## Explicit Provability And Constructive Semantics (2001)

Venue: | Bulletin of Symbolic Logic |

Citations: | 116 - 22 self |

### BibTeX

@ARTICLE{Artemov01explicitprovability,

author = {Sergei N. Artemov},

title = {Explicit Provability And Constructive Semantics},

journal = {Bulletin of Symbolic Logic},

year = {2001},

volume = {7},

pages = {1--36}

}

### Years of Citing Articles

### OpenURL

### Abstract

In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is nothing but the forgetful projection of LP. This also achieves G odel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int which resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and #-calculus.

### Citations

469 |
Knowledge and Belief
- Hintikka
- 1962
(Show Context)
Citation Context ...ard isomorphism which relates terms/types with proofs/formulas. 30 SERGEI N. ARTEMOV 11. First order case. Theories based on the first order modal logics were studied in [14], [40], [41], [49], [50], =-=[55]-=-, [81], [85], [86], [93], [95], [96], and many other papers. In the first order logic of proofs constants and proof letters depend on individual variables: u(#x), c(#x), . . . and are interpreted as p... |

337 |
An algorithmic interpretation of classical natural deduction
- Parigot
- 1992
(Show Context)
Citation Context ...tics for the latter. Note that the Curry-Howard isomorphism does not specify Int; decent #-calculi can be built for a variety of logics including proper fragments of Int, classical logic, etc. ([21], =-=[88]-=-, [89]). Abstract computational and functional semantics for Int which did not address the issue of the original BHK semantics for Int were also studied in [71], [94] and many other papers (cf. [18], ... |

334 |
Introduction to Mathematical Logic
- Mendelson
- 1964
(Show Context)
Citation Context ...he existential quantifiers on proofs by Skolem style operations on proofs. Some of those operations appeared in the proof of Gsodel's second incompleteness theorem. Within that proof (cf. [27], [29], =-=[79]-=-, [98]), in order to prove what are now known as Hilbert-Bernays-Lsob derivability conditions, one constructs computable functions m(x, y) and c(x) such that PA # Proof(s, F #G) # Proof(t, F ) # Proof... |

275 |
Foundation of Constructive Mathematics
- Beeson
- 1985
(Show Context)
Citation Context ..., [89]). Abstract computational and functional semantics for Int which did not address the issue of the original BHK semantics for Int were also studied in [71], [94] and many other papers (cf. [18], =-=[22], [106]). -=-Kuznetsov-Muravitsky-Goldblatt semantics for Int is based on a nonconstructive notion "classically true and formally provable" incompatible with the BHK semantics. In particular, it does not... |

220 |
Combinatory Logic
- Curry, Feys
- 1958
(Show Context)
Citation Context ...tics (Stone, 1937; Tarski, 1938, [76]) 3. Realizability semantics (Kleene, 1945, [56]) 4. Beth models (1956, [24]) 5. Dialectica Interpretation (Gsodel, 1958, [45]) 6. Curry-Howard isomorphism (1958, =-=[35]-=-) 7. Medvedev's logic of problems (1962, [78]) 8. Kripke models (1965, [66]) 9. Kuznetsov-Muravitsky-Goldblatt interpretation (1976, [47], [70]) 10. Categorical semantics (Goldblatt, 1979, [48]) Those... |

187 |
Semantical considerations on modal logic
- Kripke
- 1963
(Show Context)
Citation Context ...not axiomatizable. Problem 1 receives a solution in this paper (cf. also technical reports [7], [9]). The issue of provability semantics for S4 was addressed by Lemmon [72], Myhill [84], [85], Kripke =-=[65]-=-, Montague [83], Novikov [86], Mints [80], 3 Intuitionism is derivable from this. 4 So called Lsob's logic, also known under the names G, GL, K4.W, PRL. 6 SERGEI N. ARTEMOV Kuznetsov and Muravitsky [7... |

184 | Modal logic
- Chagrov, Zakharyaschev
- 1997
(Show Context)
Citation Context ...rmal, by introducing what is now known as the Brouwer-Heyting-Kolmogorov (BHK) semantics ([53], [54], [58]). The BHK semantics is widely recognized as the intended semantics for intuitionistic logic (=-=[33]-=-, [34], [43], [62], [74], [78], [104], [105], [107], [108], [111], [114]). Its description uses the unexplained primitive notions of construction and proof (Kolmogorov used the term problem solution f... |

