## Interpretability logic (1990)

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Venue: | Mathematical Logic, Proceedings of the 1988 Heyting Conference |

Citations: | 33 - 9 self |

### BibTeX

@INPROCEEDINGS{Visser90interpretabilitylogic,

author = {Albert Visser},

title = {Interpretability logic},

booktitle = {Mathematical Logic, Proceedings of the 1988 Heyting Conference},

year = {1990},

pages = {307--359},

publisher = {Plenum Press}

}

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### Abstract

Interpretations are much used in metamathematics. The first application that comes to mind is their use in reductive Hilbert-style programs. Think of the kind of program proposed by Simpson, Feferman or Nelson (see Simpson[1988], Feferman[1988], Nelson[1986]). Here they serve to compare the strength of theories, or better to prove

### Citations

123 |
Bounded Arithmetic, Bibliopolis
- Buss
- 1986
(Show Context)
Citation Context ...up to the formalization of Godel's Second Incompleteness Theorem. Complicated Rosser arguments |like the proof of Solovay's Theorem| may present di culties. S1 2 is a theory introduced by Buss in his =-=[8]-=-. It is weaker than I 0+ 1 and can be consider as the theory of P-TIME. Its provably recursive functions are precisely the P-TIME computable ones. In S1 2 we can formalize the Second Incompleteness Th... |

118 |
The Logic of Provability
- Boolos
- 1994
(Show Context)
Citation Context ...rs) of formulas of the target theory. The miracle does not always work |as is to be expected of true miracles| we get no analogous result if we try to work with modal predicate logical languages. See =-=[7]-=-. Provability Logic studies formal provability as a modality employing ideas and methods of modal logic. Interpretability Logic extends Provability Logic by adding a binary modality. This modality can... |

115 | Metamathematics of First-Order Arithmetics - Hájek, Pudlák - 1993 |

72 |
Undecidable Theories
- Tarski, Mostowski, et al.
- 1953
(Show Context)
Citation Context ...tion of interpretation satisfying certain intuitive constraints, we will study one given notion of interpretation: relative interpretability. This notion is due to Tarski, Mostowski and Robinson. See =-=[47]-=-. Roughly we demand that our interpretations commute with all the propositional connectives (including the all important ?) and with the quanti ers modulo relativization to a domain. Moreover, we rest... |

63 | Self-Reference and Modal Logic - Smorynski - 1985 |

46 |
On the scheme of induction for bounded arithmetic formulas
- Wilkie, Paris
- 1987
(Show Context)
Citation Context ...in ZF, butnotinGB. Solovay's work on this question produced the method of shortening cuts. This method was e.g. used by Paris & Wilkie, by Pudlak and various others to prove several metatheorems. See =-=[35]-=-,[37], [38]. The method was adapted by Nelson to build stronger and stronger theories in his predicativist programme. See [34]. 1 For the case of provability logic the programme of using the language ... |

35 |
Predicative Arithmetic
- Nelson
(Show Context)
Citation Context ...& Wilkie, by Pudlak and various others to prove several metatheorems. See [35],[37], [38]. The method was adapted by Nelson to build stronger and stronger theories in his predicativist programme. See =-=[34]-=-. 1 For the case of provability logic the programme of using the language to represent nontrivial reasoning was strongly advocated by Craig Smorynnski. See his book [42]. 2The expertise developed in ... |

32 |
Consistency Statements and Interpretations
- Pudlak, \Cuts
- 1985
(Show Context)
Citation Context ..., butnotinGB. Solovay's work on this question produced the method of shortening cuts. This method was e.g. used by Paris & Wilkie, by Pudlak and various others to prove several metatheorems. See [35],=-=[37]-=-, [38]. The method was adapted by Nelson to build stronger and stronger theories in his predicativist programme. See [34]. 1 For the case of provability logic the programme of using the language to re... |

