# Which of common rules of inference are rejected on some philosophical grounds?

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.

RULES OF INFERENCE

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)
8. Disjunctive Syllogism: {p∨q,¬p} ⊢ q and {p∨q,¬q} ⊢ p
9. Constructive Dilemma: {(p→q)∧(r→s),p∨r} ⊢ q∨s
• Review the Stanford Encyclopedia of Philosophy > Logical Consequence. Also search SEP using the search term “rules inference”. Dec 8, 2019 at 2:46
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Dec 8, 2019 at 11:08