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en>Ser Amantio di Nicolao |
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| In [[mathematical logic]], a '''superintuitionistic logic''' is a [[propositional logic]] extending [[intuitionistic logic]]. [[Classical logic]] is the strongest consistent superintuitionistic logic; thus, consistent superintuitionistic logics are called '''intermediate logics''' (the logics are intermediate between intuitionistic logic and classical logic).{{citation needed|date=July 2012}}
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| ==Definition==
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| A superintuitionistic logic is a set ''L'' of propositional formulas in a countable set of
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| variables ''p''<sub>''i''</sub> satisfying the following properties:
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| # all [[Intuitionistic logic#Axiomatization|axioms]] of intuitionistic logic belong to ''L'';
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| # if ''F'' and ''G'' are formulas such that ''F'' and ''F'' → ''G'' both belong to ''L'', then ''G'' also belongs to ''L'' (closure under [[modus ponens]]);
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| # if ''F''(''p''<sub>1</sub>, ''p''<sub>2</sub>, ..., ''p''<sub>''n''</sub>) is a formula of ''L'', and ''G''<sub>1</sub>, ''G''<sub>2</sub>, ..., ''G''<sub>''n''</sub> are any formulas, then ''F''(''G''<sub>1</sub>, ''G''<sub>2</sub>, ..., ''G''<sub>''n''</sub>) belongs to ''L'' (closure under substitution).
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| Such a logic is intermediate if furthermore
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| <ol><li value="4">''L'' is not the set of all formulas.</li></ol>
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| ==Properties and examples==
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| There exists a [[cardinality of the continuum|continuum]] of different intermediate logics. Specific intermediate logics are often constructed by adding one or more axioms to intuitionistic logic, or by a semantical description. Examples of intermediate logics include:
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| * intuitionistic logic ('''IPC''', '''Int''', '''IL''', '''H''')
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| * classical logic ('''CPC''', '''Cl''', '''CL'''): {{nowrap|'''IPC''' + ''p'' ∨ ¬''p''}} = {{nowrap|'''IPC''' + ¬¬''p'' → ''p''}} = {{nowrap|'''IPC''' + ((''p'' → ''q'') → ''p'') → ''p''}}
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| * the logic of the weak [[excluded middle]] ('''KC''', [[V. A. Jankov|Jankov]]'s logic, [[De Morgan's laws|De Morgan]] logic<ref>Constructive Logic and the Medvedev Lattice,
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| Sebastiaan A. Terwijn, Notre Dame J. Formal Logic, Volume 47, Number 1 (2006), 73-82.</ref>): {{nowrap|'''IPC''' + ¬¬''p'' ∨ ¬''p''}}
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| * [[Kurt Gödel|Gödel]]–[[Michael Dummett|Dummett]] logic ('''LC''', '''G'''): {{nowrap|'''IPC''' + (''p'' → ''q'') ∨ (''q'' → ''p'')}}
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| * [[Georg Kreisel|Kreisel]]–[[Hilary Putnam|Putnam]] logic ('''KP'''): {{nowrap|'''IPC''' + (¬''p'' → (''q'' ∨ ''r'')) → ((¬''p'' → ''q'') ∨ (¬''p'' → ''r''))}}
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| * [[Yuri T. Medvedev|Medvedev]]'s logic of finite problems ('''LM''', '''ML'''): defined semantically as the logic of all [[Kripke semantics|frames]] of the form <math>\langle\mathcal P(X)\setminus\{X\},\subseteq\rangle</math> for [[finite set]]s ''X'' ("Boolean hypercubes without top"), {{As of|2010|lc=on}} not known to be recursively axiomatizable
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| * [[realizability]] logics
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| * [[Dana Scott|Scott]]'s logic ('''SL'''): {{nowrap|'''IPC''' + ((¬¬''p'' → ''p'') → (''p'' ∨ ¬''p'')) → (¬¬''p'' ∨ ¬''p'')}}
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| * Smetanich's logic ('''SmL'''): {{nowrap|'''IPC''' + (¬''q'' → ''p'') → (((''p'' → ''q'') → ''p'') → ''p'')}}
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| * logics of bounded cardinality ('''BC'''<sub>''n''</sub>): <math>\textstyle\mathbf{IPC}+\bigvee_{i=0}^n\bigl(\bigwedge_{j<i}p_j\to p_i\bigr)</math>
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| * logics of bounded width, also known as the logic of bounded anti-chains ('''BW'''<sub>''n''</sub>, '''BA'''<sub>''n''</sub>): <math>\textstyle\mathbf{IPC}+\bigvee_{i=0}^n\bigl(\bigwedge_{j\ne i}p_j\to p_i\bigr)</math>
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| * logics of bounded depth ('''BD'''<sub>''n''</sub>): {{nowrap|'''IPC''' + ''p<sub>n</sub>'' ∨ (''p<sub>n</sub>'' → (''p''<sub>''n''−1</sub> ∨ (''p''<sub>''n''−1</sub> → ... → (''p''<sub>2</sub> ∨ (''p''<sub>2</sub> → (''p''<sub>1</sub> ∨ ¬''p''<sub>1</sub>)))...)))}}