31 |
Arithmetization of metamathematics in a general setting
- Feferman
- 1960
(Show Context)
Citation Context ...( ^2 ^2: ). We leave the proof of K4 to the reader. An immediate consequence of K2 is the familiar fact that `> > 2?, whichwas proved rst in Feferman's classical `Arithmetization of Metamathematics' (=-=[20]-=-). Note that this last principle is one possible `interpretation' version of the Second Incompleteness Theorem. Any theory interprets itself plus its own inconsistency and, hence, cannot prove its own... |

25 | On the length of proofs of finitistic consistency statements in first order theories - Pudlák - 1986 |

21 |
Provability logics for relative interpretability
- Jongh, Veltman
- 1990
(Show Context)
Citation Context ... in subsection 5.4 illustrates the richness of the closed fragment.) It is unknown whether IL satis es interpolation. De Jongh and Veltman prove amodal completeness theorem w.r.t. Veltman models. See =-=[12]-=-. Veltman models are explained in subsection 5.4. 5.2 Consequences of IL In our representations of reasoning we always suppress the propositional part. We may reason as follows. We will rst prove the ... |

17 | The interpretability logic of Peano Arithmetic - Berarducci - 1990 |

16 | The Unprovability of Small Inconsistency - Visser - 1991 |

13 |
A lattice of chapters of mathematics (interpretations between theorems
- Mycielski, Pudlák, et al.
- 1990
(Show Context)
Citation Context ...een local and global interpretability are presented in appendix C. For a broader discussion on local interpretability and a description of some further notions of reduction, the reader is referred to =-=[33]-=-. 3.2 Interpretations as internal models The construction of a model of two dimensional elliptic geometry on the sphere can be considered as the construction of a model of two dimensional elliptic geo... |

12 | Evaluation, provably deductive equivalence in Heyting's arithmetic of substitution instances of propositional formulas - Visser - 1985 |

11 | The logic of relative interpretability over Peano arithmetic - Shavrukov - 1988 |

11 |
The formalization of interpretability
- Visser
- 1991
(Show Context)
Citation Context ...ck blocks the derivation of the equivalence of `theorems-interpretability' and `axioms-interpretability'. The delicate points concerning formalization in a weak environment are treated extensively in =-=[56]-=-. We will ignore these subtleties here. We write: M : U >V , U interprets V via M, U >V :, for some M� M : U >V, U V :, U >V and V >U, A >U B :, (U + A) > (U + B). (We say: A interprets B over U.) We ... |

8 | Induction rules, reflection principles, and provably recursive functions - Beklemishev - 1996 |

8 | On the provability logic of bounded arithmetic - Berarducci, Verbrugge - 1993 |

8 |
The provability logics of recursively enumerable theories extending Peano Arithmetic at arbitrary theories extending Peano Arithmetic
- Visser
- 1984
(Show Context)
Citation Context ...(N )-sound. Then its provability logic is precisely Lob's Logic. In 0 case we drop the condition of 1 (N )-soundness, we get only relatively uninteresting well understood variants of Lob's Logic. See =-=[51]-=-. There are only two ways to escape the stability (keeping modal language and the interpretation of the box as provability xed). The rst is to go below I 0 + EXP, to weaker theories, where Lob's Logic... |

7 | On partially conservative sentences and interpretability - Lindstrom - 1984 |

6 |
Euclidean and non-Euclidean geometries (3 rd ed
- Greenberg
- 1993
(Show Context)
Citation Context ...own theorem that, in hyperbolic geometry, the sum of the angles of a triangle is strictly less than 180 , can be seen in one glance using Euclidean intuitions. See e.g. [27], p227. Alternatively, see =-=[21]-=-. In the Beltrami-Klein model all kinds of facts concerning incidence can be seen immediately. For example the fact that there alway is a line (asymptotically) parallel to both rays of an angle: in th... |