178 |
Labelled Deductive Systems
- Gabbay
- 1996
(Show Context)
Citation Context ...as shown that the known upper bounds on the decision procedure for LP are much better (# P 2 in the polynomial hierarchy) than the ones for S4 or Int (PSPACE). 7. Gabbay's Labelled Deductive Systems (=-=[42]-=-) may serve as a natural framework for LP. Intuitionistic Type Theory ([73], [74]) and the second order #-calculus ([43]) use the format t : F with its informal provability reading and could benefit f... |

164 | Basic Proof Theory
- Troelstra, Schwichtenberg
- 2000
(Show Context)
Citation Context ...wn as the Brouwer-Heyting-Kolmogorov (BHK) semantics ([53], [54], [58]). The BHK semantics is widely recognized as the intended semantics for intuitionistic logic ([33], [34], [43], [62], [74], [78], =-=[104]-=-, [105], [107], [108], [111], [114]). Its description uses the unexplained primitive notions of construction and proof (Kolmogorov used the term problem solution for the latter). It stipulates that . ... |

117 |
Proof theory
- TAKEUTI
- 1987
(Show Context)
Citation Context ... will use the simplified notation n for a numeral n when it is safe. Definition 6.1. We assume that the first order Peano Arithmetic PA contains terms for all primitive recursive functions (cf. [98], =-=[101]-=-), called primitive recursive terms. Formulas of the form f(#x) = 0 where f(#x) is a primitive recursive term are standard primitive recursive formulas. A standards# 1 -formula is a formula #x#(x, # y... |

90 | The semantics of reflected proof
- Allen, Constable, et al.
- 1990
(Show Context)
Citation Context ...[97], [115]. 2. A recent application of explicit provability model: stability of verification. In the framework of formal provability the stability of verification systems is not internally provable (=-=[1]-=-, [36]). Rather the reflexive provability model provides a verification mechanism with provable stability ([10]) thus fixing a certain loophole in the foundations of verification. 3. The format t is a... |

85 |
On the interpretation of intuitionistic number theory
- Kleene
- 1945
(Show Context)
Citation Context ...f since the predicate "p is a proof of F " is decidable: given a proof p we always know which one of the disjuncts it is a proof of. However, a similar condition appeared in Kleene realizabi=-=lity (cf. [56]), where i-=-t makes perfect sense, since the predicate "r realizes F " is undecidable. EXPLICIT PROVABILITY AND CONSTRUCTIVE SEMANTICS 3 and proof. Since Heyting's formalization of the axiom system for ... |

74 |
Die formalen Regeln der intuitionistischen
- Heyting
- 1930
(Show Context)
Citation Context ...ld return a proof of #. The significance of formalizing the BHK semantics extends far beyond justifying the particular choice of axioms for constructive (intuitionistic) logic made by Heyting in 1930 =-=[52]-=-. Provability and proofs as objects appear in many other areas of logic and applications such as modal logics and logics of knowledge, #-calculus and typed theories, nonmonotonic reasoning, automated ... |

72 |
Syntactical Treatments of Modality, with Corollaries on Reflexion Principles and Finite Axiomatizability. Acta Philosophica Fennica 16:153–67
- Montague
- 1963
(Show Context)
Citation Context ...le. Problem 1 receives a solution in this paper (cf. also technical reports [7], [9]). The issue of provability semantics for S4 was addressed by Lemmon [72], Myhill [84], [85], Kripke [65], Montague =-=[83]-=-, Novikov [86], Mints [80], 3 Intuitionism is derivable from this. 4 So called Lsob's logic, also known under the names G, GL, K4.W, PRL. 6 SERGEI N. ARTEMOV Kuznetsov and Muravitsky [70], Goldblatt [... |

