Questions: Model Interpretation and Satisfaction

Question 1

According to Tarski's definition, the existential sentence ∃x P(x) is satisfied in a structure M if and only if:

• A. The predicate symbol P appears in M's signature
• B. Every element of M's domain satisfies P(x)
• C. At least one element of M's domain satisfies P(x) under some variable assignment
• D. P names a non-empty set in every structure with the same signature

Answer: (C)

Tarski's clause for the existential quantifier says ∃x φ(x) is satisfied in M iff there exists some element d in the domain of M such that M satisfies φ(x) with x assigned to d. Only one witness is needed. Option B describes universal quantification (∀x P(x)). Options A and D are not truth conditions for satisfaction in a specific structure.

Question 2

A structure M is a model of sentence φ if and mainly if φ is logically valid — that is, true in most structures.

• T. True
• F. False

Answer: False

M ⊨ φ (M models φ) means φ is satisfied specifically in M — the actual objects of M's domain, under M's interpretation of symbols, make φ true. Logical validity (⊨ φ) is a strictly stronger condition meaning φ holds in every structure. A sentence like 'the domain has exactly 3 elements' can be true in some structures and false in others — it is satisfiable but not valid.

Question 3

What distinguishes the satisfaction clause for an atomic formula from the satisfaction clause for a compound formula in Tarski's framework?

Model answer: An atomic formula is satisfied by directly consulting the interpretation: check whether the tuple of objects denoted by the terms actually stands in the relation that the predicate symbol denotes in M. A compound formula is satisfied inductively: its truth is reduced to the truth of its simpler sub-formulas using the truth conditions of the connective or quantifier at the outermost level.

This reflects the architecture of Tarski's definition: base cases (atomic formulas) anchor the recursion by grounding truth in the actual domain, and the inductive clauses (for ¬, ∧, ∨, →, ∀, ∃) assemble truth for complex formulas from truth values already computed for simpler ones. The recursion bottoms out at atoms — there is no further decomposition possible.