Extremely undecidable sentences

1982 ◽  
Vol 47 (1) ◽  
pp. 191-196 ◽  
Author(s):  
George Boolos
Keyword(s):  
Modal Logic ◽  
The Other ◽  
Propositional Modal Logic ◽  
Image Position

Let ‘ϕ’, ‘χ’, and ‘ψ’ be variables ranging over functions from the sentence letters P0, P1, … Pn, … of (propositional) modal logic to sentences of P(eano) Arithmetic), and for each sentence A of modal logic, inductively define Aϕ by[and similarly for other nonmodal propositional connectives]; andwhere Bew(x) is the standard provability predicate for PA and ⌈F⌉ is the PA numeral for the Gödel number of the formula F of PA. Then for any ϕ, (−□⊥)ϕ = −Bew(⌈⊥⌉), which is the consistency assertion for PA; a sentence S is undecidable in PA iff both and , where ϕ(p0) = S. If ψ(p0) is the undecidable sentence constructed by Gödel, then ⊬PA (−□⊥→ −□p0 & − □ − p0)ψ and ⊢PA(P0 ↔ −□⊥)ψ. However, if ψ(p0) is the undecidable sentence constructed by Rosser, then the situation is the other way around: ⊬PA(P0 ↔ −□⊥)ψ and ⊢PA (−□⊥→ −□−p0 & −□−p0)ψ. We call a sentence S of PA extremely undecidable if for all modal sentences A containing no sentence letter other than p0, if for some ψ, ⊬PAAψ, then ⊬PAAϕ, where ϕ(p0) = S. (So, roughly speaking, a sentence is extremely undecidable if it can be proved to have only those modal-logically characterizable properties that every sentence can be proved to have.) Thus extremely undecidable sentences are undecidable, but neither the Godel nor the Rosser sentence is extremely undecidable. It will follow at once from the main theorem of this paper that there are infinitely many inequivalent extremely undecidable sentences.

QUANTIFIED MODAL LOGIC ON THE RATIONAL LINE

2014 ◽  
Vol 7 (3) ◽  
pp. 439-454 ◽  
Author(s):  
PHILIP KREMER
Keyword(s):  
Modal Logic ◽  
Real Line ◽  
Topological Spaces ◽  
Topological Semantics ◽  
Cantor Space ◽  
Quantified Modal Logic ◽  
Propositional Modal Logic ◽  
The Family ◽  
Modal Logic S4 ◽  
Rational Line

AbstractIn the topological semantics for propositional modal logic, S4 is known to be complete for the class of all topological spaces, for the rational line, for Cantor space, and for the real line. In the topological semantics for quantified modal logic, QS4 is known to be complete for the class of all topological spaces, and for the family of subspaces of the irrational line. The main result of the current paper is that QS4 is complete, indeed strongly complete, for the rational line.

The extensions of the modal logic K5

1985 ◽  
Vol 50 (1) ◽  
pp. 102-109 ◽  
Author(s):  
Michael C. Nagle ◽  
S. K. Thomason
Keyword(s):  
Modal Logic ◽  
Classical Logic ◽  
Abstract Characterization ◽  
Propositional Modal Logic ◽  
Normal Logic ◽  
Countably Infinite ◽  
Infinite Set

Our purpose is to delineate the extensions (normal and otherwise) of the propositional modal logic K5. We associate with each logic extending K5 a finitary index, in such a way that properties of the logics (for example, inclusion, normality, and tabularity) become effectively decidable properties of the indices. In addition we obtain explicit finite axiomatizations of all the extensions of K5 and an abstract characterization of the lattice of such extensions.This paper refines and extends the Ph.D. thesis [2] of the first-named author, who wishes to acknowledge his debt to Brian F. Chellas for his considerable efforts in directing the research culminating in [2] and [3]. We also thank W. J. Blok and Gregory Cherlin for observations which greatly simplified the proofs of Theorem 3 and Corollary 10.By a logic we mean a set of formulas in the countably infinite set Var of propositional variables and the connectives ⊥, →, and □ (other connectives being used abbreviatively) which contains all the classical tautologies and is closed under detachment and substitution. A logic is classical if it is also closed under RE (from A↔B infer □A ↔□B) and normal if it is classical and contains □ ⊤ and □ (P → q) → (□p → □q). A logic is quasi-classical if it contains a classical logic and quasi-normal if it contains a normal logic. Thus a quasi-normal logic is normal if and only if it is classical, and if and only if it is closed under RN (from A infer □A).

