The Semantics of Propositional Logic

# The Semantics of Propositional Logic

So far we have discussed propositional logical connectives and formulas. We noted that formulas derive from statements. We would now like to determine whether a formula is true or false based on the truth/falsehood of the statements that comprise the formula. We will use the letter $T$ to denote "true" and the letter $F$ to denote "false".

The semantics of propositional logic are the rules we give for our propositional connectives. They are defined in the "truth" tables below.

## Negation

$P$ $\neg P$
T F
F T

If $P$ is true then $\neg P$ is false.
If $P$ is false then $\neg P$ is true.

## Conjunction

$P$ $Q$ $P \wedge Q$
T T T
T F F
F T F
F F F

If $P$ is true and $Q$ is true then $P \wedge Q$ is true.
If $P$ is true and $Q$ is false then $P \wedge Q$ is false.
If $P$ is false and $Q$ is true then $P \wedge Q$ is false.
If $P$ is false and $Q$ is false then $P \wedge Q$ is false.

## Disjunction

$P$ $Q$ $P \vee Q$
T T T
T F T
F T T
F F F

If $P$ is true and $Q$ is true then $P \vee Q$ is true.
If $P$ is true and $Q$ is false then $P \vee Q$ is true.
If $P$ is false and $Q$ is true then $P \vee Q$ is true.
If $P$ is false and $Q$ is false then $P \vee Q$ is false.

## Exclusive Disjunction / Exclusive Or

$P$ $Q$ $P \veebar Q$
T T F
T F T
F T T
F F F

If $P$ is true and $Q$ is true then $P \veebar Q$ is false.
If $P$ is true and $Q$ is false then $P \veebar Q$ is true.
If $P$ is false and $Q$ is true then $P \veebar Q$ is true.
If $P$ is false and $Q$ is false then $P \veebar Q$ is false.

## Implication

$P$ $Q$ $P \rightarrow Q$
T T T
T F F
F T T
F F T

If $P$ is true and $Q$ is true then $P \rightarrow Q$ is true.
If $P$ is true and $Q$ is false then $P \rightarrow Q$ is false.
If $P$ is false and $Q$ is true then $P \rightarrow Q$ is true.
If $P$ is false and $Q$ is false then $P \rightarrow Q$ is true.

## Biconditional

$P$ $Q$ $P \leftrightarrow Q$
T T T
T F F
F T F
F F T

If $P$ is true and $Q$ is true then $P \leftrightarrow Q$ is true.
If $P$ is true and $Q$ is false then $P \leftrightarrow Q$ is false.
If $P$ is false and $Q$ is true then $P \leftrightarrow Q$ is false.
If $P$ is false and $Q$ is false then $P \leftrightarrow Q$ is true.