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The phrase Hilbert-style deduction system denotes a specific formalization of the notion of deduction in first-order logic [1], attributed to Gottlob Frege and David Hilbert. In fact, various slightly different formalizations are termed under this name in the literature [2]. All of them feature a characteristic trade-off between logical axioms versus rules of inference, by using Image File history File links Broom_icon. ...
Deductive reasoning is the kind of reasoning in which the conclusion is necessitated by, or reached from, previously known facts (the premises). ...
First-order logic (FOL) is a universal language in symbolic science, and is in use everyday by mathematicians, philosophers, linguists, computer scientists and practitioners of artificial intelligence. ...
Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, IPA: ) was a German mathematician who became a logician and philosopher. ...
David Hilbert (January 23, 1862, Königsberg, East Prussia – February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ...
A Tradeoff usually refers to losing one quality or aspect of something in return for gaining another quality or aspect. ...
The article Axiom may be a better treatment of the topic. ...
In logic, especially in mathematical logic, a rule of inference is a scheme for constructing valid inferences. ...
- a large number of axiom schemas ranging over the set of all possible formulae (thus, they are built on top of the underlying specific language, although in some formalizations, the specific signature is not referred to directly);
- and a small set of rules of inference.
For contrast, see natural deduction. In symbolic logic, it is sometimes inconvenient or impossible to express an axiomatic system in a finite number of axioms. ...
In mathematical logic, a formula is a formal syntactic object that expresses a proposition, except that the proposition may depend on the values of the free variables of the formula. ...
In logic, especially in mathematical logic, a rule of inference is a scheme for constructing valid inferences. ...
In mathematical logic, natural deduction is an approach to proof theory that attempts to provide a formal model of logical reasoning as it naturally occurs. ...
Formal description
With some details modified (compared to Frege's work), a detailed treatment can be read in [3]. Here in the article, some further formalizations are used (summarized as “notations”).
Logical axioms With the sole exception of the seventh axiom, the other “axioms” are in fact axiom schemas, each embracing countably infinite “instances”. The whole kit of the axioms is understood as their union. It should be proven that it is a recursive set, thus, we can “use” it as we expect (informally: we can always decide in finite time, whether any given formula belongs to the set of axioms). In symbolic logic, it is sometimes inconvenient or impossible to express an axiomatic system in a finite number of axioms. ...
In mathematics, a countable set is a set with the same cardinality (i. ...
In computability theory a countable set is called recursive, computable or decidable if we can construct an algorithm which terminates after a finite amount of time and decides whether or not a given element belongs to the set. ...
Rules of inference Modus ponens is the sole rule of inference in [3] treatment. In Logic, Modus ponens (Latin: mode that affirms) is a valid, simple argument form (often abbreviated to MP): If P, then Q. P. Therefore, Q. or in logical operator notation: P → Q P ⊢ Q where ⊢ represents the logical assertion. ...
In logic, especially in mathematical logic, a rule of inference is a scheme for constructing valid inferences. ...
Deduction Deduction can be defined - either by structural induction [3] (if parametrized over the set of formulae we begin with)
- or by using appropriate auxiliary notions (sequence, tree etc.).
Structural induction is a proof method that is used in mathematical logic (e. ...
Structural induction Note: parametrized over Γ. - Base (logical axioms, parameter are contained)
- Rule of induction (modus ponens)
- If and then
- Closure
- No alien objects
Notations used here Frege used a non-linear (two-dimensional) notation, which is not followed nowadays. See details on his symbolism in his book titled Begriffsschrift. Begriffsschrift is the title of a short book on logic by Gottlob Frege, published in 1879, and is also the name of the formal system set out in that book. ...
A right associative notation is used for material conditional (signed here as ). In mathematics, associativity is a property that a binary operation can have. ...
In propositional calculus, or logical calculus in mathematics, the material conditional or the implies operator is a binary truth-functional logical operator yielding the form If a then c, where a and c are statement variables (to be replaced by any meaningful indicative sentence of the language). ...
Here, in this article, for brevity's sake and to achieve complete formalization, some additional notations and auxiliary concepts are used, too. Most of them are the straightforward formalizations of their corresponding description mentioned in literature. E.g., universal generalization is mentioned in [3] and also in Alfred Tarski's works, and for brevity's sake and to achieve complete formalization, it is abbreviated here in the article as a function, called “UnivGen”. // Alfred Tarski (January 14, 1902, Warsaw, Russian-ruled Poland – October 26, 1983, Berkeley, California) was a logician and mathematician who spent four decades as a professor of mathematics at the University of California, Berkeley. ...
- we use it to emphasize the fact that in the case of the seventh axiom, no meta-variables (over object language variables) are needed: in this sole case, a fixed object language variable can be used.
- Var, Term, Form
- set of all variables, terms, formulas of the object language
- the set of free variables of a term or formula
- a relation for formalizing the notion of “allowed substitution”
- universal generalization of a formula (with all possible allowed variables, iterated as many times as we please)
- where
- a notation used in [4],
Further connections Axiom 1, 2 together with deduction rule modus ponens, corresponds to combinatory logic base combinators K, S together with the notion of application. See also Curry-Howard correspondence. Combinatory logic is a notation introduced by Moses Schönfinkel and Haskell Curry to eliminate the need for variables in mathematical logic. ...
The Curry-Howard correspondence is the close relationship between computer programs and mathematical proofs; the correspondence is also known as the Curry-Howard isomorphism, or the formulae-as-types correspondence. ...
Notes - ^ or sometimes its straightforward restriction to propositional calculus
- ^ Ferenczi, Miklós: Matematikai logika. Műszaki Könyvkiadó, Budapest, 2002.
- ^ a b c d Ruzsa, Imre and Máté, András: Bevezetés a modern logikába. Osiris Kiadó, Budapest, 1997.
- ^ Monk, J. Donald: Mathematical Logic. Springer-Verlag, New York • Heidelberg • Berlin, 1976.
In logic and mathematics, a propositional calculus (or a sentential calculus) is a formal system in which formulas representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules allows to establish that certain formulas are theorems of the formal system. ...
External links W.M Farmer: Propositional logic describes (among others) a part of the Hilbert-style deduction system (restricted to propositional calculus). In logic and mathematics, a propositional calculus (or a sentential calculus) is a formal system in which formulas representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules allows to establish that certain formulas are theorems of the formal system. ...
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