%% Title: Propositional Logic Created: 2022-03-13 20:57 Status: Parent: [[Resources/Maths]] Tags: Source: %% # Propositional logic - Logical symbols: - $\lnot$ (not): the negation of a symbol' a **literal** is either an atomic sentence or a negated atomic sentence. Also written as `!`. - $\land$ (and): a conjunction whose parts are conjuncts. Also written as `&`. - $\lor$ (or): a disjunction whose parts are disjuncts. Also written as `|`. - $\Rightarrow$ (implies): an implication; in the sentence $A \Rightarrow B$, $A$ is the *premise* or antecedent, and $B$ is it's _conclusion_ or consequent. Implications are also known as rules or _if-then_ statements. - $\Leftrightarrow$ (if and only if): a biconditional, also written as $\equiv$ #### Truth tables $P$ | $Q$ | $\lnot P$ | $P \land Q$ | $P \lor Q$ | $P \Rightarrow Q$ | $P \Leftrightarrow Q$ ------|-------|-------|-------|-------|-------|------ false | false | true | false | false | true | true false | true | true | false | true | true | false true | false | false | false | true | false | false true | true | false | true | true | true | true - Implication doesn't carry the notion of causation or relevance between $P$ and $Q$; any implication is true when its premise is false - $P \Rightarrow Q$ can be thought of as "I am claiming that if P is true, then Q is true; otherwise, I make no claim" ### Inferences #### Modus Tollens $ \begin{align} P \implies Q. \\ \lnot P. \\ \therefore \lnot Q.\\ \square \end{align} $