Argument
Cut rule
Overview
In mathematical logic, the cut rule is an inference rule of sequent calculus. It is a generalisation of the classical modus ponens inference rule. The meaning of the cut rule is that, if a formula A appears as a conclusion in one proof and a hypothesis in another, then another proof in which the formula A does not appear can be deduced. This applies to cases of modus ponens, such as how instances of man are eliminated from Every man is mortal, Socrates is a man to deduce Socrates is mortal.
Proofs in systems that include the cut rule often don't have the sub formula property, meaning that there can be formulas in the proof that are not sub formulas of the conclusion. This can cause issues in some proof search methods. Because of this it is often useful to reason about systems without the cut rule. This has led to many cut-elimination theorems for many proof systems in both classical logic and non-classical logic.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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Any such select set is called the set of postulates, or premises, from which the reasoning proceeds. The reasoning is nothing else than the exhibition of the whole pattern of general conditions involved in the pattern derived from the selected postulates.
Alfred North Whitehead, Science and the modern world
States what Cut rule is.
The predicate of the conclusion is called the major term of the syllogism; the subject of the conclusion is called the minor term. As there can be but three terms, the major and minor terms must each be found in one, and only one, of the premises, together with the middle term which is in them both. The premise which contains the middle term and the major term is called the major premise; that which contains the…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
States what Cut rule is.
The particular habit of mind which governs this or that inference may be formulated in a proposition whose truth depends on the validity of the inferences which the habit determines; and such a formula is called a _guiding principle_ of inference. Suppose, for example, that we observe that a rotating disk of copper quickly comes to rest when placed between the poles of a magnet, and we infer that this will happen…
Charles S. Peirce, Chance, Love, and Logic
States what Cut rule is.
Related ideas
- Modus ponens
- Mathematical logic
- Rule of inference
- Sequent calculus
- Socrates
- Classical logic
- Cut-elimination theorem
Place in the tradition
Part of Stoicism.