Semantics week 4

LING2100

What is predicate logic and how does it differ from propositional logic?

Predicate logic (also called first-order logic) breaks down propositions into subjects and predicates and allows quantification over individuals.

Unlike propositional logic, which treats whole statements as atoms (p,q), predicate logic analyses internal structure using predicates (P (x)) and variables.

What is the domain of discourse in predicate logic?

It’s the set of entities that the variables range over. All quantification and reference in the logic is restricted to this domain. It must be non-empty

What are the basic components of predicate logic syntax?

Constants: Individual names (e.g., Arthur, merlin)

Variables: Placeholders (x, y, z)

Predicates: Properties or relations (wizard(x), loves (x,y))

Logical operators: ¬, ∧, ∨, →

Quantifiers: ∀ (universal), ∃ (existential)

What is a well-formed formula (wff) in predicate logic?

A syntactically correct expression built form:

-Predicates applied to terms

-Combined using logical connectives

-Quantified using ∃ or ∀

Examples:

∃x.wizard(x)

∀x.(king(x) → noble(x))

What does the universal quantifier express?

It asserts a property or relation for all individuals in the domain.

Example: ∀x.king(x) → noble(x)

‘All kings are noble’

What does the existential quantifier express?

It asserts that the property or relation holds for at least one individual.

Example: ∃x.knight(x)

‘there is at least one knight’

What does it mean for a variable to be bound?

A variable is bound if it appears within the scope of a quantifier that declares it.

Example: ∃x.wizard(x) → x is bound.

If a variable is not bound by any quantifier, it is free, and the formula is not a sentence.