the semantic paradoxes employ semantic concepts such as truth or definability. Both often involve self-reference.

The best known logical paradox is Russell’s paradox concerning the set S of all sets x such that x is not a member of x. Russell’s paradox asks: is S a member of itself? A moment’s reflection shows that S is a member of itself if and only if S is not a member of itself – a contradiction.

Russell found this paradox by analysing the paradox of the largest cardinal. The set U of all sets has the largest cardinal number, since every set is a subset of U. But there is a cardinal number greater than that of any given set M, namely the cardinal of the power set, or set of all subsets, of M. Thus the cardinal of the power set of U is greater than that of U, a contradiction. (The paradox of the largest ordinal, discussed below, is similar in structure.)

Among the semantic paradoxes, the best known is the liar paradox, found by the ancient Greeks. A man says that he is lying. Is what he says true or false? Again, either conclusion leads to its opposite. Although this paradox was debated in medieval Europe, its modern interest stems from Russell, who placed it in the context of a whole series of paradoxes, including his own.
Set theory, different systems of
Authors: Potter Michael
Published: September 2018

To begin with we shall use the word ‘collection’ quite broadly to mean anything the identity of which is solely a matter of what its members are (including ‘sets’ and ‘classes’). Which collections exist? Two extreme positions are initially appealing. The first is to say that all do. Unfortunately this is inconsistent because of Russell’s paradox: the collection of all collections which are not members of themselves does not exist. The second is to say that none do, but to talk as if they did whenever such talk can be shown to be eliminable and therefore harmless. This is consistent but far too weak to be of much use. We need an intermediate theory.

Various theories of collections have been proposed since the start of the twentieth century. What they share is the axiom of ‘extensionality’, which asserts that any two sets which have exactly the same elements must be identical. This is just a matter of definition: objects which do not satisfy extensionality are not collections. Beyond extensionality, theories differ. The most popular among mathematicians is Zermelo–Fraenkel set theory (ZF). A common alternative is von Neumann–Bernays–Gödel class theory (NBG), which allows for the same sets but also has proper classes, that is, collections whose members are sets but which are not themselves sets (such as the class of all sets or the class of all ordinals).

Two general principles have been used to motivate the axioms of ZF and its relatives. The first is the iterative conception, according to which sets occur cumulatively in layers, each containing all the members and subsets of all previous layers. The second is the doctrine of limitation of size, according to which the ‘paradoxical sets’ (that is, the proper classes of NBG) fail to be sets because they are in some sense too big. Neither principle is altogether satisfactory as a justification for the whole of ZF: for example, the replacement schema is motivated only by limitation of size; and ‘foundation’ is motivated only by the iterative conception.

Among the other systems of set theory to have been proposed, the one that has received widespread attention is Quine’s NF (from the title of an article, ‘New Foundations for Mathematical Logic’), which seeks to avoid paradox by means of a syntactic restriction but which has not been provided with an intuitive justification on the basis of any conception of set. It is known that if NF is consistent then ZF is consistent, but the converse result has still not been proved.
Property theory
Authors: Cocchiarella Nino B.
Published: September 2018

Traditionally, a property theory is a theory of abstract entities that can be predicated of things. A theory of properties in this sense is a theory of predication – just as a theory of classes or sets is a theory of membership. In a formal theory of predication, properties are taken to correspond to some (or all) one-place predicate expressions. In addition to properties, it is usually assumed that there are n-ary relations that correspond to some (or all) n-place predicate expressions (for n≥ 2). A theory of properties is then also a theory of relations.

In this entry we shall use the traditional labels ‘realism’ and ‘conceptualism’ as a convenient way to classify theories. In natural realism, where properties and relations are the physical, or natural, causal structures involved in the laws of nature, properties and relations correspond to only some predicate expressions, whereas in logical realism properties and relations are generally assumed to correspond to all predicate expressions.