63 | Lambda Calculi With Types, Handbook of Logic - Barendregt - 1992 |

63 |
The Mathematics of Metamathematics
- Rasiowa, Sikorski
- 1970
(Show Context)
Citation Context ...otally di#erent from the provability semantics. Kleene realizers are not proofs in a formal theory, 2 Comprehensive surveys of these and other semantics for intuitionistic logic can be found in [33], =-=[92], [105]. 4-=- SERGEI N. ARTEMOV the predicate "r realizes F " is not decidable. Kleene himself denied any connection of his realizability with BHK interpretation. It is also worth mentioning that Kleene ... |

57 |
Provability interpretations of modal logics
- Solovay
- 1976
(Show Context)
Citation Context ...te Provable(F ). It was already clear, however, that 1 and 2 led to essentially di#erent models of Provability, each targeting its own set of applications (cf. 12). Problem 2 was solved by R. Solovay =-=[100]-=- who showed that the modal logic L 4 axiomatized all propositional properties of the formal provability, and by Artemov [4] and Vardanyan [112] who demonstrated that the first order logic of formal pr... |

35 |
On closed elements in closure algebras
- McKinsey, Tarski
- 1946
(Show Context)
Citation Context ...tions in Int itself. Here is the list of major known classical semantics for intuitionistic logic 2 . 1. Algebraic semantics (Birkho#, 1935, [26]) 2. Topological semantics (Stone, 1937; Tarski, 1938, =-=[76]-=-) 3. Realizability semantics (Kleene, 1945, [56]) 4. Beth models (1956, [24]) 5. Dialectica Interpretation (Gsodel, 1958, [45]) 6. Curry-Howard isomorphism (1958, [35]) 7. Medvedev's logic of problems... |

29 |
Tools and Techniques in Modal Logic
- Kracht
- 1999
(Show Context)
Citation Context ...rder to get a semantics of proofs for Int. Though neither definitions nor axiomatization 5 There are many adequate non-provability models for S4 known: algebraic, topological, Kripke, etc. (cf. [33], =-=[61]-=-). EXPLICIT PROVABILITY AND CONSTRUCTIVE SEMANTICS 7 were given, Gsodel's suggestion specified the format t : F of an expected solution of the provability semantics problem for S4 and for the BHK prob... |

27 |
On the structure of abstract algebras
- Birkho
(Show Context)
Citation Context ...ircular. In particular, BHK proofs should not denote derivations in Int itself. Here is the list of major known classical semantics for intuitionistic logic 2 . 1. Algebraic semantics (Birkho#, 1935, =-=[26]-=-) 2. Topological semantics (Stone, 1937; Tarski, 1938, [76]) 3. Realizability semantics (Kleene, 1945, [56]) 4. Beth models (1956, [24]) 5. Dialectica Interpretation (Gsodel, 1958, [45]) 6. Curry-Howa... |

23 | Intuitionistic necessity revisited
- Bierman, Paiva
- 1996
(Show Context)
Citation Context ... represented by proof polynomials, the Logic of Proofs can also emulate modal #-calculi. As it was shown in [9], [12] the intuitionistic version of LP naturally realizes the modal #-calculus for IS4 (=-=[25]-=-, [75], [91], cf. also [30]) and thus supplies modal #-terms with the standard provability semantics. This may be considered as a more general abstract version of the Curry-Howard isomorphism which re... |

22 |
On explicit counterparts of modal logics
- Brezhnev
- 2000
(Show Context)
Citation Context ...ality and provide a fresh look at modal logic and its applications in general. S4 is a lazy higher level language on top of LP. Explicit counterparts of modal logics K, K4, and S5 were found in [13], =-=[31]-=-. 5. LPmay be regarded as a basic epistemic logic with explicit justifications; a problem of finding such systems was raised by van Benthem in [109]. 6. The complexity of the Logic of Proofs has been ... |

21 | On epistemic logic with justification
- Artemov, Nogina
- 2005
(Show Context)
Citation Context ...of modality and provide a fresh look at modal logic and its applications in general. S4 is a lazy higher level language on top of LP. Explicit counterparts of modal logics K, K4, and S5 were found in =-=[13]-=-, [31]. 5. LPmay be regarded as a basic epistemic logic with explicit justifications; a problem of finding such systems was raised by van Benthem in [109]. 6. The complexity of the Logic of Proofs has... |

