Logical Thinking Intermediate

The biconditional p ↔ q (read "p if and only if q," abbreviated iff) is true when both sides have the same truth value — both true or both false.

Truth table for p ↔ q:
  p   | q   | p ↔ q
  ----+-----+-------
  ⊤   | ⊤   | ⊤
  ⊤   | ⊥   | ⊥
  ⊥   | ⊤   | ⊥
  ⊥   | ⊥   | ⊤

Equivalently, p ↔ q ≡ (p → q) ∧ (q → p). Example: "A triangle is equilateral if and only if all its sides are equal." The biconditional captures two-way entailment.

From a conditional p → q we can form three related statements:

• Converse: q → p — swap hypothesis and conclusion.
• Inverse: ¬p → ¬q — negate both.
• Contrapositive: ¬q → ¬p — swap and negate both.

Original:       p → q      (logically equivalent to its contrapositive)
Contrapositive: ¬q → ¬p   (always same truth value as original)
Converse:       q → p      (not equivalent to original in general)
Inverse:        ¬p → ¬q   (equivalent to the converse)

Key insight: the contrapositive is always logically equivalent to the original. The converse and inverse are equivalent to each other but not to the original.

Two statements are logically equivalent (≡) if they have identical truth values for every possible combination of inputs. This can be verified by building truth tables for both and checking they match column-by-column.

• p → q ≡ ¬p ∨ q (the conditional rewritten as a disjunction)
• p ↔ q ≡ (p → q) ∧ (q → p)
• ¬(¬p) ≡ p (double negation)

Verify p → q  ≡  ¬p ∨ q:
  p=⊤, q=⊤: ¬⊤ ∨ ⊤ = ⊥ ∨ ⊤ = ⊤  and  ⊤→⊤ = ⊤  ✓
  p=⊤, q=⊥: ⊥ ∨ ⊥ = ⊥          and  ⊤→⊥ = ⊥  ✓
  p=⊥, q=⊤: ⊤ ∨ ⊤ = ⊤          and  ⊥→⊤ = ⊤  ✓
  p=⊥, q=⊥: ⊤ ∨ ⊥ = ⊤          and  ⊥→⊥ = ⊤  ✓

De Morgan's laws describe how negation distributes over conjunction and disjunction:

• ¬(p ∧ q) ≡ ¬p ∨ ¬q
• ¬(p ∨ q) ≡ ¬p ∧ ¬q

In plain English: "Not (A and B)" means "Not-A or Not-B." "Not (A or B)" means "Not-A and Not-B." These are used constantly in programming (when negating compound boolean expressions) and in legal drafting.

Example — negate "it is raining AND cold":
  ¬(rain ∧ cold) ≡ ¬rain ∨ ¬cold
  "It is not raining OR it is not cold."

A tautology is a compound statement that is always true regardless of the truth values of its components. A contradiction is always false. A statement that is neither is called a contingency.

Tautology example — Law of Excluded Middle:
  p ∨ ¬p is always ⊤ (either p is true, or ¬p is true).

Contradiction example:
  p ∧ ¬p is always ⊥ (p cannot be both true and false).

Tautologies are important in mathematics as they represent universally valid inferences; contradictions indicate a logical impossibility or an inconsistent set of premises.

An argument consists of premises and a conclusion. An argument is valid if the conclusion must follow from the premises — even if the premises are false. It is sound if it is both valid and has true premises.

• A valid argument can have false premises and a false conclusion.
• A valid argument with true premises always yields a true conclusion.
• An invalid argument can accidentally have a true conclusion.

Valid but unsound:
  All birds can fly.   (false premise)
  Penguins are birds.  (true premise)
  ∴ Penguins can fly.  (false conclusion — but argument form is valid)

A fallacy is an error in reasoning that makes an argument invalid or misleading. Key formal fallacies:

• Affirming the consequent: "If p then q; q is true; therefore p is true." Invalid — q may have other causes.
• Denying the antecedent: "If p then q; p is false; therefore q is false." Invalid — q may still be true.

Common informal fallacies:

• Ad hominem: attacking the person rather than their argument.
• Straw man: misrepresenting an opponent's argument to make it easier to attack.
• False dichotomy: presenting only two options when more exist.

Affirming the consequent (invalid):
  If it rains, the ground is wet.
  The ground is wet.
  ∴ It rained.   ← WRONG — a sprinkler could explain it.