Validity, Satisfiability, And Semantic Entailment Quiz

1) A formula is unsatisfiable when:

It is false in some models

It has no variables

It cannot be written syntactically

It is false in all models

Unsatisfiability means there does not exist any model that makes the formula true. Unlike satisfiable formulas, which have at least one true model, unsatisfiable formulas fail in every possible interpretation. This is the semantic notion of contradiction within predicate logic.

Explanation

Unsatisfiability means there does not exist any model that makes the formula true. Unlike satisfiable formulas, which have at least one true model, unsatisfiable formulas fail in every possible interpretation. This is the semantic notion of contradiction within predicate logic.

2) What does the semantic entailment notation Γ ⊨ φ mean?

Γ and φ are syntactically equivalent.

Every model that satisfies all formulas in Γ also satisfies φ.

φ is true in at least one model where Γ is true.

There is a proof of φ from Γ.

Semantic entailment, written Γ ⊨ φ, means truth-preservation across models: every model that makes all formulas in Γ true also makes φ true. It is a semantic notion (about models), not a syntactic one (proofs) and not merely existence of some model.

Explanation

Semantic entailment, written Γ ⊨ φ, means truth-preservation across models: every model that makes all formulas in Γ true also makes φ true. It is a semantic notion (about models), not a syntactic one (proofs) and not merely existence of some model.

3) Is (∃x (P(x) ∧ Q(x))) → (∃x P(x)) valid?

No

Yes

Only in finite domains

Only if P ≡ Q

If a model has some a with P(a)∧Q(a), the same a witnesses ∃x P(x), making the consequent true. If no such a exists, the antecedent is false and the implication is vacuously true. Hence no model makes the antecedent true and consequent false; the formula is valid (true in all models).

Explanation

If a model has some a with P(a)∧Q(a), the same a witnesses ∃x P(x), making the consequent true. If no such a exists, the antecedent is false and the implication is vacuously true. Hence no model makes the antecedent true and consequent false; the formula is valid (true in all models).

4) The formula ∃x(P(x) ∧ Q(x)) → ∃x P(x) is:

Invalid

Unsatisfiable

Sometimes valid

Valid

If there exists an element satisfying both P and Q, then that same element certainly satisfies P. Thus the antecedent being true guarantees that the consequent is true. If the antecedent is false, the implication is automatically true. Therefore the entire formula is true in every model.

Explanation

If there exists an element satisfying both P and Q, then that same element certainly satisfies P. Thus the antecedent being true guarantees that the consequent is true. If the antecedent is false, the implication is automatically true. Therefore the entire formula is true in every model.

5) Domain {1,2}, P={1} makes ∃x P(x):

Undefined

Dependent on domain size

False

True

Existential statements require at least one witness that makes the predicate true. Since 1 is in P’s extension, P(1) is true, and therefore ∃x P(x) holds. The fact that P(2) is false is irrelevant, because an existential quantifier only needs one satisfying instance.

Explanation

Existential statements require at least one witness that makes the predicate true. Since 1 is in P’s extension, P(1) is true, and therefore ∃x P(x) holds. The fact that P(2) is false is irrelevant, because an existential quantifier only needs one satisfying instance.

6) A model for a predicate formula consists of:

A proof of the formula

A list of all true statements

The truth table for the formula

A domain and interpretation of predicates

A model assigns meaning to the symbols in predicate logic. First, it specifies a domain, which is the set of objects being talked about. Second, it interprets each predicate symbol by assigning to it a set of tuples from the domain representing when the predicate is true. This combination of domain plus interpretation tells us how to evaluate the truth or falsity of formulas. Other elements like proofs or truth tables are syntactic tools, not models.

Explanation

A model assigns meaning to the symbols in predicate logic. First, it specifies a domain, which is the set of objects being talked about. Second, it interprets each predicate symbol by assigning to it a set of tuples from the domain representing when the predicate is true. This combination of domain plus interpretation tells us how to evaluate the truth or falsity of formulas. Other elements like proofs or truth tables are syntactic tools, not models.

7) To show an argument is invalid, you must:

Show the conclusion is false

Prove the premises contradict each other

Prove the argument commits a formal fallacy

Find a countermodel where premises are true and conclusion is false

Invalidity means there exists at least one situation (one model) where the premises are all true but the conclusion fails. Producing such a model demonstrates that the conclusion does not logically follow. Merely showing the conclusion is false does not suffice, because validity depends on the relationship, not on actual truth. Likewise, contradictions or reasoning errors are not required; only the existence of a model breaking the connection is needed.

Explanation

Invalidity means there exists at least one situation (one model) where the premises are all true but the conclusion fails. Producing such a model demonstrates that the conclusion does not logically follow. Merely showing the conclusion is false does not suffice, because validity depends on the relationship, not on actual truth. Likewise, contradictions or reasoning errors are not required; only the existence of a model breaking the connection is needed.

8) Which of the following is true of the formula ∀x (P(x) → P(x))?

Unsatisfiable

Valid

Neither valid nor satisfiable

Satisfiable but not valid

For any object a in the domain, the instance P(a) → P(a) is a tautology (a formula that is true in all interpretations due to its logical form alone). Since each instance is true, the universal ∀x (P(x) → P(x)) is true in every model; therefore it is valid.

