Propositional and First Order Logic Questions and Answers

Q: What is wrong with writing some student is diligent using implication?

Writing there exists x such that x is a student implies x is diligent is made true by any object that is not a student, since a false antecedent makes the implication true, so the sentence says almost nothing. The formula is satisfied in a domain containing a single stone. Recognising vacuous satisfaction as the mechanism of the error connects it to the same phenomenon in universal restrictions in description logics.

Q: What does nesting a universal inside an existential quantifier express?

There exists an x such that for all y, x loves y asserts that one particular individual loves everybody, whereas reversing the order says only that everybody is loved by someone or other. The two are quite different claims and only one implies the other. Quantifiers of different kinds do not commute, and remembering which direction the implication runs is worth more than remembering the rule that they differ.

Q: Which implication holds between the two quantifier orders?

If one individual loves everybody then certainly everybody is loved by someone, but the converse fails since each person might be loved by a different individual. The stronger claim implies the weaker. Working out which of two similar formulas is stronger, by finding a model of one that is not a model of the other, is the standard technique for settling such questions and it is more reliable than intuition.

Q: What does the equality symbol assert in first order logic?

Equality relates terms and asserts identity of the objects they denote, which is why it permits substitution and why counting statements can be written. Truth value equality is equivalence between formulas, indiscernibility is a consequence rather than the definition, and syntactic identity is a different relation entirely. Having identity available in the logic is what allows uniqueness and cardinality to be expressed at all. A limit that survives making every external component ideal is an intrinsic one, and worth knowing for that reason.

Q: How is there is exactly one such object expressed?

Uniqueness requires both existence and the statement that anything with the property is identical to the witness, which needs equality. The shorthand notation for unique existence abbreviates exactly this. The second option denies the existence of anything else at all, and the third is not well formed. Constructing a compound claim from existence plus a uniqueness clause is a pattern that recurs whenever counting is required.

Q: Why can counting statements be expressed only with equality?

Two existentially quantified variables may be satisfied by the same object unless their distinctness is asserted, so at least two requires an inequality. Nesting, reference and domain size are unaffected. That variables do not automatically denote different objects surprises most people learning the notation, and it is precisely why explicit inequalities litter formulas that count.

Q: What is an interpretation in first order logic?

An interpretation supplies a non-empty domain of objects and assigns each constant an object, each function symbol a mapping and each predicate a relation, and a sentence is then true or false in it. Truth value assignment is the propositional analogue and is insufficient here. Proofs and translations are different objects. Specifying meaning by fixing a domain and interpreting every symbol is what makes truth in first order logic precise. Checking the units of each quantity catches a wrong formula before the arithmetic begins.

Q: Why is first order entailment only semidecidable?

A complete procedure will eventually derive any entailed sentence, but when the sentence is not entailed it may search forever without ever concluding that. That is the semidecidability result. Procedures are complete, and infinite domains and quantifiers are the reasons behind it rather than the statement. A guarantee that holds in one direction only is a genuine limitation rather than a defect of implementation, and no better algorithm will remove it.

Q: What practical consequence follows from semidecidability?

A user who has waited an hour cannot tell whether the conjecture is false or the proof is merely long, which is why provers report timeouts rather than negative answers. Theorems are provable and questions are often answered, and propositional encodings are one response among several. Knowing that a negative result is unobtainable, rather than merely slow, changes how a tool's output must be interpreted.

31.

What is wrong with writing some student is diligent using implication?

A.Any non-student satisfies it vacuously
B.It claims every student is diligent
C.It claims no student is diligent
D.It cannot be written with the existential quantifier

Answer: A

Writing there exists x such that x is a student implies x is diligent is made true by any object that is not a student, since a false antecedent makes the implication true, so the sentence says almost nothing. The formula is satisfied in a domain containing a single stone. Recognising vacuous satisfaction as the mechanism of the error connects it to the same phenomenon in universal restrictions in description logics.

Discuss in Forum

32.

What does nesting a universal inside an existential quantifier express?

A.That the relation holds between every pair of objects
B.That one object stands in the relation to every object
C.That every object stands in the relation to some object
D.That the two quantifiers may be exchanged freely

Answer: B

There exists an x such that for all y, x loves y asserts that one particular individual loves everybody, whereas reversing the order says only that everybody is loved by someone or other. The two are quite different claims and only one implies the other. Quantifiers of different kinds do not commute, and remembering which direction the implication runs is worth more than remembering the rule that they differ.

Discuss in Forum

33.

Which implication holds between the two quantifier orders?

