### IV.3 Formal semantics of First Order Logic

#### Truth Table [Slide 116]

Again one could use truth tables to deﬁne the semantics of ﬁrst order formulae but ...

Example 30.

Let A (A + 1 > A) be a formula

 A A + 1 > A 1 true 2 true …

In order to determine the truth value of A (A + 1 > A) one needs to consider all rows in above truth table.

Truth tables are not suitable to determine the semantics or reason about ﬁrst order formulae.

#### Semantics of formula [Slide 117]

We will use a denotational semantics to deﬁne the formal semantics of a ﬁrst order formula.

Formulae × State   Bool

• Formulae: set of all possible ﬁrst order formulae
• Bool: set of semantic Boolean values {tt,}
• State: a ‘snapshot’ of the semantic values of the propositional variables and integer variables in a formula

#### State [Slide 118]

A State is the union of

• an integer state Statee which is a mapping from the set of integer variables Vare to the set of integer values Val and
• a Boolean state Stateb which is a mapping from the set of propositional variable Varb to the set of Boolean values Bool.

State : (Vare   Val) (Varb   Bool)

where Vare Varb = .

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

Example 31.

Let P be a Boolean variable and A be an integer variable then σ0 s.t. σ0(P) = tt and σ0(A) = 5 is a state.

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