Logical paradoxes for many-valued systems

1954 ◽  
Vol 19 (1) ◽  
pp. 37-40 ◽  
Author(s):  
Moh Shaw-Kwei
Keyword(s):  
Propositional Variable ◽  
Natural Question ◽  
Material Implication ◽  
System L ◽  
New Interpretation ◽  
Logical Paradoxes ◽  
Image Position ◽  
Function Of Two Variables

As we know, the system of material implication is led into an inconsistency by the Russellian class, defined as λx. Nϵxx. This class, however, does no harm to many other systems, for example, the three-valued system L3 given by J. Łukasiewicz. A natural question is whether or not there exist classes which affect some of these other logical systems. The main result of the present paper is to answer this question affirmatively. At the end of this paper we point out that the interpretation given by J. Łukasiewicz for the system L3 is not satisfactory, and propose a new interpretation.B. Russell deduced the mentioned inconsistency by the aid of the notion of negation. Later on, H. B. Curry pointed out that we could get the same result without the aid of that notion. None of these results affects the system L3 and other similar systems. But these systems may be involved.To show this, we need the following definitions.A function of two variables Cpq will be called an implication when the following “implication rule” is valid:Under this definition we should note that material equivalence Epq, for example, is an implication.Let C be such an implication. Then the symbol “(Cp)iq” is defined recursively byThe class an is defined as λx. (Cϵxx)np, where p is a propositional variable. The rule of absorption of order n, denoted by (An), is:

scholarly journals Note on Continued Fractions and the Sequence of Natural Numbers

1932 ◽  
Vol 27 ◽  
pp. ix-xiii ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Continued Fractions ◽  
Natural Question ◽  
Natural Numbers ◽  
Image Position

When a student first approaches the theory of infinite continued fractions a natural question that suggests itself is how to evaluate the expression

On closure under direct product

1958 ◽  
Vol 23 (2) ◽  
pp. 149-154 ◽  
Author(s):  
C. C. Chang ◽  
Anne C. Morel
Keyword(s):  
Normal Form ◽  
Direct Product ◽  
Disjunctive Normal Form ◽  
Negative Answer ◽  
Sufficient Condition ◽  
Natural Question ◽  
Existential Quantifier ◽  
Relational Systems ◽  
Image Position

In 1951, Horn obtained a sufficient condition for an arithmetical class to be closed under direct product. A natural question which arose was whether Horn's condition is also necessary. We obtain a negative answer to that question.We shall discuss relational systems of the formwhere A and R are non-empty sets; each element of R is an ordered triple 〈a, b, c〉, with a, b, c ∈ A.1 If the triple 〈a, b, c〉 belongs to the relation R, we write R(a, b, c); if 〈a, b, c〉 ∉ R, we write (a, b, c). If x0, x1 and x2 are variables, then R(x0, x1, x2) and x0 = x1 are predicates. The expressions (x0, x1, x2) and x0 ≠ x1 will be referred to as negations of predicates.We speak of α1, …, αn as terms of the disjunction α1 ∨ … ∨ αn and as factors of the conjunction α1 ∧ … ∧ αn. A sentence (open, closed or neither) of the formwhere each Qi (if there be any) is either the universal or the existential quantifier and each αi, l is either a predicate or a negation of a predicate, is said to be in prenex disjunctive normal form.

Generalized quantifiers and elementary extensions of countable models

1977 ◽  
Vol 42 (3) ◽  
pp. 341-348 ◽  
Author(s):  
Małgorzata Dubiel
Keyword(s):  
Countable Model ◽  
Generalized Quantifiers ◽  
Natural Question ◽  
Elementary Extension ◽  
First Order ◽  
Transitive Model ◽  
Order Language ◽  
Image Position ◽  
Countable Models

