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Propositional Logic Semantics

Let ${\cal B} = \{{\it true}, {\it false}\}$ .

Interpretation for propositional logic is a function $I : V \to {\cal B}$ .

We next define evaluation function: \[

  e : F \to (I \to {\cal B})

by recursion on formula syntax tree:

\[\begin{array}{l}

e(p)(I) = I(p), \mbox{ for } p \in V \\
e({"\lnot "}\> F)(I) = \lnot (e(F)(I)) \\
e(F_1\> {"\land "}\> F_2)(I) = e(F_1)(I) \land e(F_2)(I) \\
e(F_1\> {"\lor "}\> F_2)(I) = e(F_1)(I) \lor e(F_2)(I) \\
e(F_1\> {"\rightarrow "}\> F_2)(I) = (e(F_1)(I) \rightarrow e(F_2)(I)) \\
e(F_1\> {"\leftrightarrow "}\> F_2)(I) = (e(F_1)(I) \leftrightarrow e(F_2)(I))

\end{array} \]

We wrote symbols like $" \land "$ on left in quotes to emphasize that those are syntactic entities, in contrast to symbols like $\land$ on right-hand side that denote propositional operations given by truth tables (stated in Propositional Logic Informally).