The set Γ = {P, ¬P, Q → R} is inconsistent. What does this mean for the formula 'The moon is made of cheese'?
AΓ neither entails nor refutes it — an inconsistency in Γ has no bearing on unrelated formulas
BΓ entails it — ex falso quodlibet means an inconsistent set entails every formula, including absurd ones
CΓ refutes it — an inconsistent set cannot entail any positive claim
DIt depends on whether Q → R is relevant to statements about the moon
Ex falso quodlibet ('from falsehood, anything follows') is a theorem of classical logic: if Γ ⊨ ⊥ (falsehood is a consequence of Γ), then Γ ⊨ φ for every formula φ. Semantically, this is vacuously true — there are no models of Γ, so the condition 'every model of Γ satisfies φ' is satisfied trivially by there being nothing to check. An inconsistent set is useless precisely because it entails everything: it cannot distinguish true claims from false ones, making it uninformative as a description of any state of affairs.
Question 2 Multiple Choice
To prove that a set of formulas Γ = {P, Q → R} is consistent, which of the following do you need to do?
AProve that every formula in Γ is a tautology
BShow that no formula in Γ contradicts another formula in Γ
CExhibit at least one assignment or structure under which all formulas in Γ are simultaneously true
DDerive a contradiction from Γ and then show it is avoidable
Consistency is existential: you only need one witness — a single satisfying assignment or structure. For {P, Q → R}, the assignment P = T, Q = F makes both formulas true simultaneously, so the set is consistent. Option B is a common error: formulas can 'look like they don't contradict' but still be collectively unsatisfiable (e.g., {P → Q, Q → ¬P, P} has no pairwise contradiction between any two formulas, but the three together are inconsistent). You must check global simultaneous satisfiability, not pairwise compatibility.
Question 3 True / False
A set of formulas containing both P and ¬P as members entails every formula in the language.
TTrue
FFalse
Answer: True
A set containing both P and ¬P is inconsistent — no assignment can make P and ¬P simultaneously true, so the set has no models. By ex falso quodlibet, an inconsistent set entails every formula: Γ ⊨ φ for all φ. The semantic definition of entailment ('every model of Γ satisfies φ') is vacuously true when Γ has no models. This is not a curiosity — it is a fundamental feature of classical logic that explains why inconsistency is catastrophic for any formal theory.
Question 4 True / False
If a set of formulas Γ is inconsistent, it means each individual formula in Γ, taken alone, is logically false.
TTrue
FFalse
Answer: False
Inconsistency is a property of the SET, not of individual members. Each formula in an inconsistent set may be individually satisfiable — even individually valid. For example, {P, ¬P} contains two individually satisfiable formulas that cannot be simultaneously satisfied. Similarly, {P → Q, Q → ¬P, P} contains three formulas each satisfiable on its own, but together they are inconsistent (P and Q → ¬P and P → Q force a contradiction). Inconsistency is about the impossibility of simultaneous truth, not individual falsehood.
Question 5 Short Answer
Why is ex falso quodlibet ('from falsehood, anything follows') catastrophic for a formal theory, and what condition must a theory satisfy to be useful?
Think about your answer, then reveal below.
Model answer: Ex falso quodlibet means that if a theory is inconsistent — if it has no models — then it entails every formula, including contradictions. A theory that entails everything conveys no information: it cannot distinguish true claims from false ones because it declares both true. For a theory to be useful, it must be consistent: there must exist at least one model (structure or assignment) satisfying all its axioms. Only then can the theory's entailments carry meaning — 'this follows from the theory' is informative only if the theory rules out some states of affairs.
Consistency is the minimum condition of meaningfulness for any formal theory. Without it, the theory's logical consequences are trivially all formulas, which is equivalent to saying nothing at all. This is why, in mathematics and logic, consistency proofs — showing that an axiom system has at least one model — are themselves significant results. Gödel's completeness theorem and subsequent work in model theory all rest on the prior question: is this theory consistent? A theory that is not consistent cannot be completed, extended, or used for anything.