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Intuitionistic logic encompasses the general principles of logical reasoning which have been abstracted by logicians from intuitionistic mathematics, as developed by L. Brouwer beginning in his  and . Because these principles also hold for Russian recursive mathematics and the constructive analysis of E.
Bishop and his followers, intuitionistic logic may be considered the logical basis of constructive mathematics. Although intuitionistic analysis conflicts with classical analysis, intuitionistic Heyting arithmetic is a subsystem of classical Peano arithmetic. It follows that intuitionistic propositional logic is a proper subsystem of classical propositional logic, and pure intuitionistic predicate logic is a proper subsystem of pure classical predicate logic.
Philosophically, intuitionism differs from logicism by treating logic as a part of mathematics rather than as the foundation of mathematics; from finitism by allowing constructive reasoning about uncountable structures e. In his essay Intuitionism and Formalism Brouwer correctly predicted that any attempt to prove the consistency of complete induction on the natural numbers would lead to a vicious circle.
Brouwer rejected formalism per se but admitted the potential usefulness of formulating general logical principles expressing intuitionistically correct constructions, such as modus ponens. Formal systems for intuitionistic propositional and predicate logic and arithmetic were fully developed by Heyting , Gentzen  and Kleene . Beth  and Kripke  provided semantics with respect to which intuitionistic logic is correct and complete, although the completeness proofs for intuitionistic predicate logic require some classical reasoning.
Intuitionistic logic can be succinctly described as classical logic without the Aristotelian law of excluded middle:. Brouwer  observed that LEM was abstracted from finite situations, then extended without justification to statements about infinite collections.
One may object that these examples depend on the fact that the Twin Primes Conjecture has not yet been settled. Formalized intuitionistic logic is naturally motivated by the informal Brouwer-Heyting-Kolmogorov explanation of intuitionistic truth, outlined in the entries on intuitionism in the philosophy of mathematics and the development of intuitionistic logic.
Troelstra and van Dalen  for intuitionistic first-order predicate logic. A proof is any finite sequence of formulas, each of which is an axiom or an immediate consequence, by a rule of inference, of one or two preceding formulas of the sequence. Any proof is said to prove its last formula, which is called a theorem or provable formula of first-order intuitionistic predicate logic. Thus the last two rules of inference and the last two axiom schemas are absent from the propositional subsystem.
If, in the given list of axiom schemas for intuitionistic propositional or first-order predicate logic, the law expressing ex falso sequitur quodlibet :. Since ex falso and the law of contradiction are classical theorems, intuitionistic logic is contained in classical logic.
In a sense, classical logic is also contained in intuitionistic logic; see Section 4. Decidability implies stability, but not conversely. The conjunction of stability and testability is equivalent to decidability. Here 1, 2 and 5 are axioms; 4 comes from 2 and 3 by modus ponens ; and 6 and 7 come from earlier lines by modus ponens ; so no variables have been varied.
The first such calculus was defined by Gentzen [—5], cf. Kleene . Section 4. These topics are treated in Kleene  and Troelstra and Schwichtenberg . While identity can of course be added to intuitionistic logic, for applications e. Each is capable of numeralwise expressing its own proof predicate. A fundamental fact about intuitionistic logic is that it has the same consistency strength as classical logic. For propositional logic this was first proved by Glivenko .
The negative translation of classical into intuitionistic number theory is even simpler, since prime formulas of intuitionistic arithmetic are stable. The negative translation of any instance of mathematical induction is another instance of mathematical induction, and the other nonlogical axioms of arithmetic are their own negative translations, so. Direct attempts to extend the negative interpretation to analysis fail because the negative translation of the countable axiom of choice is not a theorem of intuitionistic analysis.
Section 6. Gentzen  established the disjunction property for closed formulas of intuitionistic predicate logic. Kleene [, ] proved that intuitionistic first-order number theory also has the related cf. Friedman  existence property :. The disjunction and existence properties are special cases of a general phenomenon peculiar to nonclassical theories. The admissible rules of a theory are the rules under which the theory is closed.
For example, Harrop  observed that the rule. The fact that the intuitionistic situation is more interesting leads to many natural questions, some of which have recently been answered. Building on work of Ghilardi , Iemhoff  succeeded in proving their conjecture. Much less is known about the admissible rules of intuitionistic predicate logic. So is the implication CT corresponding to one of the most interesting admissible rules of Heyting arithmetic, let us call it the Church-Kleene Rule :.