Not all theories of predication take properties and relations to be the universals that predicates stand for in their role as predicates. The universals of conceptualism, for example, are unsaturated concepts in the sense of cognitive capacities that are exercised (saturated) in thought and speech. Properties and relations in the sense of intensional Platonic objects may still correspond to predicate expressions, as they do in conceptual intensional realism, but only indirectly as the intensional contents of the concepts that predicates stand for in their role as predicates. In that case, instead of properties and relations being what predicates stand for directly, they are what nominalized predicates denote as abstract singular terms. It is in this way that concepts – such as those that the predicate phrases ‘is wise’, ‘is triangular’ and ‘is identical with’ stand for – are distinguished from the properties and relations that are their intensional contents – such as those that are denoted by the abstract singular terms ‘wisdom’, ‘triangularity’ and ‘identity’, respectively. Once properties are represented by abstract singular terms, concepts can be predicated of them, and, in particular, a concept can be predicated of the property that is its intensional content. For example, the concept represented by ‘is a property’ can be predicated of the property denoted by the abstract noun phrase ‘being a property’, so that ‘Being a property is a property’ (or, ‘The property of being a property is a property’) becomes well-formed. In this way, however, we are confronted with Russell’s paradox of (the property of) being a non-self-predicable property, which is the intensional content of the concept represented by ‘is a non-self-predicable property’. That is, the property of being a non-self-predicable property both falls and does not fall under the concept of being a non-self-predicable property (and therefore both falls and does not fall under the concept of being self-predicable).
Logic in the early 20th century
Authors: Moore Gregory H.
Published: September 2018

The creation of modern logic is one of the most stunning achievements of mathematics and philosophy in the twentieth century. Modern logic – sometimes called logistic, symbolic logic or mathematical logic – makes essential use of artificial symbolic languages.

Since Aristotle, logic has been a part of philosophy. Around 1850 the mathematician Boole began the modern development of symbolic logic. During the twentieth century, logic continued in philosophy departments, but it began to be seriously investigated and taught in mathematics departments as well. The most important examples of the latter were, from 1905 on, Hilbert at Göttingen and then, during the 1920s, Church at Princeton.

As the twentieth century began, there were several distinct logical traditions. Besides Aristotelian logic, there was an active tradition in algebraic logic initiated by Boole in the UK and continued by C.S. Peirce in the USA and Schröder in Germany. In Italy, Peano began in the Boolean tradition, but soon aimed higher: to express all major mathematical theorems in his symbolic logic. Finally, from 1879 to 1903, Frege consciously deviated from the Boolean tradition by creating a logic strong enough to construct the natural and real numbers. The Boole–Schröder tradition culminated in the work of Löwenheim (1915) and Skolem (1920) on the existence of a countable model for any first-order axiom system having a model.

Meanwhile, in 1900, Russell was strongly influenced by Peano’s logical symbolism. Russell used this as the basis for his own logic of relations, which led to his logicism: pure mathematics is a part of logic. But his discovery of Russell’s paradox in 1901 required him to build a new basis for logic. This culminated in his masterwork, Principia Mathematica, written with Whitehead, which offered the theory of types as a solution.

Hilbert came to logic from geometry, where models were used to prove consistency and independence results. He brought a strong concern with the axiomatic method and a rejection of the metaphysical goal of determining what numbers ‘really’ are. In his view, any objects that satisfied the axioms for numbers were numbers. He rejected the genetic method, favoured by Frege and Russell, which emphasized constructing numbers rather than giving axioms for them. In his 1917 lectures Hilbert was the first to introduce first-order logic as an explicit subsystem of all of logic (which, for him, was the theory of types) without the infinitely long formulas found in Löwenheim. In 1923 Skolem, directly influenced by Löwenheim, also abandoned those formulas, and argued that first-order logic is all of logic.

Influenced by Hilbert and Ackermann (1928), Gödel proved the completeness theorem for first-order logic (1929) as well as incompleteness theorems for arithmetic in first-order and higher-order logics (1931). These results were the true beginning of modern logic.
Multiple-conclusion logic
Authors: Smiley Timothy
Published: September 2018

Ordinary arguments can have any number of premises but only one conclusion. Multiple-conclusion logic also allows for any number of conclusions in an argument, regarding them as setting out the field of possibilities among which the truth must lie if the premises are true. Such an argument counts as valid if it is impossible for all the premises to be true and all the conclusions false. Anything that can be said about premises can now be said, mutatis mutandis, about conclusions, and much of the interest of the subject comes from exploiting this duality. Putting conclusions on a par with premises reflects the idea that truth and falsity, and likewise acceptance and rejection, are polar notions standing on a par with one another.
Quantum logic
Authors: Forrest Peter
Published: September 2018

The topic of quantum logic was introduced by Birkhoff and von Neumann (1936), who described the formal properties of a certain algebraic system associated with quantum theory. To avoid begging questions, it is convenient to use the term ‘logic’ broadly enough to cover any algebraic system with formal characteristics similar to the standard sentential calculus. In that sense it is uncontroversial that there is a logic of experimental questions (for example, ‘Is the particle in region R?’ or ‘Do the particles have opposite spins?’) associated with any physical system. Having introduced this logic for quantum theory, we may ask how it differs from the standard sentential calculus, the logic for the experimental questions in classical mechanics. The most notable difference is that the distributive laws fail, being replaced by a weaker law known as orthomodularity.