21 |
The Unprovability of Consistency: An Essay in Modal Logic
- Boolos
- 1979
(Show Context)
Citation Context ...the BHK semantics. In particular, it does not contain any BHK constructionssor proofs whatsoever. As far as S4 is concerned the KuznetsovMuravitsky -Goldblatt semantics turned out not to be adequate (=-=[27]-=-, [29]). An attempt to formalize the BHK semantics directly was made by Kreisel in his theory of constructions ([62], [64]). The original variant of the theory was inconsistent, and di#culties there o... |

19 |
On the complexity of explicit modal logics
- Kuznets
- 2000
(Show Context)
Citation Context ...be regarded as a basic epistemic logic with explicit justifications; a problem of finding such systems was raised by van Benthem in [109]. 6. The complexity of the Logic of Proofs has been studied in =-=[69]-=- where it was shown that the known upper bounds on the decision procedure for LP are much better (# P 2 in the polynomial hierarchy) than the ones for S4 or Int (PSPACE). 7. Gabbay's Labelled Deductiv... |

18 | The basic logic of proofs
- Artëmov, Straßen
- 1993
(Show Context)
Citation Context ...ms may be regarded as multi-conclusion by assuming that a proof derives all formulas assigned to the nodes of the proof tree. The logic of strictly single-conclusion proof systems was studied in [6], =-=[16]-=-, and in [67], [68] where it received a complete axiomatization (system FLP). However, FLP is not compatible with any modal logic. For example, FLP derives (x : ##x : (###)), which has the forgetful p... |

18 |
Metamathematical Extensibility for Theorem Verifiers and Proof-Checkers
- Davis, Schwartz
- 1979
(Show Context)
Citation Context ... [115]. 2. A recent application of explicit provability model: stability of verification. In the framework of formal provability the stability of verification systems is not internally provable ([1], =-=[36]-=-). Rather the reflexive provability model provides a verification mechanism with provable stability ([10]) thus fixing a certain loophole in the foundations of verification. 3. The format t is a proof... |

18 | Modality and SelfReference - SMULLYAN - 1985 |

15 |
Semantic construction of intuitionistic logic. Mededelingen Koninklijke Nederlandse Akademie van Wetenschappen
- Beth
- 1956
(Show Context)
Citation Context ... intuitionistic logic 2 . 1. Algebraic semantics (Birkho#, 1935, [26]) 2. Topological semantics (Stone, 1937; Tarski, 1938, [76]) 3. Realizability semantics (Kleene, 1945, [56]) 4. Beth models (1956, =-=[24]-=-) 5. Dialectica Interpretation (Gsodel, 1958, [45]) 6. Curry-Howard isomorphism (1958, [35]) 7. Medvedev's logic of problems (1962, [78]) 8. Kripke models (1965, [66]) 9. Kuznetsov-Muravitsky-Goldblat... |

14 |
Coming to Terms with Modal Logic: On the interpretation of modalities in typed -calculus
- Borghuis
- 1995
(Show Context)
Citation Context ...omials, the Logic of Proofs can also emulate modal #-calculi. As it was shown in [9], [12] the intuitionistic version of LP naturally realizes the modal #-calculus for IS4 ([25], [75], [91], cf. also =-=[30]-=-) and thus supplies modal #-terms with the standard provability semantics. This may be considered as a more general abstract version of the Curry-Howard isomorphism which relates terms/types with proo... |

13 | Arithmetical Necessity, Provability and Intuitionistic Logic - Goldblatt - 1978 |

13 |
The calculus of compatibility of propositions
- Orlov
- 1928
(Show Context)
Citation Context ...e a definitive formulation." 3. Defining intuitionistic logic in classical provability logic. One of the first papers on provability semantics for intuitionistic logic was published in 1928 by Or=-=lov ([87]-=-) who suggested prefixing all subformulas of a formula by a provability operator. Gsodel in 1933 ([44]) introduced the modal calculus of provability (essentially equivalent to the Lewis modal system S... |

13 |
Kolmogorov and mathematical logic
- Uspensky
- 1992
(Show Context)
Citation Context ...ting-Kolmogorov (BHK) semantics ([53], [54], [58]). The BHK semantics is widely recognized as the intended semantics for intuitionistic logic ([33], [34], [43], [62], [74], [78], [104], [105], [107], =-=[108]-=-, [111], [114]). Its description uses the unexplained primitive notions of construction and proof (Kolmogorov used the term problem solution for the latter). It stipulates that . a proof of A # B cons... |

