Section 1.1: Problem 1 Solution »

# Section 1.1: The Language of Sentential Logic

An infinite sequence of distinct symbols to define wff:
 logical symbols sentential connective symbols $\neg$ (negation symbol) $\wedge$ (conjunction symbol) $\vee$ (disjunction symbol) $\rightarrow$ (conditional symbol) $\leftrightarrow$ (biconditional symbol) punctuation symbols $($ (left parenthesis) $)$ (right parenthesis) non-logical symbols sentence (propositional) symbols $A_{n}$ ($n$ th sentence symbol)
• The role of logical symbols in translation from/into English does not change.
• The meaning of non-logical symbols (parameters) is not fixed.
• It is assumed that no symbol is a finite sequence of other symbols.
• English sentences are different from their translations into the formal language, for two reasons, first same sentence symbols can be used for different English sentences in different contexts, and, second, English sentences are usually presumed to be either true or false, while sentence symbols may have different true/false values depending on their interpretations in different contexts.
An expression is a finite sequence of symbols, that can be expressed as a finite sequence of symbols and other expressions.
Rules to form a grammatically correct expression, i.e. a wff or simply formula:
1. A sentence symbol is a wff.
2. If $\alpha$ and $\beta$ are wffs, then so are $\neg\alpha$ , $\alpha\wedge\beta$ , $\alpha\vee\beta$ , $\alpha\rightarrow\beta$ , $\alpha\leftrightarrow\beta$ .
An alternative way is to consider a construction sequence:
1. A wff $\alpha$ is the expression obtained at the last step of a finite sequence of expressions $<\epsilon_{1},\ldots,\epsilon_{n}>$ ,
2. where $\epsilon_{i}$ is either a sentence symbol, or an expression obtained by a formula-building operation, i.e. $\mathcal{E}_{\neg}(\epsilon_{j})$ or $\mathcal{E}_{\square}(\epsilon_{j},\epsilon_{k})$ , where $j,k and $\square\in\{\wedge,\vee,\rightarrow,\leftrightarrow\}$ .
(Induction principle) If a set $S$ of wffs contains all the sentence symbols and is closed under all the five sentential connectives (formula-building operations), then $S$ is a set of all wffs.