On the logic of incomplete answers

1965 ◽  
Vol 30 (1) ◽  
pp. 65-68 ◽  
Author(s):  
M. J. Cresswell
Keyword(s):  
Formal Analysis ◽  
Complete List ◽  
Strict Implication ◽  
Complete Answer ◽  
Questions And Answers ◽  
Image Position

I have argued in [1] that a concept bearing some resemblance to ‘p is the answer to d’ (p a proposition and d a question) can be defined wherever d has the form,‘For which a's is it the case that A (a)?’ (Qa)A(a)where a is a variable and A a wff containing a. To say that p is the true and complete answer to (Qa)A(a) is expressed as saying that p is logically equivalent to the true conjunction of A(a) or ~A(a) for each a. It is defined as;Such a concept of answer is like Belnap's [2] direct true answer to a complete list question, or like Harrah's use [3] (p. 43) of the notion of a state description. The main difference between my approach and that of Belnap and Harrah is that while they are concerned to develop a formal metalanguage for discussion of questions and answers I am concerned to express, as far as possible in existing systems, certain interrogative statements; in particular statements of the form ‘— is the (an) answer to —’.While the account in [1] does give a formal analysis of one ‘answer’ concept there are respects in which it is inadequate.1. Since it uses entailment (or strict implication) to define the relation between p the answer and d the question we can shew that if p is the answer to d and q is logically equivalent to p then q is the answer to d.

Three-phase boundary motions under constant velocities. I: The vanishing surface tension limit

1996 ◽  
Vol 126 (4) ◽  
pp. 837-865 ◽  
Author(s):  
Fernando Reitich ◽  
H. Mete Soner
Keyword(s):  
Surface Tension ◽  
Formal Analysis ◽  
Local Equilibrium ◽  
Convergence Result ◽  
Normal Velocity ◽  
Infinitely Many Solutions ◽  
Weak Formulation ◽  
Material Interfaces ◽  
Internal Surfaces ◽  
Image Position

In this paper, we deal with the dynamics of material interfaces such as solid–liquid, grain or antiphase boundaries. We concentrate on the situation in which these internal surfaces separate three regions in the material with different physical attributes (e.g. grain boundaries in a polycrystal). The basic two-dimensional model proposes that the motion of an interface Гij between regions i and j (i, j = 1, 2, 3, i ≠ j) is governed by the equationHere Vij, kij, μij and fij denote, respectively, the normal velocity, the curvature, the mobility and the surface tension of the interface and the numbers Fij stand for the (constant) difference in bulk energies. At the point where the three phases coexist, local equilibrium requires thatIn case the material constants fij are small, and ε ≪ 1, previous analyses based on the parabolic nature of the equations (0.1) do not provide good qualitative information on the behaviour of solutions. In this case, it is more appropriate to consider the singular case with fij = 0. It turns out that this problem, (0.1) with fij = 0, admits infinitely many solutions. Here, we present results that strongly suggest that, in all cases, a unique solution—‘the vanishing surface tension (VST) solution’—is selected by letting ε→0. Indeed, a formal analysis of this limiting process motivates us to introduce the concept of weak viscosity solution for the problem with ε = 0. As we show, this weak solution is unique and is therefore expected to coincide with the VST solution. To support this statement, we present a perturbation analysis and a construction of self-similar solutions; a rigorous convergence result is established in the case of symmetric configurations. Finally, we use the weak formulation to write down a catalogue of solutions showing that, in several cases of physical relevance, the VST solution differs from results proposed previously.

