truth-value semantics for intuitionistic propositional logic

truth-value semantics for intuitionistic propositional logic

A

truth-value semantic

system

for intuitionistic

propositional logic

consists of the set

V

n

:=

{

0

,

1

,

…

,

n

}

, where

n

≥

1

, and a

function

v

from the set of wff’s (

well-formed formulas

) to

V

n

satisfying the following

properties

:

1.

v

⁢

(

A

∧

B

)

=

min

⁡

{

v

⁢

(

A

)

,

v

⁢

(

B

)

}

2.

v

⁢

(

A

∨

B

)

=

max

⁡

{

v

⁢

(

A

)

,

v

⁢

(

B

)

}

3.

v

(

A

→

B

)

=

n

if

v

⁢

(

A

)

≤

v

⁢

(

B

)

, and

v

⁢

(

B

)

otherwise

4.

v

⁢

(

¬

⁢

A

)

=

n

if

v

⁢

(

A

)

=

0

, and

0

otherwise.

This function

v

is called an

interpretation

for the propositional logic. A wff

A

is said to be

true

for

(

V

n

,

v

)

if

v

⁢

(

A

)

=

n

, and a

tautology

for

V

n

if

A

is true for

(

V

n

,

v

)

for all interpretations

v

. When

A

is a tautology for

V

n

, we write

⊧

n

A

. It is not hard see that any truth-value semantic system is sound, in the sense that

⊢

i

A

(

A

is a

theorem

) implies

⊧

n

A

, for any

n

. A proof of this fact can be found here (

http://planetmath.org/truthvaluesemanticsforintuitionisticpropositionallogicissound

).

(

V

n

,

v

)

is a

generalization

of the truth-value semantics for classical propositional logic. Indeed, when

n

=

1

, we have the truth-value system for classical propositional logic.

However, unlike the truth-value semantic system for classical propositional logic, no truth-value semantic systems for intuitionistic propositional logic are

complete

: there are tautologies that are not theorems for each

n

. For example, for each

n

, the wff

⋁

j

=

1

n

+

2

⋁

i

=

j

n

+

1

(

p

j

↔

p

i

+

1

)

is a tautology for

V

n

that is not a theorem, where each

p

i

is a propositional letter. The

formula

⋁

k

=

1

m

A

i

is the abbreviation for

(

⋯

⁢

(

A

1

∨

A

2

)

∨

⋯

)

∨

A

m

, where each

A

i

is a formula. The following is a proof of this fact:

Proof.

Let

A

be the

⋁

j

=

1

n

+

2

⋁

i

=

j

n

+

1

(

p

j

↔

p

i

+

1

)

. Note that

p

1

,

…

,

p

n

+

2

are all the

proposition

letters in

A

. However, there are only

n

+

1

elements

in

V

n

, so for every interpretation

v

, there are some

p

i

and

p

j

such that

v

⁢

(

p

i

)

=

v

⁢

(

p

j

)

by the

pigeonhole principle

. Then

v

(

p

i

↔

p

j

)

=

n

, and hence

v

⁢

(

A

)

=

n

, implying that

A

is a tautology for

V

n

. However,

A

is not a tautology for

V

n

+

1

: let

v

be the interpretation that maps

p

i

to

i

-

1

. Then

v

(

p

i

↔

p

j

)

=

min

{

i

,

j

}

-

1

, so that

v

⁢

(

A

)

=

n

≠

n

+

1

. Therefore,

A

is not a theorem.
∎

Nevertheless, the truth-value semantic systems are useful in showing that certain theorems of the classical propositional logic are not theorems of the intuitionistic propositional logic. For example, the wff

p

∨

¬

⁢

p

(law of the excluded middle) is not a theorem, because it is not a tautology for

V

2

, for if

v

⁢

(

p

)

=

1

, then

v

⁢

(

p

∨

¬

⁢

p

)

=

1

≠

2

. Similarly, neither

¬

⁢

¬

⁢

p

→

p

(

law of double negation

) nor

(

(

p

→

q

)

→

p

)

→

p

(Peirce’s law) are theorems of the intuitionistic propositional logic.

Remark

. The

linearly ordered set

V

n

:=

{

0

,

1

,

…

,

n

}

turns into a

Heyting algebra

if we define the relative pseudocomplementation

operation

→

by

x

→

y

:=

n

if

x

≤

y

and

x

→

y

:=

y

otherwise. Then the

pseudocomplement

x

*

of

x

is just

x

→

0

. This points to the connection of the intuitionistic propositional logic and Heyting algebra.

Title

truth-value semantics for intuitionistic propositional logic

Canonical

name

TruthvalueSemanticsForIntuitionisticPropositionalLogic

Date of creation

2013-03-22 19:31:04

Last modified on

2013-03-22 19:31:04

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

21

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B20

Related topic

AxiomSystemForIntuitionisticLogic

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