13 |
Intuitionistic logic, Handbook of Philosophical Logic (2d
- Dalen
- 2002
(Show Context)
Citation Context ...onistically justified semantics for intuitionistic logic have been introduced by Kreisel, Kripke, Dyson, van Dalen, Leivant, Veldman, de Swart, Dummet, Troelstra, H. Friedman, Visser, and others (cf. =-=[110]). Th-=-ose studies lie outside the scope of this paper. We note however that they do not seem to produce a satisfactory formalization of the BHK semantics. Here is a summary from van Dalen's chapter "In... |

12 | A language and axioms for explicit mathematics in Algebra and Logic - Feferman - 1975 |

11 | Constructivism in mathematics - Troelstra, Dalen - 1988 |

10 |
Logics of Programs. Handbook of Theoretical Computer Science, v.B, Elsilver and The
- Kozen, Tiuryn
- 1990
(Show Context)
Citation Context ...t : " in LP are not normal modalities since they do not satisfy the property t : (P #Q)# (t : P # t : Q). This makes LP essentially di#erent from polymodal logics, e.g. the dynamic logic of progr=-=ams ([60]-=-), where the modality is upgraded by some additional features. Rather in the Logic of Proofs the modality has been decomposed into a family of proof polynomials (see 9). 6. Standard provability interp... |

9 |
On Modal Systems having arithmetical interpretations
- Avron
- 1984
(Show Context)
Citation Context ...roof variables and the corresponding constant specification is injective. Theorem 9.4. If S4 # F then LP # F r for some normal realization r. Proof. Consider a cut-free sequent formulation of S4 (cf. =-=[19]-=-, [80]), with sequents #=# #, where # and # are finite multisets of modal formulas. Axioms are sequents of the form S =#S, where S is a propositional letter, and the sequent #=# . Along with the usual... |

9 | A notion of classical pure type system
- Barthe, Hatcliff, et al.
- 1997
(Show Context)
Citation Context ... semantics for the latter. Note that the Curry-Howard isomorphism does not specify Int; decent #-calculi can be built for a variety of logics including proper fragments of Int, classical logic, etc. (=-=[21]-=-, [88], [89]). Abstract computational and functional semantics for Int which did not address the issue of the original BHK semantics for Int were also studied in [71], [94] and many other papers (cf. ... |

9 |
Operational logic of proofs with functionality condition on proof predicate
- Krupski
- 1997
(Show Context)
Citation Context ...arded as multi-conclusion by assuming that a proof derives all formulas assigned to the nodes of the proof tree. The logic of strictly single-conclusion proof systems was studied in [6], [16], and in =-=[67]-=-, [68] where it received a complete axiomatization (system FLP). However, FLP is not compatible with any modal logic. For example, FLP derives (x : ##x : (###)), which has the forgetful projection (##... |

9 |
1980], Constructive mathematics and computer programming
- of
(Show Context)
Citation Context ...uch better (# P 2 in the polynomial hierarchy) than the ones for S4 or Int (PSPACE). 7. Gabbay's Labelled Deductive Systems ([42]) may serve as a natural framework for LP. Intuitionistic Type Theory (=-=[73]-=-, [74]) and the second order #-calculus ([43]) use the format t : F with its informal provability reading and could benefit from the corresponding formal semantics. Acknowledgements. This work has ben... |

8 |
Finite problems
- Medvedev
- 1962
(Show Context)
Citation Context ...ow known as the Brouwer-Heyting-Kolmogorov (BHK) semantics ([53], [54], [58]). The BHK semantics is widely recognized as the intended semantics for intuitionistic logic ([33], [34], [43], [62], [74], =-=[78]-=-, [104], [105], [107], [108], [111], [114]). Its description uses the unexplained primitive notions of construction and proof (Kolmogorov used the term problem solution for the latter). It stipulates ... |

