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 ⊃.

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.

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]).

Implicational formulas in intuitionistic logic

1974 ◽  
Vol 39 (4) ◽  
pp. 661-664 ◽  
Author(s):  
Alasdair Urquhart
Keyword(s):  
Propositional Calculus ◽  
Modus Ponens ◽  
Equivalence Classes ◽  
Alternative Proof ◽  
Hilbert Algebra ◽  
Hilbert Algebras ◽  
Free Generators ◽  
Finite Set ◽  
Image Position ◽  
The Right

In [1] Diego showed that there are only finitely many nonequivalent formulas in n variables in the positive implicational propositional calculus P. He also gave a recursive construction of the corresponding algebra of formulas, the free Hilbert algebra In on n free generators. In the present paper we give an alternative proof of the finiteness of In, and another construction of free Hilbert algebras, yielding a normal form for implicational formulas. The main new result is that In is built up from n copies of a finite Boolean algebra. The proofs use Kripke models [2] rather than the algebraic techniques of [1].Let V be a finite set of propositional variables, and let F(V) be the set of all formulas built up from V ⋃ {t} using → alone. The algebra defined on the equivalence classes , by settingis a free Hilbert algebra I(V) on the free generators . A set T ⊆ F(V) is a theory if ⊦pA implies A ∈ T, and T is closed under modus ponens. For T a theory, T[A] is the theory {B ∣ A → B ∈ T}. A theory T is p-prime, where p ∈ V, if p ∉ T and, for any A ∈ F(V), A ∈ T or A → p ∈ T. A theory is prime if it is p-prime for some p. Pp(V) denotes the set of p-prime theories in F(V), P(V) the set of prime theories. T ∈ P(V) is minimal if there is no theory in P(V) strictly contained in T. Where X = {A1, …, An} is a finite set of formulas, let X → B be A1 →····→·An → B (ϕ → B is B). A formula A is a p-formula if p is the right-most variable occurring in A, i.e. if A is of the form X → p.

Note on a problem of Paul Bernays

1955 ◽  
Vol 20 (02) ◽  
pp. 109-114 ◽  
Author(s):  
Bolesław Sobociński
Keyword(s):  
Propositional Calculus ◽  
Modus Ponens ◽  
The Law ◽  
Image Position ◽  
Axiom Systems

In this Journal, vol. 18 (1953), p. 350 (Problem 7), Prof. P. Bernays proposed the following problem on propositional calculus: What is the smallest number n such that the propositional calculus, formulated with substitution and modus ponens as the only rules of inference, can be based on a set of initial formulas each of which contains at most n propositional letters (counted with multiplicity) ? In this note I give a solution to this problem, viz., that this number n = 5. For a system of propositional calculus in which the primitive functors are “C” (implication) and “N” (negation) and in which there are only two rules of inference, i.e. the rules of substitution and detachment (modus ponens), the following can be proved. (1) A set of propositional theses each of which contains at most 4 propositional letters is inadequate to give the complete bi-valued calculus of propositions. (2) There are axiom systems for this calculus in which each axiom contains at most 5 propositional letters. § 1. Consider the following normal metrix, in which the designated value is I: This satisfies the two rules of inference, and the following. (a) The law of commutation, i.e. the thesis CCpCqrCqCpr. (b) The following theses: Furthermore, in this matrix “N” is defined in such a way that: (c) For any well-formed formula α and any value m of this matrix, α = m if and only if NNα = m.

Formalisations of further ℵ0-valued Łukasiewicz propositional calculi

1978 ◽  
Vol 43 (2) ◽  
pp. 207-210 ◽  
Author(s):  
Alan Rose
Keyword(s):  
Propositional Calculus ◽  
Rational Numbers ◽  
Modus Ponens ◽  
Truth Values ◽  
Rules Of Procedure ◽  
C And N ◽  
Image Position ◽  
Propositional Calculi

It has been shown that, for all rational numbers r such that 0≤ r ≤ 1, the ℵ0-valued Łukasiewicz propositional calculus whose designated truth-values are those truth-values x such that r ≤ x ≤ 1 may be formalised completely by means of finitely many axiom schemes and primitive rules of procedure. We shall consider now the case where r is rational, 0≥r≤1 and the designated truth-values are those truth-values x such that r≤x≤1.We note that, in the subcase of the previous case where r = 1, a complete formalisation is given by the following four axiom schemes together with the rule of modus ponens (with respect to C),the functor A being defined in the usual way. The functors B, K, L will also be considered to be defined in the usual way. Let us consider now the functor Dαβ such that if P, Dαβ take the truth-values x, dαβ(x) respectively, α, β are relatively prime integers and r = α/β thenIt follows at once from a theorem of McNaughton that the functor Dαβ is definable in terms of C and N in an effective way. If r = 0 we make the definitionWe note first that if x ≤ α/β then dαβ(x)≤(β + 1)α/β − α = α/β. HenceLet us now define the functions dnαβ(x) (n = 0,1,…) bySinceit follows easily thatand thatThus, if x is designated, x − α/β > 0 and, if n > − log(x − α/β)/log(β + 1), then (β + 1)n(x−α/β) > 1.

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.

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.

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.

Extending the Curry-Howard interpretation to linear, relevant and other resource logics

1992 ◽  
Vol 57 (4) ◽  
pp. 1319-1365 ◽  
Author(s):  
Dov M. Gabbay ◽  
Ruy J. G. B. de Queiroz
Keyword(s):  
Inference Rule ◽  
Modus Ponens ◽  
Inference Rules ◽  
Good Account ◽  
Resource Logics ◽  
Ticket Entailment ◽  
Image Position

The so-called Curry-Howard interpretation (Curry [1934], Curry and Feys [1958], Howard [1969], Tait [1965]) is known to provide a rather neat term-functional account of intuitionistic implication. Could one refine the interpretation to obtain an almost as good account of other neighbouring implications, including the so-called ‘resource’ implications (e.g. linear, relevant, etc.)?We answer this question positively by demonstrating that just by working with side conditions on the rule of assertability conditions for the connective representing implication (‘→’) one can characterise those ‘resource’ logics. The idea stems from the realisation that whereas the elimination rule for conditionals (of which implication is a particular case) remains virtually unchanged no matter what kind of conditional one has (i.e. linear, relevant, intuitionistic, classical, etc., all have modus ponens), the corresponding introduction rule carries an element of vagueness which can be explored in the characterisation of several sorts of conditionals. The rule of →-introduction is classified as an ‘improper’ inference rule, to use a terminology from Prawitz [1965]. Now, the so-called improper rules leave room for manoeuvre as to how a particular logic can be obtained just by imposing conditions on the discharge of assumptions that would correspond to the particular logical discipline one is adopting (linear, relevant, ticket entailment, intuitionistic, classical, etc.). The side conditions can be ‘naturally’ imposed, given that a degree of ‘vagueness’ is introduced by the presentation of those improper inference rules, such as the rule of →-introduction:which says: starting from assumption ‘A’, and arriving at ‘B’ via an unspecified number of steps, one can discharge the assumption and conclude that ‘A’ implies ‘B’.

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