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.