Recursive realizability interpretations, on the other hand, attempt to effectively implement the B-H-K explanation of intuitionistic truth. Say that. For applications to intuitionistic arithmetic, normal models those in which equality is interpreted by identity at each node suffice because equality of natural numbers is decidable. Each terminal node or leaf of a Kripke model is a classical model, because a leaf forces every formula or its negation.
Familiar non-intuitionistic logical schemata correspond to structural properties of Kripke models, for example. Kripke models and Beth models which differ from Kripke models in detail, but are intuitionistically equivalent are a powerful tool for establishing properties of intuitionistic formal systems; cf. Troelstra and van Dalen , Smorynski , de Jongh and Smorynski , Ghilardi  and Iemhoff , .
Kreisel  suggested that GDK may eventually be provable on the basis of as yet undiscovered properties of intuitionistic mathematics. An arbitrary formula is realizable if some number realizes its universal closure. The fundamental result is.
Some nonintuitionistic principles can be shown to be realizable. But realizability is a fundamentally nonclassical interpretation. In Kleene and Vesley  and Kleene , functions replace numbers as realizing objects, establishing the consistency of formalized intuitionistic analysis and its closure under a second-order version of the Church-Kleene Rule. A uniform assignment of simple existential formulas to predicate letters suffices to prove.
Smorynski . See van Oosten  for a historical exposition and a simpler proof of the full theorem, using abstract realizability with Beth models instead of Kripke models. At present there are several other entries in this encyclopedia treating intuitionistic logic in various contexts, but a general treatment of weaker and stronger propositional and predicate logics appears to be lacking.
Many such logics have been identified and studied. Here are a few examples. This result contrasts with. Section 5. Kolmogorov  showed that this fragment already contains a negative interpretation of classical logic retaining both quantifiers, cf. Leivant . Goudsmit  is a thorough study of the admissible rules of intermediate logics, with a comprehensive bibliography.
Jankov  used an infinite sequence of finite rooted Kripke frames to prove that there are continuum many intermediate logics. For these results and more, see Citkin [, Other Internet Resources]. See also Mints . Kripke models for modal logic predated those for intuitionistic logic. Alternatives to Kripke and Beth semantics for intuitionistic propositional and predicate logic include the topological interpretation of Stone , Tarski  and Mostowski  cf.
Rasiowa and Sikorski , Rasiowa  , which was extended to intuitionistic analysis by Scott  and Krol . Hyland  defined the effective topos Eff and proved that its logic is intuitionistic.
For a very informative discussion of semantics for intuitionistic logic and mathematics by W. Ruitenberg, and an interesting new perspective by G. Bezhanishvili and W. Holliday, see Other Internet Resources below. The interpretation was extended to analysis by Spector ; cf. Howard . Kleene , Vesley  and Moschovakis . Concrete and abstract realizability semantics for a wide variety of formal systems have been developed and studied by logicians and computer scientists; cf.
Troelstra  and van Oosten  and . Variations of the basic notions are especially useful for establishing relative consistency and relative independence of the nonlogical axioms in theories based on intuitionistic logic; some examples are Moschovakis , Lifschitz , and the realizability notions for constructive and intuitionistic set theories developed by Rathjen [, ] and Chen .
Kohlenbach, Avigad and others have developed realizability interpretations for parts of classical mathematics. See also Artemov and Iemhoff . The entry on L. The best way to learn more is to read some of the original papers. Brouwer: Collected Works , edited by Heyting. Veldman  and  are authentic modern examples of traditional intuitionistic mathematical practice. Troelstra  places intuitionistic logic in its historical context as the common foundation of constructive mathematics in the twentieth century.
Bezhanishvili and de Jongh [, Other Internet Resources] includes recent developments in intuitionistic logic. Troelstra and Schwichtenberg  presents the proof theory of classical, intuitionistic and minimal logic in parallel, focusing on sequent systems.
Section 3. Over the years, many readers have offered corrections and improvements.
From Heyting Arithmetic to Peano Arithmetic
In mathematical logic , Heyting arithmetic sometimes abbreviated HA is an axiomatization of arithmetic in accordance with the philosophy of intuitionism. Heyting arithmetic adopts the axioms of Peano arithmetic PA , but uses intuitionistic logic as its rules of inference. In particular, the law of the excluded middle does not hold in general, though the induction axiom can be used to prove many specific cases. Heyting arithmetic should not be confused with Heyting algebras , which are the intuitionistic analogue of Boolean algebras. From Wikipedia, the free encyclopedia. Non-classical logic.