Infinite-valued Logics

Editorial

This is a short “stub” of a section on infinite-valued L ukasiewicz logic.

Infinite and finite valued Lukasiewicz matrices

Infinite-valued L ukasiewicz logic $\LogLuk[\infty]$source is defined using the matrix:

1. The standard propositional language $\Lang L_0$source with $\lnot$source, $\land$source, $\lor$source, $\lif$source.

2. The set of truth values $V_\infty$source.

3. $1$source is the only designated value, i.e., $V^+ = \{1\}$source.

4. Truth functions are given by the following functions:

   $$\tf{\lnot}[\LogLuk](x) & = 1 - x\\ \tf{\land}[\LogLuk](x,y) & = \min(x,y)\\ \tf{\lor}[\LogLuk](x,y) & = \max(x,y)\\ \tf{\lif}[\LogLuk](x,y) & = \min(1,1-(x-y)) = \begin{cases} 1 & \text{if } x \le y\\ 1-(x-y) & \text{otherwise.} \end{cases}$$source

$m$source-valued L ukasiewicz logic is defined the same, except $V = V_m$source.

Agreement of the two three valued Lukasiewicz definitions

The logic $\LogLuk[3]$source defined by the earlier three valued Lukasiewicz matrix definition is the same as $\LogLuk[3]$source defined by the infinite and finite valued Lukasiewicz matrix definition.

Proof

This can be seen by comparing the truth tables for the connectives given in the earlier three valued Lukasiewicz matrix definition with the truth tables determined by the equations in the infinite and finite valued Lukasiewicz matrix definition:

From infinite to finite Lukasiewicz consequence

If $\Gamma \Entails[\LogLuk[\infty]] !B$source then $\Gamma \Entails[\LogLuk[m]] !B$source for all $m \ge 2$source.

Proof

Exercise.

Exercise on restricting Lukasiewicz truth values

Prove the proposition on restricting infinite Lukasiewicz consequence to finitely many values.

Exercise on Lukasiewicz prelinearity

Show that $(p \lif q) \lor (q \lif p)$source is a tautology of $\LogLuk[\infty]$source.

Editorial

This is a short “stub” of a section on infinite-valued Gödel logic.

Infinite and finite valued Goedel matrices

Infinite-valued Gödel logic $\LogGod[\infty]$source is defined using the matrix:

1. The standard propositional language $\Lang L_0$source with $\lfalse$source, $\lnot$source, $\land$source, $\lor$source, $\lif$source.

2. The set of truth values $V_\infty$source.

3. $1$source is the only designated value, i.e., $V^+ = \{1\}$source.

4. Truth functions are given by the following functions:

   $$\tf{\lfalse} & = 0\\ \tf{\lnot}[\LogGod](x) & = \begin{cases} $1$ & \text{if } x =0\\ $0$ & \text{otherwise} \end{cases}\\ \tf{\land}[\LogGod](x,y) & = \min(x,y)\\ \tf{\lor}[\LogGod](x,y) & = \max(x,y)\\ \tf{\lif}[\LogGod](x,y) & = \begin{cases} 1 & \text{if } x \le y\\ y & \text{otherwise.} \end{cases}$$source

$m$source-valued Gödel logic is defined the same, except $V = V_m$source.

Agreement of the two three valued Goedel definitions

The logic $\LogGod[3]$source defined by the earlier three valued Goedel matrix definition is the same as $\LogGod[3]$source defined by the infinite and finite valued Goedel matrix definition.

Proof

This can be seen by comparing the truth tables for the connectives given in the earlier three valued Goedel matrix definition with the truth tables determined by the equations in the infinite and finite valued Goedel matrix definition:

From infinite to finite Goedel consequence

If $\Gamma \Entails[\LogGod[\infty]] !B$source then $\Gamma \Entails[\LogGod[m]] !B$source for all $m \ge 2$source.

Proof

Exercise.

Exercise on restricting Goedel truth values

Prove the proposition on restricting infinite Goedel consequence to finitely many values.

Exercise on Goedel prelinearity

Show that $(p \lif q) \lor (q \lif p)$source is a tautology of $\LogGod[\infty]$source.

Exercise separating finite and infinite Goedel tautologies

Show that $(p \lif q) \lor (q \lif r) \lor (r \lif s)$source, which is a tautology of $\LogGod[3]$source, is not a tautology of $\LogGod[\infty]$source.