Search results
Results from the WOW.Com Content Network
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. [1] This contrasts with Hilbert-style systems , which instead use axioms as much as possible to express the logical laws of deductive reasoning .
In proof theory and mathematical logic, sequent calculus is a family of formal systems sharing a certain style of inference and certain formal properties. The first sequent calculi systems, LK and LJ, were introduced in 1934/1935 by Gerhard Gentzen [1] as a tool for studying natural deduction in first-order logic (in classical and intuitionistic versions, respectively).
The Löwenheim–Skolem theorem implies that infinite structures cannot be categorically axiomatized in first-order logic. For example, there is no first-order theory whose only model is the real line: any first-order theory with an infinite model also has a model of cardinality larger than the continuum.
The definition of a deduction is such that it is finite and that it is possible to verify algorithmically (by a computer, for example, or by hand) that a given sequence (or tree) of formulae is indeed a deduction. A first-order formula is called logically valid if it is true in every structure for the language of the formula (i.e. for any ...
Prior to the work of these three, term logic (syllogistic logic) was widely considered adequate for formal deductive reasoning. Inferences in term logic can all be represented in the monadic predicate calculus. For example the argument All dogs are mammals. No mammal is a bird. Thus, no dog is a bird.
The deduction theorem for predicate logic is similar, but comes with some extra constraints (that would for example be satisfied if is a closed formula). In general a deduction theorem needs to take into account all logical details of the theory under consideration, so each logical system technically needs its own deduction theorem, although ...
A graphical representation of a partially built propositional tableau. In proof theory, the semantic tableau [1] (/ t æ ˈ b l oʊ, ˈ t æ b l oʊ /; plural: tableaux), also called an analytic tableau, [2] truth tree, [1] or simply tree, [2] is a decision procedure for sentential and related logics, and a proof procedure for formulae of first-order logic. [1]
It is an open problem whether this theory is decidable, but if Schanuel's conjecture holds then the decidability of this theory would follow. [ 2 ] [ 3 ] In contrast, the extension of the theory of real closed fields with the sine function is undecidable since this allows encoding of the undecidable theory of integers (see Richardson's theorem ).