On Certain Polynomials of Gaussian Type

1979 ◽  
Vol 31 (2) ◽  
pp. 274-281 ◽  
Author(s):  
Daniel Reich
Keyword(s):  
Group Action ◽  
Lie Group ◽  
Orbit Space ◽  
Partition Functions ◽  
Poincaré Polynomial ◽  
Complete Answer ◽  
Lie Group Action ◽  
Image Position ◽  
Positive Coefficients ◽  
Positive Integers

Introduction. We shall consider functions of the formwhere {ri} and {si} are sets of positive integers. Such functions were studied by E. Grosswald in [2], who took {si} to be pairwise relatively prime, and asked the following two questions:(a) When is ƒ(t) a polynomial?(b) When does ƒ(t) have positive coefficients?These questions arise naturally from the work of Allday and Halperin, who show in [1] that under suitable circumstance ƒ(t) will be the Poincare polynomial of the orbit space of a certain Lie group action. Grosswald gives a complete answer to (a), but (b) is a much harder question, and a complete answer is provided only for the case m = 2. His treatment involves the representation of the coefficients of ƒ(t) by partition functions, and uses a classical description by Sylvester of the semigroup generated by {si}.

A note concerning the paradoxes of strict implication and Lewis's system SI

1948 ◽  
Vol 13 (3) ◽  
pp. 138-139 ◽  
Author(s):  
Sören Halldén
Keyword(s):  
Strict Implication ◽  
Image Position

It has been shown by Lewis and Langford that the postulate B8,is not deducible in SI. From this it follows that neither are the paradoxes of strict implication deducible in that system. However, the following weaker—but perhaps philosophically equally important—analogues are deducible:

A reduction theorem for predicate logic

1972 ◽  
Vol 37 (2) ◽  
pp. 352-354 ◽  
Author(s):  
M. H. Löb
Keyword(s):  
Predicate Logic ◽  
Complete List ◽  
Reduction Theorem ◽  
Logical Connectives ◽  
Image Position

It is well known that in prepositional logic not every propositional connective is definable in terms of equivalence nor indeed in terms of equivalence together with negation. It is the purpose of this note to show that in a certain sense this is, however, possible within predicate logic.Let F1 be the class of predicate logical formulae containing only the logical connectives ¬ (not), ∨ (or), E (exists), and F2 the class of predicate logical formulae containing only the logical connectives ≡ (if and only if) and E.Our object will be to find a method for reducing any formulae ϕ of F1 to a formula ψ of F2 such that ϕ is valid if and only if ψ is valid.Note first that for every domain D containing at least two individuals we can for every n-place relation R defined over D find an n + 1-place relation S defined over D such that for all x1, …, xn ∈ D,holds. For instance, let S(y, x1, …, xn) be the relation y = x1 & R(x1, …, xn).Now let ϕ be an arbitrary formula of F1 and A1, …, A3 a complete list of predicate letters occurring in ϕ of degree g1, …, gi, respectively. Let ϕ′ be the formula obtained from ϕ by replacing every prime formula Ai(x1, …, xgi) byThen it follows immediately from the previous observation that (a) ϕ is satisfiable in D if and only if ϕ′ is satisfiable in D.

On the interpretation of the sign ‘⊃’

1953 ◽  
Vol 18 (1) ◽  
pp. 60-62 ◽  
Author(s):  
John Myhill
Keyword(s):  
The Other ◽  
Strict Implication ◽  
Material Implication ◽  
Strict Interpretation ◽  
False Proposition ◽  
Positive Logic ◽  
Law Of Excluded Middle ◽  
Image Position ◽  
Formal Fallacies ◽  
Excluded Middle

The sign ‘⊃’ (or ‘→’ or ‘C’) functions in many logical systems in a way which precludes its interpretation as either strict or material implication. For example, in the systems of Heyting, Johansson, Fitch and Bernays (positive logic), the following are theorems:Now if ‘⊃’ were interpreted as strict implication, ⊃2 would mean ‘if p is true, then p is strictly implied by every proposition’, i.e. ‘if p is true, it is necessarily true’, which is false for contingently true p. If on the other hand ‘⊃’ were interpreted as material implication, ⊃1 would reduce to ‘~p ∨ p’, i.e. to the law of excluded middle, which is conspicuously lacking in the systems mentioned. The reader is likely in practice to veer between these two interpretations. Thus in Fitch or Heyting on realizing that ‘~p⊃▪ p⊃q’ is a theorem, one thinks of it as meaning ‘a false proposition implies everything’ and regards the implication as material; but the presence of ‘p⊃p’ as a theorem, even for choices of p which do not satisfy excluded middle, inclines one again to the strict interpretation. This vacillation, while it need not lead to the commission of any formal fallacies, tends to hamstring one's intuition and thus waste time. The purpose of this paper is to suggest an interpretation of ‘⊃’ which will prevent such havering.Let two formulae A and B be called interdeducible if A ⊢ B and B ⊢ A.

