New foundations for Lewis modal systems

1957 ◽  
Vol 22 (2) ◽  
pp. 176-186 ◽  
Author(s):  
E. J. Lemmon
Keyword(s):  
Modal Logic ◽  
Propositional Calculus ◽  
Modal System ◽  
New Foundations ◽  
Image Position ◽  
Classical Propositional Calculus ◽  
Modal Systems

The main aims of this paper are firstly to present new and simpler postulate sets for certain well-known systems of modal logic, and secondly, in the light of these results, to suggest some new or newly formulated calculi, capable of interpretation as systems of epistemic or deontic modalities. The symbolism throughout is that of [9] (see especially Part III, Chapter I). In what follows, by a Lewis modal system is meant a system which (i) contains the full classical propositional calculus, (ii) is contained in the Lewis system S5, (iii) admits of the substitutability of tautologous equivalents, (iv) possesses as theses the four formulae:We shall also say that a system Σ1 is stricter than a system Σ2, if both are Lewis modal systems and Σ1 is contained in Σ2 but Σ2 is not contained in Σ1; and we shall call Σ1absolutely strict, if it possesses an infinity of irreducible modalities. Thus, the five systems of Lewis in [5], S1, S2, S3, S4, and S5, are all Lewis modal systems by this definition; they are in an order of decreasing strictness from S1 to S5; and S1 and S2 alone are absolutely strict.

scholarly journals COMPLETE ADDITIVITY AND MODAL INCOMPLETENESS

2019 ◽  
Vol 12 (3) ◽  
pp. 487-535
Author(s):  
WESLEY H. HOLLIDAY ◽  
TADEUSZ LITAK
Keyword(s):  
Modal Logic ◽  
Propositional Calculus ◽  
Provability Logic ◽  
Modal Algebras ◽  
Naturally Occurring ◽  
Bimodal Logic ◽  
Incompleteness Theorems ◽  
Image Position ◽  
Classical Propositional Calculus ◽  
Open Question

AbstractIn this article, we tell a story about incompleteness in modal logic. The story weaves together an article of van Benthem (1979), “Syntactic aspects of modal incompleteness theorems,” and a longstanding open question: whether every normal modal logic can be characterized by a class ofcompletely additivemodal algebras, or as we call them,${\cal V}$-baos. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem’s article resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to${\cal V}$-baos, namely the provability logic$GLB$(Japaridze, 1988; Boolos, 1993). We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is${\cal V}$-complete. After these results, we generalize the Blok Dichotomy (Blok, 1978) to degrees of${\cal V}$-incompleteness. In the end, we return to van Benthem’s theme of syntactic aspects of modal incompleteness.

On finite and infinite modal systems

1938 ◽  
Vol 3 (2) ◽  
pp. 77-82 ◽  
Author(s):  
C. West Churchman
Keyword(s):  
Modal Logic ◽  
Symbolic Logic ◽  
Modal Operator ◽  
Image Position ◽  
Modal Systems ◽  
Complex Modal

In Oskar Becker's Zur Logik der Modalitäten four systems of modal logic are considered. Two of these are mentioned in Appendix II of Lewis and Langford's Symbolic logic. The first system is based on A1–8 plus the postulate,From A7: ∼◊p⊰∼p we can prove the converse of C11 by writing ∼◊p for p, and hence deriveThe addition of this postulate to A1–8, as Becker points out, allows us to “reduce” all complex modal functions to six, and these six are precisely those which Lewis mentions in his postulates and theorems: p, ∼p, ◊p, ∼◊p, ∼◊∼p, and ◊∼p This reduction is accomplished by showingwhere ◊n means that the modal operator ◊ is repeated n times; e.g., ◊3p = ◊◊◊p. Then it is shown thatBy means of (1), (2), and (3) any complex modal function whatsoever may be reduced to one of the six “simple” modals mentioned above.It might be asked whether this reduction could be carried out still further, i.e., whether two of the six “irreducible” modals could not be equated. But such a reduction would have to be based on the fact that ◊p = p which is inconsistent with the set B1–9 of Lewis and Langford's Symbolic logic and independent of the set A1–8. Hence for neither set would such a reduction be possible.

A note on some intermediate propositional calculi

1984 ◽  
Vol 49 (2) ◽  
pp. 329-333 ◽  
Author(s):  
Branislav R. Boričić
Keyword(s):  
Positive Solution ◽  
Natural Deduction ◽  
Propositional Calculus ◽  
Conservative Extension ◽  
Image Position ◽  
Classical Propositional Calculus ◽  
Propositional Calculi

This note is written in reply to López-Escobar's paper [L-E] where a “sequence” of intermediate propositional systems NLCn (n ≥ 1) and corresponding implicative propositional systems NLICn (n ≥ 1) is given. We will show that the “sequence” NLCn contains three different systems only. These are the classical propositional calculus NLC1, Dummett's system NLC2 and the system NLC3. Accordingly (see [C], [Hs2], [Hs3], [B 1], [B2], [Hs4], [L-E]), the problem posed in the paper [L-E] can be formulated as follows: is NLC3a conservative extension of NLIC3? Having in mind investigations of intermediate propositional calculi that give more general results of this type (see V. I. Homič [H1], [H2], C. G. McKay [Mc], T. Hosoi [Hs 1]), in this note, using a result of Homič (Theorem 2, [H1]), we will give a positive solution to this problem.NLICnand NLCn. If X and Y are propositional logical systems, by X ⊆ Y we mean that the set of all provable formulas of X is included in that of Y. And X = Y means that X ⊆ Y and Y ⊆ X. A(P1/B1, …, Pn/Bn) is the formula (or the sequent) obtained from the formula (or the sequent) A by substituting simultaneously B1, …, Bn for the distinct propositional variables P1, …, Pn in A.Let Cn(n ≥ 1) be the string of the following sequents:Having in mind that the calculi of sequents can be understood as meta-calculi for the deducibility relation in the corresponding systems of natural deduction (see [P]), the systems of natural deductions NLCn and NLICn (n ≥ 1), introduced in [L-E], can be identified with the calculi of sequents obtained by adding the sequents Cn as axioms to a sequential formulation of the Heyting propositional calculus and to a system of positive implication, respectively (see [C], [Ch], [K], [P]).