All this can be discussed without deciding whether quantum logic is a genuine logic, in the sense of a system of deduction. Putnam argued that quantum logic was indeed a genuine logic, because taking it as such solved various problems, notably that of reconciling the wave-like character of a beam of, say, electrons, as it passes through two slits, with the thesis that the electrons in the beam go through one or other of the two slits. If Putnam’s argument succeeds this would be a remarkable case of the empirical defeat of logical intuitions. Subsequent discussion, however, seems to have undermined his claim.
Infinitary logics
Authors: Buldt Bernd
Published: September 2018

An infinitary logic arises from ordinary first-order logic when one or more of its finitary properties is allowed to become infinite, for example, by admitting infinitely long formulas or infinitely long or infinitely branched proof figures. The need to extend first-order logic became pressing in the late 1950s when it was realized that many of the fundamental notions of mathematics cannot be expressed in first-order logic in a way that would allow for their logical analysis. Because infinitary logics often do not suffer the same limitation, they have become an essential tool in mathematical logic.
Dialogical logic
Authors: Krabbe Erik C.W.
Published: September 2018

Dialogical logic characterizes logical constants (such as ‘and’, ‘or’, ‘for all’) by their use in a critical dialogue between two parties: a proponent who has asserted a thesis and an opponent who challenges it. For each logical constant, a rule specifies how to challenge a statement that displays the corresponding logical form, and how to respond to such a challenge. These rules are incorporated into systems of regimented dialogue that are games in the game-theoretical sense. Dialogical concepts of logical consequence can then be based upon the concept of a winning strategy in a (formal) dialogue game: B is a logical consequence of A if and only if there is a winning strategy for the proponent of B against any opponent who is willing to concede A. But it should be stressed that there are several plausible (and non-equivalent) ways to draw up the rules.
Mereology
Authors: Forrest Peter
Published: September 2018

Mereology is the theory of the part–whole relation and of derived operations such as the mereological sum. (The sum of several things is the smallest thing of which they are all parts.) It was introduced by Leśniewski to avoid Russell’s paradox.

Unlike the set-membership relation, the part–whole relation is transitive. This makes mereology much weaker than set theory, but gives the advantage of ontological parsimony. For example, mereology does not posit the proliferation of entities found in set theory, such as ∅⁣,{∅⁣},{{∅⁣}},….

Mereology has occasioned controversy: over whether many things really have a mereological sum if they are either scattered or, even worse, of different categories; over the uniqueness of sums; and over Lewis’ claim that the non-empty subsets of a set are literally parts of it.
Antirealism in the philosophy of mathematics
Authors: Moore A.W.
Published: September 2018

Realism in the philosophy of mathematics is the position that takes mathematics at face value. According to realists, mathematics is the science of mathematical objects (numbers, sets, lines and so on); mathematicians, to use the old metaphor, are discoverers, not inventors. Moreover, just as there may be truths about physical reality which we can never know, so too, realists say, there may be truths about mathematical reality which we can never know.

It is this claim in particular which antirealists find unacceptable. Equating what can be known in mathematics with what can be proved, they insist that only what can be proved is true. (Only what can be proved: different accounts of what this ‘can’ means, facing different difficulties, generate different positions.) This leads antirealists to recoil not only from realism but also from the practice of mathematicians themselves. For the orthodox assumption that every mathematical statement is either true or false would be invalidated, on the antirealist view, by a statement that was neither provable nor disprovable. Not that antirealists themselves can see it in these terms. For if a statement were neither provable nor disprovable, that would itself be an unprovable truth about mathematical reality. Antirealists must learn how to be circumspect even in defence of their own circumspection.
Category theory, applications to the foundations of mathematics
Authors: McLarty Colin
Published: September 2018

Since the 1960s Lawvere has distinguished two senses of the foundations of mathematics. Logical foundations use formal axioms to organize the subject. The other sense aims to survey ‘what is universal in mathematics’. The ontology of mathematics is a third, related issue.

Moderately categorical foundations use sets as axiomatized by the elementary theory of the category of sets (ETCS) rather than Zermelo–Fraenkel set theory (ZF). This claims to make set theory conceptually more like the rest of mathematics than ZF is. And it suggests that sets are not ‘made of’ anything determinate; they only have determinate functional relations to one another. The ZF and ETCS axioms both support classical mathematics.

Other categories have also been offered as logical foundations. The ‘category of categories’ takes categories and functors as fundamental. The ‘free topos’ (see Lambek and Couture 1991) stresses provability. These and others are certainly formally adequate. The question is how far they illuminate the most universal aspects of current mathematics.

