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.