truthvalue semantics for intuitionistic propositional logic
A truthvalue semantic system for intuitionistic propositional logic^{} consists of the set ${V}_{n}:=\{0,1,\mathrm{\dots},n\}$, where $n\ge 1$, and a function $v$ from the set of wff’s (wellformed formulas) to ${V}_{n}$ satisfying the following properties:

1.
$v(A\wedge B)=\mathrm{min}\{v(A),v(B)\}$

2.
$v(A\vee B)=\mathrm{max}\{v(A),v(B)\}$

3.
$v(A\to B)=n$ if $v(A)\le v(B)$, and $v(B)$ otherwise

4.
$v(\mathrm{\neg}A)=n$ 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 ${\vDash}_{n}A$. It is not hard see that any truthvalue semantic system is sound, in the sense that ${\u22a2}_{i}A$ ($A$ is a theorem^{}) implies ${\vDash}_{n}A$, for any $n$. A proof of this fact can be found here (http://planetmath.org/truthvaluesemanticsforintuitionisticpropositionallogicissound).
$({V}_{n},v)$ is a generalization^{} of the truthvalue semantics for classical propositional logic. Indeed, when $n=1$, we have the truthvalue system for classical propositional logic.
However, unlike the truthvalue semantic system for classical propositional logic, no truthvalue 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
$$\underset{j=1}{\overset{n+2}{\bigvee}}\underset{i=j}{\overset{n+1}{\bigvee}}({p}_{j}\leftrightarrow {p}_{i+1})$$ 
is a tautology for ${V}_{n}$ that is not a theorem, where each ${p}_{i}$ is a propositional letter. The formula^{} ${\bigvee}_{k=1}^{m}{A}_{i}$ is the abbreviation for $(\mathrm{\cdots}({A}_{1}\vee {A}_{2})\vee \mathrm{\cdots})\vee {A}_{m}$, where each ${A}_{i}$ is a formula. The following is a proof of this fact:
Proof.
Let $A$ be the ${\bigvee}_{j=1}^{n+2}{\bigvee}_{i=j}^{n+1}({p}_{j}\leftrightarrow {p}_{i+1})$. Note that ${p}_{1},\mathrm{\dots},{p}_{n+2}$ 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 $v({p}_{i}\leftrightarrow {p}_{j})=n$, 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 $i1$. Then $v({p}_{i}\leftrightarrow {p}_{j})=\mathrm{min}\{i,j\}1$, so that $v(A)=n\ne n+1$. Therefore, $A$ is not a theorem. ∎
Nevertheless, the truthvalue 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 $p\vee \mathrm{\neg}p$ (law of the excluded middle) is not a theorem, because it is not a tautology for ${V}_{2}$, for if $v(p)=1$, then $v(p\vee \mathrm{\neg}p)=1\ne 2$. Similarly, neither $\mathrm{\neg}\mathrm{\neg}p\to p$ (law of double negation) nor $((p\to q)\to p)\to p$ (Peirce’s law) are theorems of the intuitionistic propositional logic.
Remark. The linearly ordered set ${V}_{n}:=\{0,1,\mathrm{\dots},n\}$ turns into a Heyting algebra^{} if we define the relative pseudocomplementation operation^{} $\to $ by $x\to y:=n$ if $x\le y$ and $x\to y:=y$ otherwise. Then the pseudocomplement ${x}^{*}$ of $x$ is just $x\to 0$. This points to the connection of the intuitionistic propositional logic and Heyting algebra.
Title  truthvalue semantics for intuitionistic propositional logic 

Canonical name  TruthvalueSemanticsForIntuitionisticPropositionalLogic 
Date of creation  20130322 19:31:04 
Last modified on  20130322 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 