Logic cheat sheet

Validity and soundness

A deductive argument is valid when its conclusion must follow if its premises are true. Soundness also requires true premises.

A valid form can contain a false premise.

Modus ponens

From “if P then Q” and P, infer Q. This is a valid conditional inference.

If it is a square, it has four sides; it is a square; therefore it has four sides.

Invalid reversal

From “if P then Q” and Q, P does not necessarily follow. Other conditions may also produce Q.

Having four sides does not establish that a shape is a square.

Quantifiers

Universal and existential claims have different logical force. The scope of a quantifier affects meaning.

“Every student read a book” need not mean they all read the same book.

Truth-functional connectives

A conjunction P ∧ Q is true only when both statements are true. Inclusive disjunction P ∨ Q is true when at least one is true.

“Or” in formal logic is inclusive unless an exclusive meaning is explicitly defined.

Material implication

In classical truth-functional logic, P → Q is false only when P is true and Q is false. It does not by itself assert a causal relationship.

A truth table evaluates logical form, not whether one event physically produces another.

Contraposition

P → Q is logically equivalent to ¬Q → ¬P. It is not generally equivalent to Q → P or ¬P → ¬Q.

If every square is a rectangle, something that is not a rectangle cannot be a square.

Negating quantifiers

Negating “every” produces an existential counterexample; negating “some” produces a universal negative. Keep the domain of discussion explicit.

¬∀x P(x) means ∃x ¬P(x).

Worked example: modus tollens

If the alarm is functioning, it sounds during the test. It does not sound during the test. Under the stated conditional, conclude that it is not functioning.

The inference uses P → Q and ¬Q to infer ¬P; whether the conditional is factually adequate is a separate issue.

Common mistake: absence of proof proves absence

Failure to establish a claim is not automatically proof of its negation. An argument may remain undecided because the evidence is incomplete.

Distinguish “P is false” from “P has not been demonstrated”.

Check your understanding

What is the negation of “all birds in this enclosure are blue”?

At least one bird in this enclosure is not blue. It is not “all are not blue”.

Curriculum connections

Sources