Interpretetions of formulas in the first-order logic

In the propositional logic, an interpretation is an assignment of truth values to atoms.

In an interpretation the first-order logic formula we have to specify:

- the domain

- an assignment of values to constants, function symbols, and predicate symbols occurring in the formula.

Example

Let us consider the formulas

Let an interpretation be as follows: Domain: D={1,2},

Assignment for P:

It should be easy for the reader to confirm that is F in this interpretation, because P(x) is not T for both x=1 and x=2. On other hand, since ~P(2) is T in this interpretation, is T in this interpretation.

Example

Consider the formula

Let us define an interpretation as follows: D={1,2}

If x=1 we can see that there is a y, namely 1, such that P (1,y) is T.

If x=2 there is y = 2, such that P (2, y) is T. Therefore, in the above interpretation, for every x in D, there is a y such that P (x, y) is T; that is is T in this interpretation.

Definition: A formula G is inconsistent (unsatisfiable, a contradiction) if and only if there exists no interpretation that brings value true to G.

Definition: A formula G is valid if and only if every interpretation of G brings value true to G.

Definition: A formula G is a logical consequence of formulas F₁, F₂, …, F_n if and only if for every interpretation ¡, if F₁ ʌ … ʌ F_n is TRUE in ¡, G is also true in I.

Ex. Consider formulas

F₁: (∀x)(MAN(x) → MORTAL(x)),

F₂: MAN(Confucius).

G: MORTAL(Confucius)

We will now prove that formula MORTAL(Confucius) is a logical consequence of F₁ and F₂. Consider any interpretation ¡ that brings value TRUE to (∀x)(MAN(x) → MORTAL(x)) Ù MAN(Confucius). Certainly in this interpretation, MAN(Confucius) is TRUE.

Let us assume MORTAL(Confucius) is not TRUE in this interpretation. Then MAN(Confucius) → MORTAL(Confucius) is (TRUE → FALSE) = FALSE in ¡. This means that (∀x)(MAN(Confucius) → MORTAL(Confucius)) is FALSE in ¡, which is impossible. Therefore, MORTAL(Confucius) must be TRUE in every interpretation that satisfies (∀x)(MAN(x) → MORTAL(x)) Ù MAN(Confucius).

This means that MORTAL(Confucius) is a logical consequence of F₁ and F₂. We proved it by contradiction.