III.—Sedgwick Museum Notes. New Fossils from the Dufton Shales

1910 ◽  
Vol 7 (5) ◽  
pp. 211-220 ◽  
Author(s):  
F. R. Cowper Reed
Keyword(s):  
New Species ◽  
Complete List ◽  
Small Collection ◽  
Image Position ◽  
A New Species

A small collection of fossils was made a few years ago by Mr. V. M. Turnbull from a cutting on the Alston Road, near Melmerby, and the author has already described a new species of Lichas (L. melmerbiensis) which was included amongst them. Several more new species of trilobites and other groups are now described, and the complete list of the fauna is as follows:—.

Independent axiom schemata for S5

1956 ◽  
Vol 21 (3) ◽  
pp. 255-256
Author(s):  
Alan Ross Anderson
Keyword(s):  
Strict Implication ◽  
Material Implication ◽  
Independent Axiom ◽  
Image Position

Leo Simons has shown that H1—H6 below constitute a set of independent axiom schemata for S3, with detachment for material implication “→” as the only primitive rule. He also showed that addition of the scheme (◇ ◇ α ⥽ ◇ α) yields S4, and that these schemata for S4 are independent. The question for S5 was left open. We shall show (presupposing familiarity with Simons' paper) that H1—H6 and S, below, constitute a set of independent axiom schemata for S5, with detachment for material implication as the only primitive rule.Let S5′ be the system generated from H1—H6 and S with the help of the primitive rule. It is easy to see that Simons' derivations of the rules (a) adjunction, (b) detachment for strict implication, and (c) intersubstitutability of strict equivalents, may be carried out for S5′. We know that (1) (∼ ◇ ∼ α ⥽ ◇ α) is provable in S2, hence also in S3 and S5′; and (1) and S yield (2) (α ⥽ ∼ ◇ ∼ ◇ α). Perry has shown that addition of (2) to S3 yields S5, so S5 is a subsystem of S5′. And it is easy to prove S in S5; hence the systems are equivalent.

Périodicité des théories élémentaires des corps de séries formelles itérées

1986 ◽  
Vol 51 (2) ◽  
pp. 334-351
Author(s):  
Françoise Delon
Keyword(s):  
Complete Answer ◽  
Image Position

AbstractC. U. Jensen suggested the following construction, starting from a fieldK:and asked when two fieldsKαandKβare equivalent. We give a complete answer in the case of a fieldKof characteristic 0.

Modalities in the Survey system of strict implication

1939 ◽  
Vol 4 (4) ◽  
pp. 137-154 ◽  
Author(s):  
William Tuthill Parry
Keyword(s):  
Finite Number ◽  
Symbolic Logic ◽  
Strict Implication ◽  
Survey System ◽  
Image Position

Professor C. I. Lewis, in Lewis and Langford's Symbolic logic, designates the system (S2) determined by the postulates used in Chapter VI—namely, 11.1–7 (B1–7) andas the system of strict implication. For certain reasons, he prefers it to either the earlier system (S3) determined by the stronger set of postulates of his Survey of symbolic logic, as emended, namely, A1–7 andor the system (S1) determined by the weaker set of postulates B1–7 or A1–7.But Lewis and others, following O. Becker, have also given consideration to systems which contain some additional principle effecting the reduction of complex modalities to simpler ones. Notable are the system (S4) determined by B1–7 pluswhich includes (is stronger than) S3; and the system (S5) determined by B1–7 pluswhich includes S4.It seems worth while to investigate further the system of the Survey (emended), S3, which is intermediate between S2 and S4. This is the purpose of the present paper. We first prove some additional theorems in S2 and S3. These enable us to reduce all the complex modalities in S3 to a finite number, viz. 42; and it is shown that no further reduction is possible. Finally, several systems which include S3 are considered.

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.

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