Radically categorical foundations say mathematics has no one starting point; each mathematical structure exists in its own right and can be described intrinsically. The most flexible way to do this to date is categorically. From this point of view various structures have their own logic. Sets have classical logic, or rather the topos Set has classical logic. But differential manifolds, for instance, fit neatly into a topos Spaces with nonclassical logic. This view urges a broader practice of mathematics than classical.

This article assumes knowledge of category theory on the level of Category theory, introduction to §1.
Kronecker, Leopold (1823–91)
Authors: Majer Ulrich
Published: September 2018

Leopold Kronecker was one of the most influential German mathematicians of the late nineteenth century. He exercised a strong sociopolitical influence on the development of mathematics as an academic institution. From a philosophical point of view, his main significance lies in his anticipation of a new and rigorous epistemological perspective with regard to the foundations of mathematics: Kronecker became the father of intuitionism or constructivism, which stands in strict opposition to the methods of classical mathematics and their canonization by set theory.
Cantor, Georg (1845–1918)
Authors: Majer Ulrich
Published: September 2018

Georg Cantor and set theory belong forever together. Although Dedekind had already introduced the concept of a set and naïve set theory in 1872, it was Cantor who single-handedly created transfinite set theory as a new branch of mathematics. In a series of papers written between 1874 and 1885, he developed the fundamental concepts of abstract set theory and proved the most important of its theorems. Although today set theory is accepted by the majority of scientists as an autonomous branch of mathematics, and perhaps the most fundamental, this was not always the case. Indeed, when Cantor set out to develop his conception of sets and to argue for its acceptance, he initiated an inquiry into the infinite which raised questions that have still not been completely resolved today.
Non-monotonic logic
Authors: Fuhrmann Andre
Published: September 2018

A relation of inference is ‘monotonic’ if the addition of premises does not undermine previously reached conclusions; otherwise the relation is non-monotonic. Deductive inference, at least according to the canons of classical logic, is monotonic: if a conclusion is reached on the basis of a certain set of premises, then that conclusion still holds if more premises are added.

By contrast, everyday reasoning is mostly non-monotonic because it involves risk: we jump to conclusions from deductively insufficient premises. We know when it is worthwhile or even necessary (for example, in medical diagnosis) to take the risk. Yet we are also aware that such inference is ‘defeasible’ – that new information may undermine old conclusions. Various kinds of defeasible but remarkably successful inference have traditionally captured the attention of philosophers (theories of induction, Peirce’s theory of abduction, inference to the best explanation, and so on). More recently logicians have begun to approach the phenomenon from a formal point of view. The result is a large body of theories at the interface of philosophy, logic and artificial intelligence.
Boolean algebra
Authors: Bell J.L.
Published: September 2018

Boolean algebra, or the algebra of logic, was devised by the English mathematician George Boole (1815–64) and embodies the first successful application of algebraic methods to logic.

Boole seems to have had several interpretations for his system in mind. In his earlier work he thinks of each of the basic symbols of his ‘algebra’ as standing for the mental operation of selecting just the objects possessing some given attribute or included in some given class; later he conceives of these symbols as standing for the attributes or classes themselves. In each of these interpretations the basic symbols are conceived as being capable of combination under certain operations: ‘multiplication’, corresponding to conjunction of attributes or intersection of classes; ‘addition’, corresponding to (exclusive) disjunction or (disjoint) union; and ‘subtraction’, corresponding to ‘excepting’ or difference. He also recognizes that the algebraic laws he proposes are satisfied if the basic symbols are interpreted as taking just the number values 0 and 1.

Boole’s ideas have since undergone extensive development, and the resulting concept of Boolean algebra now plays a central role in mathematical logic, probability theory and computer design.
Tarski, Alfred (1901–83)
Authors: Murawski Roman
Published: September 2018

Alfred Tarski was a Polish mathematician and logician. He worked in metamathematics and semantics, set theory, algebra and the foundations of geometry. Some of his logical works, in particular his definition of truth, were also significant contributions to philosophy. He was a successful teacher and a master of writing simply and with great clarity about complicated matters.
Akan philosophical psychology
Authors: Wiredu Kwasi
Published: September 2018

The word Akan refers to the Twi-speaking people of southern and central Ghana. Akan traditional philosophy is essentially a philosophy of the person. It has cosmological ramifications, but the basic concepts emerge from the analysis of the human personality. That analysis is extremely sensitive to the complexity of the human psyche and the social dimensions of individual consciousness. These considerations explain and justify the prominent position occupied by the concept of a person in contemporary Akan philosophy.