Explanation

For any object a in the domain, the instance P(a) → P(a) is a tautology (a formula that is true in all interpretations due to its logical form alone). Since each instance is true, the universal ∀x (P(x) → P(x)) is true in every model; therefore it is valid.

9) Which argument is invalid?

∀x P(x); therefore ∃x P(x)

P(a); Q(a); therefore P(a) ∧ Q(a)

∀x P(x); therefore P(a)

∃x P(x); therefore ∀x P(x)

From “there exists an x with property P,” one cannot conclude that all x have property P. A model where exactly one element satisfies P and others do not is enough to show invalidity. The other listed arguments are valid: universal instantiation, existential weakening, and conjunction introduction all preserve truth.

Explanation

From “there exists an x with property P,” one cannot conclude that all x have property P. A model where exactly one element satisfies P and others do not is enough to show invalidity. The other listed arguments are valid: universal instantiation, existential weakening, and conjunction introduction all preserve truth.

10) When is an argument in predicate logic valid?

The conclusion is true in every model.

At least one premise is true in every model.

Whenever all premises are true in a model, the conclusion is also true in that model.

The premises are true in at least one model.

Validity is a semantic relation between premises and conclusion: an argument is valid exactly when there is no model in which all the premises are true and the conclusion false. Equivalently, in every model where all premises hold, the conclusion must hold as well. Choice A is false because the conclusion need not be true in every model indiscriminately; it need only follow from the premises. Choice B and D describe unrelated properties.

Explanation

Validity is a semantic relation between premises and conclusion: an argument is valid exactly when there is no model in which all the premises are true and the conclusion false. Equivalently, in every model where all premises hold, the conclusion must hold as well. Choice A is false because the conclusion need not be true in every model indiscriminately; it need only follow from the premises. Choice B and D describe unrelated properties.

11) A formula is valid when:

It contains valid predicates

It has a proof

It is satisfiable

It is true in all models

A formula is valid if it cannot possibly be false—meaning every possible model assigns it the truth value true. Satisfiable formulas only need to be true in at least one model, not all. Proofs indicate derivability, but in semantic terms, validity means truth across every model without exception.

Explanation

A formula is valid if it cannot possibly be false—meaning every possible model assigns it the truth value true. Satisfiable formulas only need to be true in at least one model, not all. Proofs indicate derivability, but in semantic terms, validity means truth across every model without exception.

12) Which is satisfiable but not valid?

∀x (P(x) ∨ ¬P(x)) is satisfiable but not valid.

(∀x P(x)) ∨ (∀x ¬P(x)) is satisfiable but not valid.

Both are valid.

Both are unsatisfiable.

∀x (P(x) ∨ ¬P(x)) is valid by the law of excluded middle applied to each element: for every a either P(a) or ¬P(a). On the other hand (∀x P(x)) ∨ (∀x ¬P(x)) is true in models where P holds of every element or of none, so it is satisfiable in some models but false in mixed models, hence not valid. The two formulas are distinct.

Explanation

∀x (P(x) ∨ ¬P(x)) is valid by the law of excluded middle applied to each element: for every a either P(a) or ¬P(a). On the other hand (∀x P(x)) ∨ (∀x ¬P(x)) is true in models where P holds of every element or of none, so it is satisfiable in some models but false in mixed models, hence not valid. The two formulas are distinct.

13) Can two different models satisfy the same formula?

No

Only if domains are identical

Only if the formula is valid

Yes

Many formulas are satisfied by numerous different models. For example, ∃x P(x) is satisfied by any model in which at least one element is in the extension of P, regardless of domain size or structure. Only valid formulas must be satisfied by all models, but non-valid ones can be satisfied by many or few, and there is no uniqueness requirement.

Explanation

Many formulas are satisfied by numerous different models. For example, ∃x P(x) is satisfied by any model in which at least one element is in the extension of P, regardless of domain size or structure. Only valid formulas must be satisfied by all models, but non-valid ones can be satisfied by many or few, and there is no uniqueness requirement.

14) If φ is valid, then ¬φ is:

Unsatisfiable

Valid

Neither valid nor unsatisfiable

Satisfiable

A valid formula is true in all models. Its negation must therefore be false in all models, which means the negation has no model that makes it true. A formula with no satisfying model is unsatisfiable. Thus validity and unsatisfiability are semantic opposites.

Explanation

A valid formula is true in all models. Its negation must therefore be false in all models, which means the negation has no model that makes it true. A formula with no satisfying model is unsatisfiable. Thus validity and unsatisfiability are semantic opposites.

15) Given domain {1,2} with P={1}, the sentence ∀x P(x) is:

Undefined

Dependent on other predicates

True

False

To satisfy ∀x P(x) every domain element must be in P’s extension. Here P(2) is false because 2 ∉ {1}, so the universal formula fails in this model. Since at least one element fails to satisfy P, the universal statement is false.

Explanation

To satisfy ∀x P(x) every domain element must be in P’s extension. Here P(2) is false because 2 ∉ {1}, so the universal formula fails in this model. Since at least one element fails to satisfy P, the universal statement is false.

Validity, Satisfiability, and Semantic Entailment Quiz