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One of the paradoxes of material implication in relevance logic is

P → (Q → P)

A proof of this statement in classical propositional logic is:

  1. P [OSC1]
  2. P ∨ ¬Q [1; addition]
  3. ¬Q ∨ P [2; commutativity of disjunction]
  4. Q → P [3; df. of if]
  5. P → (Q → P) [1-4; CSC1]

So if you have the law of addition, then you can prove this supposed paradox. But, it's my observation that if you have reductio ad absurdum in your logic, then you have the law of addition, as the following derived Rule of inference shows:

Derived Rule: A; therefore A ∨ B

  1. A
  2. ¬(A ∨ B) [OSC1]
  3. ¬¬(¬A ∧ ¬B) [2; df. of or]
  4. ¬A ∧ ¬ B [3; double negation]
  5. ¬A [4; simplification 1]
  6. A ∧ ¬A [1,5; conjunction]
  7. ¬(A ∨ B) → contradiction [2-6; CSC1]
  8. ¬¬(A ∨ B) [reductio ad absurdum]
  9. A ∨ B [8; double negation]

So my question is, do the various relevance logics invalidate reductio ad absurdum?

From Wikipedia:

The early developments in relevance logic focused on the stronger systems. The development of the Routley–Meyer semantics brought out a range of weaker logics. The weakest of these logics is the relevance logic B. It is axiomatized with the following axioms and rules.

One of the axioms is

A → (A ∨ B)

This axiom leads to addition as a derived Rule of inference, so it seems to me you can prove the paradoxes of material implication in the weakest of all the relevance logics. And if you remove that axiom, still if RAA is valid you can derive the law of addition. What's going on?

2 Answers 2

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You cannot prove P → (Q → P) from disjunctive addition alone, without other rules. Your proof using classical logic relies on the equivalence of ¬Q ∨ P with Q → P, which does not hold in general for relevant implication.

Depending on which logic you are using, there may be restrictions on deriving P ∨ ¬Q from P. Many versions of relevance logic require P and Q to have shared proposition or predicate terms, without which it does not hold.

Some advocates of relevance logic (e.g. Stephen Read) distinguish two different kinds of disjunction: an extensional version that permits the inference from P to P or Q, and an intensional one that does not. One of the motivations behind this is to block one of the standard ways of proving the principle of explosion, since explosion does not hold in relevance logics.

1. P ∧ ¬P          Premise
2. P               1, conjunctive simplification
3. P ∨ Q           2, disjunctive addition
4. ¬P              1, conjunctive simplification
5. Q               3,4 disjunctive syllogism

Here, we have a classical derivation of an arbitrary Q from the contradiction P ∧ ¬P. Read says this is equivocating in its use of the disjunction. We are using an extensional disjunction at 3 to derive P ∨ Q from P, and an intensional disjunction at 5 to derive Q by disjunctive syllogism. The idea behind this intensional disjunction is that it holds only when there is some relevant connection between P and Q.

You mention the principle of reductio ad absurdum, which can also be used to prove explosion:

1. P ∧ ¬P          Premise
2. |  ¬Q           Assumption
3. |  P ∧ ¬P       1, reiteration
4. ¬¬Q             2,3 negation introduction, discharging the assumption 
5. Q               4, DNE 

Again, we have a classical proof of explosion. Here, this is invalid in relevance logic because there is no relevant connection between 2 and 3. So in relevance logics, the reductio principle does not hold without restrictions. What does hold is the weaker (P → (P ∧ ¬P)) ⊢ ¬P.

This has serious consequences for understanding what negation means in relevance logics. Since reductio does not hold in general, the logic needs to cope with theoretical situations in which contradictions are true. Also, since P → (Q ∨ ¬Q) is not a theorem, the logic needs to cope with situations in which bivalence fails. Relevance logicians do not typically allow that any contradictions are actually true. Unlike dialetheists, their motivation for rejecting explosion is based on semantic considerations concerning the nature of implication.

A common approach to treating negation in relevance logic is to handle it using Routley-Meyer semantics. This involves defining a Routley star operator on worlds such that ¬P holds at a world A if and only if P does not hold at A*. This can then be combined with the conventional approach of relating the axioms of the logic with frame conditions on worlds. It is more complex than using Kripke semantics in classical modal logic because you have to allow for non-normal or inconsistent worlds that correspond to logical fictions.

The upshot is that logics that attempt to avoid explosion tend to be more weird than explosion itself.

Stephen Read, Relevant Logic. (1988)

J. M. Dunn, Relevance logic and entailment. In D. Gabbay and F. Geunthner (eds.), Handbook of Philosophical Logic III, pp. 117–224. (1986).

B. J. Copeland, On when a semantics is not a semantics: some reasons for disliking the Routley–Meyer semantics for relevance logic. Journal of Philosophical Logic, 8, pp. 399–413. (1979).

Greg Restall, Negation in Relevant Logics. In Dov Gabbay and Heinrich Wansing (eds.), What is Negation?, pp. 53-76 (1999).

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Note that (A->B):=(~A v B) doesn’t work for all logics, e.g. Intuitionistic Logic and relevance logics.

The main thrust of (~P v Q)->(P->Q) is that Q->(P->Q) and ~P->(P->Q) are both true in Classical and Intuitionistic Logic. Since relevance logics still validate a version of proof by cases, neither of these formulas hold in general for relevance logics.

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  • how is implication defined in relevance logic? Also, does relevance logic have the deduction theorem?
    – lee pappas
    Commented May 17 at 20:10
  • Basically, an implication is true at a state if the antecedent is enough to require the consequent at states that the initial state accesses for the state in which the antecedent holds. Symbolically, A->B is true at x if for all y,z, if Rxyz and A is true at y, then B is true at z. I’m not sure about what restrictions need to be placed on the deduction theorem, but I do know that it needs to be weakened in order to prevent proofs of A->(B->A) that depend on it, e.g. natural deduction proofs.
    – PW_246
    Commented May 17 at 20:31

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