Questions: Propositional Logic Syntax

Question 1

Which of the following strings is NOT a well-formed formula (WFF) of propositional logic?

• A. ¬(p ∧ q)
• B. p → (q ∨ r)
• C. p ∧ ∧ q
• D. (p ↔ q) ∧ ¬r

Answer: (C)

The string 'p ∧ ∧ q' is not a WFF because ∧ is a binary connective that requires a complete WFF on each side. After the first ∧, the parser expects a WFF, but encounters another ∧ instead — so the grammar rule cannot be applied. The other three strings all parse correctly according to the inductive definition: they are built from atomic propositions by applying connectives to sub-WFFs.

Question 2

The formula ¬p ∧ q is logically equivalent to ¬(p ∧ q).

• T. True
• F. False

Answer: False

By precedence, ¬ binds more tightly than ∧, so ¬p ∧ q parses as (¬p) ∧ q — meaning 'p is false AND q is true.' The formula ¬(p ∧ q) means 'it is not the case that both p and q are true,' which by De Morgan's law is equivalent to ¬p ∨ ¬q. These are different formulas with different truth tables. This is one of the most common precedence errors in formal logic.

Question 3

The formula p ∧ ¬p is syntactically well-formed even though it is always false. Explain why being a contradiction does not affect syntactic well-formedness.

Model answer: Syntax only checks grammatical structure, not truth values. By the inductive definition: p is a WFF (atomic proposition), ¬p is a WFF (negation applied to a WFF), and p ∧ ¬p is a WFF (conjunction of two WFFs). Whether the formula is true, false, or contingent is a semantic question determined by assigning truth values — a completely separate layer of analysis. Syntax and semantics are deliberately kept apart in formal logic.

The syntax/semantics distinction is foundational in formal logic. A grammar defines which strings are legal formulas; separately, a semantics assigns meanings (truth values under interpretations) to those formulas. Contradictions, tautologies, and contingent formulas are all equally well-formed — the grammar does not filter by logical status.