6 |
provably deductive equivalence in heyting's arithmetic of substitution instances of propositional formulas
- Evaluation
- 1985
(Show Context)
Citation Context ...a reasonable one. Ihave two tentative examples. The rst is the notion of -preservativity, whichis proposed as a metamathematical tool in the study of Heyting's Arithmetic, HA, and its extensions. See =-=[52]-=-, [59] and subsection 10.4 of this paper. The second is the formula of the Friedman Characterization of interpretability in sequential, nitely axiomatized theories applied to in nitely axiomatized the... |

5 | Sentences implying their own provability - Guaspari, M |

5 |
Modal Analysis of Generalized Rosser Sentences
- Svejdar
- 1983
(Show Context)
Citation Context ...an be formalized in the modal logics. The gain here is perspicuity and generality. 1 Representing some substantial reasoning in modal terms was the main aim of Viteslav Svejdar in his classical paper =-=[45]-=-. Svejdar employs a combination of interpretability andwitness comparisons for certain special formulations of proof predicates. The true potential of the ideas of Svejdar's paper is still not fully e... |

5 |
Efficient metamathematics. ILLC-disseration series 1993-3
- Verbrugge
- 1993
(Show Context)
Citation Context ...ability. See [56]. 299.1 Weak theories We knowthat the provability logics of I 0 + 1 and S1 2 both extend Lob's Logic. It is a great open question whether these logics are equal to Lob's Logic. (See =-=[48]-=- and [6].) In the light of this great open problem the question what the interpretability logics of I 0 + 1 and S1 2 are, seems to be de nitely immodest. However, you never know. Sometimes it is easie... |

5 | Peano’s smart children: A provability logical study of systems with built-in consistency - Visser - 1989 |

5 |
An Inside View of Exp
- Visser
- 1992
(Show Context)
Citation Context ...onsequences, not only of Godelian Incompleteness, but also of Godelian Completeness in the form of the interpretation version of the Model Existence Lemma. It seems to me that the results of my paper =-=[57]-=- throw some light on Nelson's program for founding predicative mathematics using interpretations, see [34]. Speci cally, the result that I 0 + EXP is equivalent modulo interpretability with I 0 + 1 + ... |

5 |
On the proofs of arithmetical completeness of interpretability logic. The Notre Dame Journal of Formal Logic, 35:542{551
- Zambella
- 1992
(Show Context)
Citation Context ...of such theories are: I 0 + SUPEXP, I n for n = 1� 2� ..., ACA0 and GB. The proof of arithmetical completeness is given in [55]. For a somewhat di erent proof, not using simpli ed Veltman models, see =-=[62]-=-. For a proof of modal and arithmetical completeness for the closely related ILP ! , see [15]. Since ILP is arithmetically complete for some class of reasonable theories, it must, a fortiori, imply W ... |

4 |
On some orderings of extensions of arithmetic
- Bennet
- 1986
(Show Context)
Citation Context ...50] itisshown that feasible interpretability over PA is 2-complete. For more information on complexity, see subsection A.2. 10.3 Functional Relative Consistency Christian Bennet studied in his thesis =-=[4]-=- the following notion of strong relative consistency. Let T and U be extensions of PA in the language of PA. LetPR be the set of primitive recursive terms. De ne: T >src U :,9t2PR 2 PA(8x (ProofU (x� ... |

4 |
Preliminary notes on interpretability logic
- Visser
- 1988
(Show Context)
Citation Context ... transitive, re exive relation on K satisfying R We can recover the Sx's in the new setting by taking: S ySxz :, xRy� xRz and ySz: The clauses for forcing are as before, now using the de ned Sx's. In =-=[53]-=- it is shown that we can unravel each Veltman model to a bisimulating simpli ed one. (The relevant notion of bisimulation is speci ed below.) So we havealso completeness for IL in simpli ed models. Th... |

4 |
Induction rules, re ection principles, and provably recursive functions
- Beklemishev
- 1996
(Show Context)
Citation Context ...tates that if wehave 0 proved the premiss of 1-induction, then we maydrawits conclusion. We will call this theory R 0 1 .4 For an extensive discussion of formulations of PRA and related theories, see =-=[2]-=-. RfnT is the local re ection principle for T ,i.e. the schema 2T A ! A, for the sentences A of LT . RFNT is the global or uniform re ection principle for T , i.e. the schema 8~x (2T A(~x) ! A(~x)), f... |

