Interactive problem sets for deductive logic courses.
Developed for Harvard's introductory course.

Example problems below

Truth table, truth-functional paraphrase, implication, disjunctive normal form, natural language argument, quantificational paraphrase, quantificational interpretation.

In progress

Deduction problems.

Construct a truth table for p ∨ r ≡ q ∨ r

p	q	r	p ∨ r ≡ q ∨ r
⊤	⊤	⊤	⊤ ⊥
⊤	⊤	⊥	⊤ ⊥
⊤	⊥	⊤	⊤ ⊥
⊤	⊥	⊥	⊤ ⊥
⊥	⊤	⊤	⊤ ⊥
⊥	⊤	⊥	⊤ ⊥
⊥	⊥	⊤	⊤ ⊥
⊥	⊥	⊥	⊤ ⊥

Construct a truth table for –p ∨ q ≡ (p ⊃ r)

p	q	r	–p ∨ q ≡ (p ⊃ r)
⊤	⊤	⊤	⊤ ⊥
⊤	⊤	⊥	⊤ ⊥
⊤	⊥	⊤	⊤ ⊥
⊤	⊥	⊥	⊤ ⊥
⊥	⊤	⊤	⊤ ⊥
⊥	⊤	⊥	⊤ ⊥
⊥	⊥	⊤	⊤ ⊥
⊥	⊥	⊥	⊤ ⊥

3.

Paraphrase the following sentence in logical notation:

If Serbia is forced to submit, then Austria-Hungary will control the Balkans and threaten Constantinople if and only if England does not intervene.

–

∙

∨

⊃

≡

∀

∃

Does schema (1) imply schema (2)?

1. p ≡ q ≡ r

2. p ∙ –q ∨ –p ∙ r

If implication fails to hold, provide an interpretation that witnesses this fact.

Show Truth Table

Implies

Does not imply...

Determine whether schema (1) and schema (2) are equivalent:

1. p ≡ q ≡ r

2. p ∙ q ∙ r ∨ –p ∙ –q ∙ –r

If equivalence fails to hold, provide an interpretation that witnesses this fact.

Show Truth Table

Equivalent

Not equivalent...

6.

Transform the following schema into disjunctive normal form:

(p ≡ q) ∙ (q ≡ r)

–

∙

∨

⊃

≡

∀

∃

7.

For the following argument, paraphrase the premises and conclusion and also determine whether the premises truth-functionally imply the conclusion.

1. If Jones did not meet Smith last night, then either Smith was the murderer or Jones is lying.
2. If Smith wasn't the murderer, then Jones did not meet Smith last night and the murder took place after midnight.
3. If the murder took place after midnight, then either Smith was the murderer or Jones is lying.

C. Therefore, Smith was the murderer.

1.

–

∙

∨

⊃

≡

∀

∃

2.

–

∙

∨

⊃

≡

∀

∃

3.

–

∙

∨

⊃

≡

∀

∃

C.

–

∙

∨

⊃

≡

∀

∃

Implies

Does not imply

8.

Paraphrase the following sentence in logical notation:

There are sopranos who respect only those tenors who are louder than they.

Use the following predicates:

S = "(1) is a soprano"
T = "(1) is a tenor"
L = "(1) is louder than (2)"
R = "(1) respects (2)"

–

∙

∨

⊃

≡

∀

∃

9.

Specify an interpretation that makes the following schema true:

(∀x)(Fx ⊃ (Gx ≡ Hx)) ∙ –(∀x)(Fx ∙ Gx ≡ Hx) ∙ (∃x)Fx

Universe = {1}

F = { }
G = { }
H = { }

10.

Specify an interpretation that makes the following schema true, and an intrepretation that makes it false:

(∀x)(∀y)(Fxy ⊃ (∃z)(Fxz ∙ Fyz))

True interpretation:

Universe = {1}

F = { }

False interpretation:

Universe = {1}

F = { }

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p	q	r	[p ≡ q ≡ r] ⊃ [p ∙ –q ∨ –p ∙ r]
⊤	⊤	⊤	⊤ ⊥
⊤	⊤	⊥	⊤ ⊥
⊤	⊥	⊤	⊤ ⊥
⊤	⊥	⊥	⊤ ⊥
⊥	⊤	⊤	⊤ ⊥
⊥	⊤	⊥	⊤ ⊥
⊥	⊥	⊤	⊤ ⊥
⊥	⊥	⊥	⊤ ⊥

p	q	r	[p ≡ q ≡ r] ≡ [p ∙ q ∙ r ∨ –p ∙ –q ∙ –r]
⊤	⊤	⊤	⊤ ⊥
⊤	⊤	⊥	⊤ ⊥
⊤	⊥	⊤	⊤ ⊥
⊤	⊥	⊥	⊤ ⊥
⊥	⊤	⊤	⊤ ⊥
⊥	⊤	⊥	⊤ ⊥
⊥	⊥	⊤	⊤ ⊥
⊥	⊥	⊥	⊤ ⊥