Let L be a countable first-order language and L(Q) be obtained by adjoining an additional quantifier Q. Q is a generalization of the quantifier “there exists uncountably many x such that…” which was introduced by Mostowski in [4]. The logic of this latter quantifier was formalized by Keisler in [2]. Krivine and McAloon [3] considered quantifiers satisfying some but not all of Keisler's axioms. They called a formula φ(x) countable-like iffor every ψ. In Keisler's logic, φ(x) being countable-like is the same as ℳ⊨┐Qxφ(x). The main theorem of [3] states that any countable model ℳ of L[Q] has an elementary extension N, which preserves countable-like formulas but no others, such that the only sets definable in both N and M are those defined by formulas countable-like in M. Suppose C(x) in M is linearly ordered and noncountable-like but with countable-like proper segments. Then in N, C will have new elements greater than all “old” elements but no least new element — otherwise it will be definable in both models. The natural question is whether it is possible to use generalized quantifiers to extend models elementarily in such a way that a noncountable-like formula C will have a minimal new element. There are models and formulas for which it is not possible. For example let M be obtained from a minimal transitive model of ZFC by letting Qxφ(x) mean “there are arbitrarily large ordinals satisfying φ”.

scholarly journals ENAYAT MODELS OF PEANO ARITHMETIC

2018 ◽  
Vol 83 (04) ◽  
pp. 1501-1511 ◽  
Author(s):  
ATHAR ABDUL-QUADER
Keyword(s):  
Linear Order ◽  
Peano Arithmetic ◽  
Countable Model ◽  
Natural Question ◽  
Conservative Extension ◽  
Image Position ◽  
Definable Elements

AbstractSimpson [6] showed that every countable model ${\cal M} \models PA$ has an expansion $\left( {{\cal M},X} \right) \models P{A^{\rm{*}}}$ that is pointwise definable. A natural question is whether, in general, one can obtain expansions of a nonprime model in which the definable elements coincide with those of the underlying model. Enayat [1] showed that this is impossible by proving that there is ${\cal M} \models PA$ such that for each undefinable class X of ${\cal M}$, the expansion $\left( {{\cal M},X} \right)$ is pointwise definable. We call models with this property Enayat models. In this article, we study Enayat models and show that a model of $PA$ is Enayat if it is countable, has no proper cofinal submodels and is a conservative extension of all of its elementary cuts. We then show that, for any countable linear order γ, if there is a model ${\cal M}$ such that $Lt\left( {\cal M} \right) \cong \gamma$, then there is an Enayat model ${\cal M}$ such that $Lt\left( {\cal M} \right) \cong \gamma$.

Two incomplete anti-realist modal epistemic logics

1990 ◽  
Vol 55 (1) ◽  
pp. 297-314 ◽  
Author(s):  
Timothy Williamson
Keyword(s):  
Propositional Variable ◽  
Crude Model ◽  
Rational Acceptance ◽  
Rational Acceptability ◽  
Image Position ◽  
Cognitive Attitude ◽  
Realist Conception ◽  
Sentential Operator

Many phrases have been used to express what are sometimes called anti-realist conceptions of truth: ‘verifiability’, ‘knowability’, ‘rational acceptability’, ‘warranted assertability’. In spite of their obvious differences, all four of these phrases have a common form; each is a cognitive attitude modified by ‘-ability’. They speak of the possibility of verification, knowledge, rational acceptance or warranted assertion. Schematically, it seems to be claimed that it is true that A if and only if it is possible that it is E'd that A, where ‘E’ is to be replaced by some cognitive verb and ‘A’ by any indicative sentence of the class to which the anti-realist conception is being claimed to apply. Since truth is redundant as a sentential operator, this boils down to the following thesis, where ‘p’ is a propositional variable and ‘M’ expresses the appropriate kind of possibility:(*) also formalizes views such as Putnam's: ‘To claim a statement is true is to claim it could be justified’ [11, p. 56]. It is no doubt a crude model for anti-realism, but one has to start somewhere; by seeing how and why more sophisticated versions of anti-realism differ from (*) one should be able to understand them better too. Moreover, if an anti-realist rejects the equation of truth with, say, warranted assertibility, arguing that truth is rather to be identified with the possibility of getting into a position in which one's warrant to assert somehow cannot be overturned, the form of (*) is preserved, for truth is still being identified with the possibility of something.