4 | Unary interpretability logic. The Notre Dame - Rijke - 1992 |

3 |
An orey sentence for predicative arithmetic
- Kalsbeek
- 1989
(Show Context)
Citation Context ...some examples. 1. Let 4 stand for tableaux-provability (a version of cut-free provability). Then the godelsentence of 4I 0+ 1 is an oreysentence. This was veri ed in Marianne Kalsbeeks masters thesis =-=[29]-=-. 2. Let T be sequential. A rossersentence R of 4T , tableaux provability in T , is a sentence such that I 0 + 1 ` R $ T :R T R. Any rossersentence of 4T is an oreysentence of T . This is immediate by... |

3 |
A smart child of Peano’s. Notre Dame
- Shavrukov
- 1994
(Show Context)
Citation Context ...eveloped in proving arithmetical completeness theorems for Interpretability Logic was used with good result by Shavrukov in the study of the combined logic for provability andaFeferman predicate. See =-=[40]-=-. Dick de Jongh and Duccio Pianigiani, in their[11], used the work of Hajek and Montagna ([23],[24]) to solve an open problem posed by Guaspari in [22]. 1.4 Internal interest The study of Interpretabi... |

3 | Oninterpretability in set theories. Unpublished manuscript - Solovay - 1976 |

3 |
A Generalized Notion of Weak Interpretability and the Corresponding Modal Logic
- Dzhaparidze
- 1993
(Show Context)
Citation Context ...ility of U in T means that U is interpretable in some consistent extension of T . Tolerance is a generalization of this notion to the n-ary case. The reader is referred to Japaridze's papers [17] and =-=[18]-=-. 11.4 1-Interpolability 1-interpolabilKonstantin Ignatiev, in his paper [28], characterizes the logic of ity over PA. 1-interpolability over PA is de ned as follows. A PA B :$9S2 0 1 -sentences (2 PA... |

2 | The logic of \Pi 1 -conservativity continued, Archiv fur Mathematische Logik und Grundlagenforschung - ajek, Montagna, et al. - 1992 |

2 |
Some independence results in interpretability logic
- Svejdar
- 1991
(Show Context)
Citation Context ... ) L1� L2� J1� J2 ! > ( ^: ) J5� J2 ! (( ^: )_ ) > ( ^: ) J1� J2� J3 ! > ( ^: ) J1� J2 23F easily follows from KW2. Our principle KW2 is interderivable with Svejdar's KW1 0 . Viteslav Svejdar in his =-=[46]-=- established that KW2 (in his paper KW1 0 ) is not derivable from F over IL. He also shows that W is not derivable from KW2. Mladen Vukovicshows that W is not derivable from M0. See his [61]. It is ea... |

2 |
astudy in intuitionistic propositional logic
- NILL
- 1995
(Show Context)
Citation Context ... just a testimony ofthe non-derivability of excluded third in HA! Regrettably the beautiful logic 1-5 does not exhaust the 1-preservativity principles valid in HA. The reader is referred to [52],[59],=-=[60]-=- for more information. 3411 Extensions of the language 11.1 Witness comparisons Hajek and Montagna characterize the 1-conservativity logic for extensions of I 1 with the witness comparison order adde... |

2 |
Iterated local re ection vs. iterated consistency. Logic Group Preprint Series 118
- Beklemishev
- 1994
(Show Context)
Citation Context ...lized statement ofits own re exivity, etcetera. E.g. the formalized version of local uniform re exivity for T is: 8A2SentLT 8x 2T (2T�xA ! A). For an extensive discussion of re ection principles, see =-=[1]-=-,[2],[3]. By the Second Incompleteness Theorem, locally re exive theories cannot be nitely axiomatizable. If a theory is sequential and satis es full induction, then it will be veri ably uniformly ess... |

