Natural deduction in the context of "Formal proof"

Gerhard Karl Erich Gentzen (24 November 1909 – 4 August 1945) was a German mathematician and logician. He made major contributions to the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus. He died of starvation in a Czech prison camp in Prague in 1945.

Fitch notation, also known as Fitch diagrams (named after Frederic Fitch), is a method of presenting natural deduction proofs in propositional calculus and first-order logics using a structured, line-by-line format that explicitly shows assumptions, inferences, and their scope. It was invented by Frederic Brenton Fitch in the 1930s and later popularized through his textbook Symbolic Logic (1952). Fitch notation is notable for its use of indentation or boxes to indicate the scope of subordinate assumptions, making it one of the most pedagogically accessible systems for teaching formal logic.

In mathematical logic, a judgment (or judgement) or assertion is a statement or enunciation in a metalanguage. For example, typical judgments in first-order logic would be that a string is a well-formed formula, or that a proposition is true. Similarly, a judgment may assert the occurrence of a free variable in an expression of the object language, or the provability of a proposition. In general, a judgment may be any inductively definable assertion in the metatheory.

Judgments are used in formalizing deduction systems: a logical axiom expresses a judgment, premises of a rule of inference are formed as a sequence of judgments, and their conclusion is a judgment as well (thus, hypotheses and conclusions of proofs are judgments). A characteristic feature of the variants of Hilbert-style deduction systems is that the context is not changed in any of their rules of inference, while both natural deduction and sequent calculus contain some context-changing rules. Thus, if we are interested only in the derivability of tautologies, not hypothetical judgments, then we can formalize the Hilbert-style deduction system in such a way that its rules of inference contain only judgments of a rather simple form. The same cannot be done with the other two deductions systems: as context is changed in some of their rules of inferences, they cannot be formalized so that hypothetical judgments could be avoided—not even if we want to use them just for proving derivability of tautologies.

Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic.

The completeness theorem applies to any first-order theory: If T is such a theory, and φ is a sentence (in the same language) and every model of T is a model of φ, then there is a (first-order) proof of φ using the statements of T as axioms. One sometimes says this as "anything true in all models is provable". (This does not contradict Gödel's incompleteness theorem, which is about a formula φ_u that is unprovable in a certain theory T but true in the "standard" model of the natural numbers: φ_u is false in some other, "non-standard" models of T.)

In logic, more specifically proof theory, a Hilbert system, sometimes called Hilbert calculus, Hilbert-style system, Hilbert-style proof system, Hilbert-style deductive system or Hilbert–Ackermann system, is a type of formal proof system attributed to Gottlob Frege and David Hilbert. These deductive systems are most often studied for first-order logic, but are of interest for other logics as well.

It is defined as a deductive system that generates theorems from axioms and inference rules, especially if the only postulated inference rule is modus ponens. Every Hilbert system is an axiomatic system, which is used by many authors as a sole less specific term to declare their Hilbert systems, without mentioning any more specific terms. In this context, "Hilbert systems" are contrasted with natural deduction systems, in which no axioms are used, only inference rules.