Tense logic for discrete future time

1970 ◽  
Vol 35 (1) ◽  
pp. 105-118 ◽  
Author(s):  
Patrick Schindler
Keyword(s):  
Propositional Calculus ◽  
Tense Logic ◽  
Finite Model ◽  
Logical System ◽  
Model Property ◽  
Future Time ◽  
Finite Model Property ◽  
Complete Basis ◽  
Image Position ◽  
Classical Propositional Calculus

Prior has conjectured that the tense-logical system Gli obtained by adding to a complete basis for the classical propositional calculus the primitive symbol G, the definitionsDf. F: Fα = NGNαDf. L: Lα = KαGα,and the postulatesis complete for the logic of linear, infinite, transitive, discrete future time. In this paper it is demonstrated that that conjecture is correct and it is shown that Gli has the finite model property: see [4]. The techniques used are in part suggested by those used in Bull [2] and [3]:Gli can be shown to be complete for the logic of linear, infinite, transitive, discrete future time in the sense that every formula of Gli which is true of such time can be proved as a theorem of Gli. For this purpose the notion of truth needs to be formalized. This formalization is effected by the construction of a model for linear, infinite, transitive, discrete future time.

Modality and quantification in S5

1956 ◽  
Vol 21 (1) ◽  
pp. 60-62 ◽  
Author(s):  
A. N. Prior
Keyword(s):  
Present Note ◽  
Propositional Calculus ◽  
Strict Implication ◽  
Quantification Theory ◽  
Normal Bases ◽  
Complete Basis ◽  
Starting Point ◽  
Image Position ◽  
Classical Propositional Calculus ◽  
Barcan Marcus

In the first of her papers on functional calculi based on strict implication, Ruth Barcan Marcus takes as her starting point the Lewis systems S2 and S4, supplemented by one of the normal bases for quantification theory, and by one special axiom for the mixture, asserting that if possibly something φ's then something possibly φ's. In the symbolism of Łukasiewicz, which will be used here, this axiom is expressible as CMΣxφxΣxMφx. In the present note I propose to show that if S5 had been taken as a startingpoint rather than S2 or S4, this formula need not have been laid down as an axiom but could have been deduced as a theorem.It has been shown by Gödel that a system equivalent to S5 may be obtained if we add to any complete basis for the classical propositional calculus a pair of symbols for ‘Necessarily’ and ‘Possibly,’ which here will be ‘L’ and ‘M’; the axiomsthe ruleRL: If α is a thesis, so is Lα;and the definitionDf. M: M = NLN.

Modus ponens under hypotheses

1965 ◽  
Vol 30 (1) ◽  
pp. 26-26 ◽  
Author(s):  
A. F. Bausch
Keyword(s):  
Mathematical Logic ◽  
Propositional Calculus ◽  
Modus Ponens ◽  
Inference Rules ◽  
Classical Calculus ◽  
New Formulation ◽  
Image Position ◽  
Classical Propositional Calculus

The Stoic “indemonstrables” were inference rules; a rule about rules was the synthetic theorem: if from certain premisses a conclusion follows and from that conclusion and certain further premisses a second conclusion follows, then the second conclusion follows from all the premisses together. Similar things occur as medieval “rules of consequence”, although not usually on a metametalevel; and (with the same proviso) the following might be deemed a contemporary avatar of that Stoic theorem.If every formula which occurs once or more often in the list A1, A2, …, An, B1, B2, …, Bm occurs also at least once in the list C1, C2, …, Cr then:This rule [Church: Introduction to Mathematical Logic, 1956, pp. 94, 165], which may be called the rule of modus ponens under hypotheses (MPH), is worthy of attention for the following reasons:A. MPH and the axioms A ⊃ A yield precisely the positive implicative calculus (and very easily, too).B. MPH and the axioms A ⊃ f ⊃ f ⊃ A yield a new formulation of the full classical propositional calculus (in terms of f and ⊃).C. MPH and the axioms ∼A ⊃ A ⊃ A and A ⊃. ∼A ⊃ B yield the classical calculus in terms of ∼ and ⊃.

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.

Post completeness in modal logic

1972 ◽  
Vol 37 (4) ◽  
pp. 711-715 ◽  
Author(s):  
Krister Segerberg
Keyword(s):  
Modal Logic ◽  
Upper Bound ◽  
Modus Ponens ◽  
Proper Subset ◽  
Proper Extension ◽  
Image Position

Let ⊥, →, and □ be primitive, and let us have a countable supply of propositional letters. By a (modal) logic we understand a proper subset of the set of all formulas containing every tautology and being closed under modus ponens and substitution. A logic is regular if it contains every instance of □A ∧ □B ↔ □(A ∧ B) and is closed under the ruleA regular logic is normal if it contains □⊤. The smallest regular logic we denote by C (the same as Lemmon's C2), the smallest normal one by K. If L and L' are logics and L ⊆ L′, then L is a sublogic of L', and L' is an extension of L; properly so if L ≠ L'. A logic is quasi-regular (respectively, quasi-normal) if it is an extension of C (respectively, K).A logic is Post complete if it has no proper extension. The Post number, denoted by p(L), is the number of Post complete extensions of L. Thanks to Lindenbaum, we know thatThere is an obvious upper bound, too:Furthermore,.

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