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| * logics of bounded top width ('''BTW'''<sub>''n''</sub>): <math>\textstyle\mathbf{IPC}+\bigvee_{i=0}^n\bigl(\bigwedge_{j<i}p_j\to\neg\neg p_i\bigr)</math>
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| * logics of bounded branching ('''T'''<sub>''n''</sub>, '''BB'''<sub>''n''</sub>): <math>\textstyle\mathbf{IPC}+\bigwedge_{i=0}^n\bigl(\bigl(p_i\to\bigvee_{j\ne i}p_j\bigr)\to\bigvee_{j\ne i}p_j\bigr)\to\bigvee_{i=0}^np_i</math>
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| * Gödel ''n''-valued logics ('''G'''<sub>''n''</sub>): '''LC''' + '''BC'''<sub>''n''−1</sub> = '''LC''' + '''BD'''<sub>''n''−1</sub>
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| Superintuitionistic or intermediate logics form a [[complete lattice]] with intuitionistic logic as the [[bottom element|bottom]] and the inconsistent logic (in the case of superintuitionistic logics) or classical logic (in the case of intermediate logics) as the top. Classical logic is the only [[atom (order theory)|coatom]] in the lattice of superintuitionistic logics; the lattice of intermediate logics also has a unique coatom, namely '''SmL'''.
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| The tools for studying intermediate logics are similar to those used for intuitionistic logic, such as [[Kripke semantics]]. For example, Gödel–Dummett logic has a simple semantic characterization in terms of [[total order]]s.
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| ==Semantics==
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| Given a [[Heyting algebra]] ''H'', the set of [[propositional formula]]s that are valid in ''H'' is an intermediate logic. Conversely, given an intermediate logic it is possible to construct its [[Lindenbaum algebra]] which is a Heyting algebra.
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| An intuitionistic [[Kripke frame]] ''F'' is a [[partially ordered set]], and a Kripke model ''M'' is a Kripke frame with valuation such that <math>\{x\mid M,x\Vdash p\}</math> is an [[upper set|upper subset]] of ''F''. The set of propositional formulas that are valid in ''F'' is an intermediate logic. Given an intermediate logic ''L'' it is possible to construct a Kripke model ''M'' such that the logic of ''M'' is ''L'' (this construction is called ''canonical model''). A Kripke frame with this property may not exist, but a [[general frame]] always does.
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| ==Relation to modal logics==
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| {{main|Modal companion}}
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| Let ''A'' be a propositional formula. The ''Gödel–Tarski translation'' of ''A'' is defined recursively as follows:
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| *<math> T(p_n) = \Box p_n </math>
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| *<math> T(\neg A) = \Box \neg T(A) </math>
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| *<math> T(A \and B) = T(A) \and T(B) </math>
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| *<math> T(A \vee B) = T(A) \vee T(B) </math>
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| *<math> T(A \to B) = \Box (T(A) \to T(B)) </math>
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| If ''M'' is a [[modal logic]] extending '''S4''' then {{nowrap begin}}ρ''M'' = {''A'' | ''T''(''A'') ∈ ''M''}{{nowrap end}} is a superintuitionistic logic, and ''M'' is called a ''modal companion'' of ρ''M''. In particular:
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| *'''IPC''' = ρ'''S4'''
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| *'''KC''' = ρ'''S4.2'''
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| *'''LC''' = ρ'''S4.3'''
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| *'''CPC''' = ρ'''S5'''
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| For every intermediate logic ''L'' there are many modal logics ''M'' such that ''L'' = ρ''M''.
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| ==References==
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| <references />
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| *Toshio Umezawa. On logics intermediate between intuitionistic and classical predicate logic. Journal of Symbolic Logic, 24(2):141–153, June 1959.
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| *Alexander Chagrov, Michael Zakharyaschev. Modal Logic. Oxford University Press, 1997.
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| [[Category:Systems of formal logic]]
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| [[Category:Propositional calculus]]
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| [[Category:Non-classical logic]]
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