My question is: is there a mathematical or philosophical basis for rejecting any of the following rules of inference? If yes, then what is the argument for rejecting any of them? I am asking this question because I would like to know if there is any non-classical logic system that rejects some of these rules of inference.

I had a first contact with these rules of inference in the book "Philosophical Foundations for a Christian Worldview" in which the author says that these rules are essential to making a valid argument. By analyzing the truth tables of the connectives "∨", "∧" and "→" I realized that these rules of inference are implicit in mathematical arguments.

I suspect that there is some basis for rejecting certain of these rules of inference because I have heard of, for example, intuitionistic logic which disagrees with some axioms of classical logic.


These rules can be found at the following link: Rules of Inference.

OBS.: {p,q} ⊢ r means that if p and q are true then r is also true.

  1. Modus Ponens: {p→q,p} ⊢ q
  2. Modus Tollens: {p→q,¬q} ⊢ ¬p
  3. Hypothetical Syllogism: {p→q,q→r} ⊢ p→r
  4. Conjunction: {p,q} ⊢ p∧q
  5. Simplification: {p∧q} ⊢ p and {p∧q} ⊢ q
  6. Absorption: {p→q} ⊢ p→(p∧q)
  7. Addition: {p} ⊢ p∨q
  8. Disjunctive Syllogism: {p∨q,¬p} ⊢ q and {p∨q,¬q} ⊢ p
  9. Constructive Dilemma: {(p→q)∧(r→s),p∨r} ⊢ q∨s
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1 Answer 1


These rules are intuitionistically valid, but...

Intuitionists accept modus tollens in the form given (2), but not {¬q→¬p, p} ⊢ q, see Can one prove by contraposition in intuitionistic logic? A contradiction out of ¬q only gives us ¬¬q, and removing the double negation is something they reject. This is because p → q is interpreted as "given a proof of p a proof of q can be constructed", and impossibility of constructing a proof of ¬q does not produce a proof of q. They generally reject, for the same reason, proofs by contradiction with a positive conclusion (negation introduction {p→q, p→¬q} ⊢ ¬p is intuitionistically valid).

Constructive dilemma (9) is similarly loaded. It is intuisionistically valid in this form only because for intuitionists proving p∨r can only be done by proving one of the disjuncts. They would reject the form of the dilemma where p∨r is only known in the usual sense (classically), i.e. when it may not be known if p or r is true, and only ¬¬(p∨r) is proved.

Disjunction introduction {p} ⊢ p∨q (7) is rejected in relevance logics because it introduces a term (q) which may be completely irrelevant to what is given (p). The tradition of treating arguments as explanatory, and hence subject to relevance restrictions, goes all the way back to Aristotle and the Stoics, see What were the historical interpretations of Aristotle's definition of validity/logical consequence?

Disjunction introduction jointly with disjunctive syllogism {p∨q,¬p} ⊢ q (8) derive the law of explosion {p,¬p} ⊢ q, so all paraconsistent logics discard one or the other (or both), see Milne, Disjunction and Disjunctive Syllogism for discarding the latter.

The philosophical basis for this is provided by dialetheism, a position that true contradictions exist. Some candidates are the Liar sentence (I am false), vague sentences (50 grains are and are not a heap), and dialectic sentences in the style of Heraclitus and Hegel (we do and do not enter the same river twice). Since we still do not want everything altogether to be derivable (trivialism) the law of explosion must be invalid.

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