Semantics
The semantics of PL involve defining a domain of values and specifying how constants, functions, and predicates are interpreted within that domain.
Semantics of First-Order Predicate Logic (PL):
In first-order predicate logic PL, the semantics define how logical formulas are interpreted and evaluated within a given domain of discourse. The semantics of PL involve assigning meanings to constants, functions, and predicates, as well as determining the truth value of logical formulas under various interpretations.
Interpretation of Formulas: The interpretation of a formula F in PL involves the following components:
Domain of Values D:
A non-empty set that serves as the domain of values for variables and objects in the logical system.
Assignment of Values:
Constants: Every constant in the formula F is assigned a specific value from the domain D
Functions: For each function symbol f of arity n , there is a correspondence that maps tuples of n values from D to a resulting value in D .
Predicates: For each predicate symbol P of arity n , there is a correspondence that maps tuples of n values from D to either true or false, indicating whether the predicate holds true for those values.
Example: Consider the following interpretation in PL:
Domain D : {1, 2, 3}
Constants: a is assigned the value 1, b is assigned the value 2.
Functions: f(x) is defined as x + 1 .
Predicates: P(x) is true if x is even.
Under this interpretation:
a is assigned the value 1, b is assigned the value 2.
f(1) evaluates to 2, f(2) evaluates to 3.
P(1) is false, P(2) is true.
Evaluation of Formulas: Once the interpretation is established, logical formulas can be evaluated to determine their truth value under that interpretation. This involves substituting the assigned values for constants, applying functions, and evaluating predicates based on the provided correspondence.
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