$L_{1p}$ (called **propositional logic**) is a sublanguage of [[L1 (FOL)]]. # Symbols >[!note] Sentence letters > The upper case letters A-E and P-S, potentially followed by a numerical index. E.g. $P, Q,R_{1},\dots$ > [!Definition] Symbols for [[Connective|connectives]] >- Negation: $\neg$; >- Conjunction: $\land$; >- Disjunction: $\lor$; >- Material conditional: $\to$; >- Biconditional symbol: $\leftrightarrow$. > > [!Note] Brackets > (, ) # Syntax >[!note] Definition (atomic sentence of $L_{1p}$) >Every sentence letter is an atomic sentence of $L_{1p}$. >[!note] Definition (sentence of $L_{1p}$) >1. All atomic sentences of $L_{1p}$ are sentences of $L_{1p}$. >2. If $\varphi$ is a sentence of $L_{1p}$, then $\lnot \varphi$ is a sentence of $L_{1p}$. >3. If $\varphi$ and $\psi$ are sentences of $L_{1p}$ then $(\phi \land \psi)$, $(\phi \lor \psi)$, $(\phi \to \psi)$, $(\phi \leftrightarrow \psi)$ are sentences of $L_{1p}$. # Semantics The **truth value** of an atomic sentence (sentence letter) in a situation is the value assigned to it in that situation - T or F. A **possible situation** is an assignment of truth values - T (true) or F (false) – to the sentence letter. For $n$ sentence letters we have $2^{n}$ different possible assignments. The semantics for the complex sentences is given by their [[Truth Table|truth tables]]: | $\varphi$ | $\psi$ | $\lnot \varphi$ | $(\varphi \land \psi)$ | $(\varphi \lor \psi)$ | $(\varphi \to \psi)$ | $(\varphi \leftrightarrow \psi)$ | | ---- | ---- | ---- | ---- | ---- | ---- | ---- | | T | T | F | T | T | T | T | | T | F | F | F | T | F | F | | F | T | T | F | T | T | F | | F | F | T | F | F | T | T | # Properties An $L_{1p}$ [[Argument|argument]] is [[Logically Valid Argument|logically valid]] iff in joint truth table for all its premises and conclusions there is no row in which its premises are all $T$ and its conclusion $F$.