Could the ' If P then Q ' be interpreted as 'If P is assumed to be true then Q is true '? This could explain why a false proposition implies any proposition , according to Raymond Smullyan. If P is a false proposition and P is assumed to be true ( when it is actually false) this introduces a paradox from which any proposition can be implied.

  • Nothing is 'proven'. Consider this: if x>5, then x>3. Suppose we're given that x<5. Is x>3? – prash May 3 '14 at 2:18

There are two notions to be distinguished here: logical consequence (⊢) vs material implication (→), each of which, in classical logic, has the unusual property of explosion, which we can summarize as:

⊢–Explosion. {S,¬S} ⊢ Q, for any Q.

→–Explosion. Under assignment v(S) = ⊥: S → Q, for any Q.

1 The →–Explosion follows simply from the definition of → in terms of disjunction and negation:

(φ → ψ) =df (¬φ ∨ ψ),

because whenever the truth-assignment v is such that v(φ) = ⊥, then φ ∨ ψ follows, for any ψ. Proof theoretically, whenever ¬φ is proved, you can ∨-introduce ¬φ ∨ ψ where ψ can be any sentence whatsoever. The reason why the classical consequence explodes is a little more interesting.

2 According to the usual Tarskian interpretation of logical consequence:

Logical Consequence. Γ ⊢ φ is true iff it is impossible to make all ψ ∈ Γ true and φ false.

Whenever you have a sentence S ∈ Γ s.t. ¬S is also in Γ, you have the sentence (S ∧ ¬S) ∈ Γ. Consider an arbitrary sentence Q; is Q a logical consequence of Γ, given that Γ contains such an inconsistent conjunction? Let's appeal to the definition above:

{S, ¬S,...,Sn} ⊢ Q is true iff it is impossible to make all of S, ¬S,...,Sn true and Q false.

Since S and ¬S are inconsistent they cannot both be made true, therefore, whatever else may be contained in Γ and whatever Q may be, it follows that {S, ¬S,...,Sn} ⊢ Q. It is an immediate byproduct of the classical definition of consequence.

There are, of course, many non-explosive logics, such as Belnap and Anderson's relevance logic where from a contradiction an arbitrary Q isn't allowed to follow because implication has to meet special 'relevance' requirements. Look at that last SEP article for the details on how exactly that works. Most relevantly, there are many paraconsistent logics specifically created to handle this so-called 'paradox' of material implication. Worth checking out.

  • @user128932 - Pefect answer ! I like to add only one comment. The two notion are very different (as explained by Hunan) : The conditional (or material implication (→)) is a connective "inside" the language, logical consequence (⊢) is a relation between sentences of the language (thus, defined "outside" it). Their "independence" can be simply "viewed" in this way: we can have e.g.propositional languages without the → connective and the definition of consequence is the same. But there is a strong link between the two. 1/2 – Mauro ALLEGRANZA May 3 '14 at 8:11
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    @MauroALLEGRANZA You're too kind. Thanks for your comments. In the second one, is there a typo in the very last sequent (the one you called 'mp')? – Hunan Rostomyan May 3 '14 at 8:24
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    One of the "basic" rules of inference is modus ponens and it exploit the truth-functional characteristics of (→) to licence us to infer ψ from φ and (φ→ψ). In other way, mp says that : (φ→ψ), φ ⊢ ψ . – Mauro ALLEGRANZA May 3 '14 at 8:26

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