### II.4 Formal semantics of Propositional Logic

#### Formal Semantics [Slide 32]

Formal (denotational) semantics of Propositional Logic:

Propositions × State   Bool

• Propositions: set of all possible propositions
• Bool: set of semantic Boolean values {tt,}
• State: a ‘snapshot’ of the semantic values of the propositional variables in a formula

#### Formal Semantics [Slide 33]

Example 3.

proposition: P (P Q)

state σ: semantic value of P is tt and of Q is

 P Q (P ∨ Q) P ∧ (P ∨ Q) tt ﬀ tt tt
• State in the truth table corresponds to the ﬁrst two elements in a row.
• The semantic value of the proposition w.r.t. a state is the last element in a row of the truth table.

#### State [Slide 34]

A state is a mapping State from the set of propositional variables Varb to the set of Boolean values Bool {tt,}.

State : Varb   Bool

We will use σ012, to denote states and Σ to denote the set of all possible states.

Example 4.

Let σ0 be a state such that

 σ0(P) = tt σ0(Q) = ﬀ

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