7 |
Godel's functional ("Dialectica") interpretation, Handbook of Proof Theory
- Avigad, Feferman
- 1998
(Show Context)
Citation Context ..., [88], [89]). Abstract computational and functional semantics for Int which did not address the issue of the original BHK semantics for Int were also studied in [71], [94] and many other papers (cf. =-=[18], [22], [1-=-06]). Kuznetsov-Muravitsky-Goldblatt semantics for Int is based on a nonconstructive notion "classically true and formally provable" incompatible with the BHK semantics. In particular, it do... |

7 |
A theory of constructions equivalent to arithmetic
- Goodman
- 1970
(Show Context)
Citation Context ...rectly was made by Kreisel in his theory of constructions ([62], [64]). The original variant of the theory was inconsistent, and di#culties there occurred already at the propositional level. Goodman (=-=[51]-=-) in 1970 fixed that gap but his solution involved a stratification of constructions into levels which ruined the BHK character of this semantics. In particular, a proof of A#B was no longer a constru... |

6 |
Constructive validity, in Symposium on Automatic Demonstration
- Scott
- 1970
(Show Context)
Citation Context ...Int, classical logic, etc. ([21], [88], [89]). Abstract computational and functional semantics for Int which did not address the issue of the original BHK semantics for Int were also studied in [71], =-=[94] and many -=-other papers (cf. [18], [22], [106]). Kuznetsov-Muravitsky-Goldblatt semantics for Int is based on a nonconstructive notion "classically true and formally provable" incompatible with the BHK... |

5 |
An Overview of Interpretability Logic, Advances in modal logic ’96
- Visser
- 1997
(Show Context)
Citation Context ...izations of the second Gsodel incompleteness theorem, Lsob theorem and fixed point theorem in the propositional language. L finds decent applications in traditional proof theory (cf. [3], [23], [37], =-=[113]-=-). B. G odel's provability calculus S4 and its decoding LP with the reflection principle #F # F . Within this model proofs are represented explicitly by computable terms. This model gives solutions to... |

4 |
The Logic of Provability, Handbook of Proof Theory
- Jongh, Japaridze
- 1998
(Show Context)
Citation Context ...formalizations of the second Gsodel incompleteness theorem, Lsob theorem and fixed point theorem in the propositional language. L finds decent applications in traditional proof theory (cf. [3], [23], =-=[37]-=-, [113]). B. G odel's provability calculus S4 and its decoding LP with the reflection principle #F # F . Within this model proofs are represented explicitly by computable terms. This model gives solut... |

4 |
New foundations for Lewis's modal systems
- Lemmon
- 1957
(Show Context)
Citation Context ...logic of formal provability was not axiomatizable. Problem 1 receives a solution in this paper (cf. also technical reports [7], [9]). The issue of provability semantics for S4 was addressed by Lemmon =-=[72]-=-, Myhill [84], [85], Kripke [65], Montague [83], Novikov [86], Mints [80], 3 Intuitionism is derivable from this. 4 So called Lsob's logic, also known under the names G, GL, K4.W, PRL. 6 SERGEI N. ART... |

4 |
Some remarks on the notion of proof
- Myhill
- 1960
(Show Context)
Citation Context ...al provability was not axiomatizable. Problem 1 receives a solution in this paper (cf. also technical reports [7], [9]). The issue of provability semantics for S4 was addressed by Lemmon [72], Myhill =-=[84]-=-, [85], Kripke [65], Montague [83], Novikov [86], Mints [80], 3 Intuitionism is derivable from this. 4 So called Lsob's logic, also known under the names G, GL, K4.W, PRL. 6 SERGEI N. ARTEMOV Kuznetso... |

3 |
On the first order logic of proofs
- Artemov, Sidon-Yavorskaya
- 1999
(Show Context)
Citation Context ...f valid principles accompanied by their plain modal projections. c(y) : (#x A(x)#A(y)) #(#x A(x)#A(y)) u : #x A(x)#(c(y) u) : A(y) ##x A(x)##A(y) u : #x A(x)##y ((c(y) u) : A(y)) ##x A(x)##y #A(y) In =-=[15]-=- it was shown that the first order logic of proofs is hyperarithmetical (in fact, # 0 1 (TA)-complete, where TA is the set of all true arithmetical sentences) . In particular, this means that this log... |