S. K. Thomason. Noncompactness in propositional modal logic. The journal of symbolic logic, vol. 37 no. 4 (for 1972, pub. 1973), pp. 716–720. - Kit Fine. An incomplete logic containing S4. Theoria, vol. 40 (1974), pp. 23–29. - S. K. Thomason. An incompleteness theorem in modal logic. Theoria, vol. 40 (1974), pp. 30–34. - Martin Gerson. The inadequacy of the neighbourhood semantics for modal logic. The journal of symbolic logic, vol. 40 (1975), pp. 141–148. - Martin Sebastian Gerson. An extension of S4 complete for the neighbourhood semantics but incomplete for the relational semantics. Studio logica, vol. 34 (1975), pp. 333–342. - Martin Gerson. A neighbourhood frame for T with no equivalent relational frame. Zeitschrift für mathematische Logik und Grundlugen der Mathematik, vol. 22 (1976), pp. 29–34. - V. B. Šehtman. On incomplete propositional logics. Soviet mathematics, vol. 18 (1977), pp. 985–989. (English translation by B. F. Wells of O népolnyh logikah vyskazyvanij, Doklady Akadémii Nauk SSSR, vol. 235 (1977), pp. 542–545.) - J. F. A. K. van Benthem. Two simple incomplete modal logics. Theoria, vol. 44 (1978), pp. 25–37. - J. F. A. K. van Benthem and W. J. Blok. Transitivity follows from Dummett's axiom. Theoria, vol. 44 (1978), pp. 117–118. - J. F. A. K. van Benthem. Syntactic aspects of modal incompleteness theorems. Theoria, vol. 44 (1978), vol. 45 (1979). pp. 63–77.

1983 ◽  
Vol 48 (2) ◽  
pp. 488-495 ◽  
Author(s):  
R. A. Bull
Keyword(s):  
Modal Logic ◽  
English Translation ◽  
Nauk Sssr ◽  
Symbolic Logic ◽  
Relational Semantics ◽  
Akademii Nauk Sssr ◽  
Doklady Akademii Nauk Sssr ◽  
Propositional Modal Logic ◽  
Relational Frame ◽  
Incompleteness Theorems

Propositional Modal Logic

Rough Sets ◽  
2002 ◽  
pp. 159-172
Author(s):  
Lech Polkowski
Keyword(s):  
Modal Logic ◽  
Propositional Modal Logic

scholarly journals COMPLETENESS OF SECOND-ORDER PROPOSITIONAL S4 AND H IN TOPOLOGICAL SEMANTICS

2018 ◽  
Vol 11 (3) ◽  
pp. 507-518
Author(s):  
PHILIP KREMER
Keyword(s):  
Modal Logic ◽  
Intuitionistic Logic ◽  
Second Order ◽  
Topological Semantics ◽  
First Order ◽  
Propositional Modal Logic ◽  
Propositional Quantifiers ◽  
Modal Logic S4 ◽  
Natural Way

AbstractWe add propositional quantifiers to the propositional modal logic S4 and to the propositional intuitionistic logic H, introducing axiom schemes that are the natural analogs to axiom schemes typically used for first-order quantifiers in classical and intuitionistic logic. We show that the resulting logics are sound and complete for a topological semantics extending, in a natural way, the topological semantics for S4 and for H.

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