2 |
Explicit xed points in interpretability logic
- Jongh, Visser
- 1991
(Show Context)
Citation Context ... Note that we donot have ` ( > ^ > ) ! > ( ^ ). This principle is invalid as is illustrated by the existence of oreysentences. De Jongh and Visser prove thatIL has unique and explicit xed points. See =-=[13]-=-. No characterization of the closed fragment of IL has been given. (The counterexample to the nite model property for simpli ed models in subsection 5.4 illustrates the richness of the closed fragment... |

2 | Bi-unary interpretability logic - Rijke - 1990 |

2 |
The logic of 1-conservativity continued. Archiv fur Mathematische Logik und Grundlagenforschung
- Hajek, Montagna
- 1992
(Show Context)
Citation Context ...d result by Shavrukov in the study of the combined logic for provability andaFeferman predicate. See [40]. Dick de Jongh and Duccio Pianigiani, in their[11], used the work of Hajek and Montagna ([23],=-=[24]-=-) to solve an open problem posed by Guaspari in [22]. 1.4 Internal interest The study of Interpretability has also some internal interest in ProvabilityLogic considered as a project. It is this. Solov... |

2 |
The provability logic for # 1 -interpolability
- Ignatiev
- 1993
(Show Context)
Citation Context ... T . Tolerance is a generalization of this notion to the n-ary case. The reader is referred to Japaridze's papers [17] and [18]. 11.4 1-Interpolability 1-interpolabilKonstantin Ignatiev, in his paper =-=[28]-=-, characterizes the logic of ity over PA. 1-interpolability over PA is de ned as follows. A PA B :$9S2 0 1 -sentences (2 PA(A ! S) ^ 2 PA(S ! B)). Interpolability can be viewed as a kind of dual of 1-... |

2 |
A°2 operators and alternating sentences in arithmetic
- Manevitz, Stavi
- 1980
(Show Context)
Citation Context ... at the cradle of modeltheory, the alternative interpretations of geometry, can all be viewed as interpretations. 12 In the next subsection we provide some salient examples of interpretations. 11 See =-=[32]-=- for a paper in which interpretations are studied as operators on models. 12 Let Mod(U) be the set of models satisfying U. Then, K : U >V yields a function from Mod(U) toMod(V ). Considering K as a mo... |

2 |
On the lengths of proofs of ¯nitistic consistency statements in ¯rst order theories
- Pudl¶ak
- 1986
(Show Context)
Citation Context ...otinGB. Solovay's work on this question produced the method of shortening cuts. This method was e.g. used by Paris & Wilkie, by Pudlak and various others to prove several metatheorems. See [35],[37], =-=[38]-=-. The method was adapted by Nelson to build stronger and stronger theories in his predicativist programme. See [34]. 1 For the case of provability logic the programme of using the language to represen... |

2 |
Interpreting re exive theories in nitely many axioms
- Shavrukov
- 1995
(Show Context)
Citation Context ...) in the intersection of ILM and ILP, but (ii) is not arithmetically valid in PRA. P ` > 3 ! 2( > 3 ) To show that this principle is not PRA-valid, we need the following theorem due to Shavrukov (see =-=[41]-=-). 31Theorem 9.1 Suppose a consistent, nitely axiomatized, sequential theory F interprets a re exive theory R that extends I 0 + EXP. Then fP2 1 j F > (R + P )g is complete 0 3. We reason as follows.... |

2 |
Propositional combinations of {sentences in heyting's arithmetic
- Visser
- 1994
(Show Context)
Citation Context ...y the logic. It is well known that if we consider the provability logic of Heyting Arithmetic2 , HA, we nd a new, rich, weird and wonderful landscape of wild and surprising modal principles. See e.g. =-=[59]-=-. Of course, extending the modal language might alsobeaway toescapethe stability ofSolovay's result. It turns out that for Interpretability we do get good modal logics and Solovaystyle completeness re... |

1 | Iterated local reflection vs iterated consistency - Beklemishev - 1994 |