A.Neither order implies the other in general
B.The two orders are logically equivalent
C.Exists then forall implies forall then exists
D.Forall then exists implies exists then forall

Answer: C

If one individual loves everybody then certainly everybody is loved by someone, but the converse fails since each person might be loved by a different individual. The stronger claim implies the weaker. Working out which of two similar formulas is stronger, by finding a model of one that is not a model of the other, is the standard technique for settling such questions and it is more reliable than intuition.

Discuss in Forum

34.

What does the equality symbol assert in first order logic?

A.That two formulas have the same truth value
B.That two objects have all the same properties
C.That two terms are syntactically identical
D.That two terms denote the same object

Answer: D

Equality relates terms and asserts identity of the objects they denote, which is why it permits substitution and why counting statements can be written. Truth value equality is equivalence between formulas, indiscernibility is a consequence rather than the definition, and syntactic identity is a different relation entirely. Having identity available in the logic is what allows uniqueness and cardinality to be expressed at all. A limit that survives making every external component ideal is an intrinsic one, and worth knowing for that reason.

Discuss in Forum

35.

How is there is exactly one such object expressed?

A.There exists x with the property, and any y with it equals x
B.There exists x with the property and no other object exists
C.For every x with the property, x is unique in the domain
D.There exists exactly one x, using a special quantifier symbol

Answer: A

Uniqueness requires both existence and the statement that anything with the property is identical to the witness, which needs equality. The shorthand notation for unique existence abbreviates exactly this. The second option denies the existence of anything else at all, and the third is not well formed. Constructing a compound claim from existence plus a uniqueness clause is a pattern that recurs whenever counting is required.

Discuss in Forum

36.

Why can counting statements be expressed only with equality?

A.Without it, the domain of discourse would be infinite
B.Without it, distinct variables may denote the same object
C.Without it, quantifiers cannot be nested inside one another
D.Without it, no formula can refer to more than one object

Answer: B

Two existentially quantified variables may be satisfied by the same object unless their distinctness is asserted, so at least two requires an inequality. Nesting, reference and domain size are unaffected. That variables do not automatically denote different objects surprises most people learning the notation, and it is precisely why explicit inequalities litter formulas that count.

Discuss in Forum

37.

What is an interpretation in first order logic?

A.A translation of the sentence into natural language
B.An assignment of truth values to the atomic sentences
C.A domain together with meanings for the symbols
D.A proof that a sentence follows from the premises

Answer: C

An interpretation supplies a non-empty domain of objects and assigns each constant an object, each function symbol a mapping and each predicate a relation, and a sentence is then true or false in it. Truth value assignment is the propositional analogue and is insufficient here. Proofs and translations are different objects. Specifying meaning by fixing a domain and interpreting every symbol is what makes truth in first order logic precise. Checking the units of each quantity catches a wrong formula before the arithmetic begins.

Discuss in Forum

38.

Why must the domain of an interpretation be non-empty?

A.Otherwise the constants would have nothing to denote
B.Otherwise the logic would become decidable
C.Otherwise equality could not be interpreted at all
D.Otherwise a universally quantified sentence would be trivially true

Answer: D

Both reasons apply, but the convention is adopted chiefly so that from a universal statement an existential one may be inferred, which fails in an empty domain. Constants needing denotations is the other half of the same point. Decidability and equality are unaffected. A convention adopted to preserve a desired inference, rather than for any deep reason, is worth recognising as such since free logics that drop it exist and behave differently. The quantity that varies and the quantity that does not are worth naming before any substitution is attempted.

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39.

Why is first order entailment only semidecidable?

A.A proof will be found if one exists but not otherwise
B.The proof procedures available are all incomplete
C.The domain of an interpretation may be infinite
D.Quantifiers cannot be eliminated from a formula

Answer: A

A complete procedure will eventually derive any entailed sentence, but when the sentence is not entailed it may search forever without ever concluding that. That is the semidecidability result. Procedures are complete, and infinite domains and quantifiers are the reasons behind it rather than the statement. A guarantee that holds in one direction only is a genuine limitation rather than a defect of implementation, and no better algorithm will remove it.

Discuss in Forum

40.

What practical consequence follows from semidecidability?

A.Only propositional encodings may be used in practice
B.A prover may run indefinitely without reporting failure
C.No theorem in first order logic can ever be proved
D.Every first order question must be answered eventually somehow

Answer: B

A user who has waited an hour cannot tell whether the conjecture is false or the proof is merely long, which is why provers report timeouts rather than negative answers. Theorems are provable and questions are often answered, and propositional encodings are one response among several. Knowing that a negative result is unobtainable, rather than merely slow, changes how a tool's output must be interpreted.

Discuss in Forum

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