Non Commutative Lp Spaces II

1982 ◽  
Vol 34 (5) ◽  
pp. 1208-1214 ◽  
Author(s):  
A. Katavolos
Keyword(s):  
Measure Space ◽  
Linear Functional ◽  
Von Neumann Algebra ◽  
Linear Mapping ◽  
Natural Question ◽  
Von Neumann ◽  
Lp Spaces ◽  
Image Position ◽  
Natural Way

Let M be a w*-algebra (Von Neumann algebra), τ a semifinite, faithful, normal trace on M. There exists a w*-dense (i.e., dense in the σ(M, M*)-topology, where M* is the predual of M) *-ideal J of M such that τ is a linear functional on J, and(where |x| = (x*x)1/2) is a norm on J. The completion of J in this norm is Lp(M, τ) (see [2], [8], [7], and [4]).If M is abelian, in which case there exists a measure space (X, μ) such that M = L∞(X, μ), then Lp(X, τ) is isometric, in a natural way, to Lp(X, μ). A natural question to ask is whether this situation persists if M is non-abelian. In a previous paper [5] it was shown that it is not possible to have a linear mapping

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.

scholarly journals COMPUTABILITY THEORY, NONSTANDARD ANALYSIS, AND THEIR CONNECTIONS

2019 ◽  
Vol 84 (4) ◽  
pp. 1422-1465 ◽  
Author(s):  
DAG NORMANN ◽  
SAM SANDERS
Keyword(s):  
Basic Property ◽  
Nonstandard Analysis ◽  
Binary Sequence ◽  
Finite Sequence ◽  
Computability Theory ◽  
Natural Question ◽  
Compactness Property ◽  
Cantor Space ◽  
Theory Yield ◽  
Image Position

AbstractWe investigate the connections between computability theory and Nonstandard Analysis. In particular, we investigate the two following topics and show that they are intimately related.(T.1) A basic property of Cantor space$2^ $ is Heine–Borel compactness: for any open covering of $2^ $, there is a finite subcovering. A natural question is: How hard is it to compute such a finite subcovering? We make this precise by analysing the complexity of so-called fan functionals that given any $G:2^ \to $, output a finite sequence $\langle f_0 , \ldots ,f_n \rangle $ in $2^ $ such that the neighbourhoods defined from $\overline {f_i } G\left( {f_i } \right)$ for $i \le n$ form a covering of $2^ $.(T.2) A basic property of Cantor space in Nonstandard Analysis is Abraham Robinson’s nonstandard compactness, i.e., that every binary sequence is “infinitely close” to a standard binary sequence. We analyse the strength of this nonstandard compactness property of Cantor space, compared to the other axioms of Nonstandard Analysis and usual mathematics.Our study of (T.1) yields exotic objects in computability theory, while (T.2) leads to surprising results in Reverse Mathematics. We stress that (T.1) and (T.2) are highly intertwined, i.e., our study is holistic in nature in that results in computability theory yield results in Nonstandard Analysis and vice versa.

Correction to A note on the Entscheidungsproblem

1936 ◽  
Vol 1 (3) ◽  
pp. 101-102 ◽  
Author(s):  
Alonzo Church
Keyword(s):  
Propositional Function ◽  
Free Variables ◽  
System L ◽  
Universal Quantifiers ◽  
Individual Variables ◽  
The Individual ◽  
Image Position

In A note on the Entscheidungsproblem the author gave a proof of the unsolvability of the general case of the Entscheidungsproblem of the engere Funktionenkalkül. This proof, however, contains an error, in order to correct which it is necessary to modify the “additional axioms” of the system L so that they contain no free variables (either free individual variables or free propositional function variables).The additional axioms of L other than x=y→[F(x)→F(y)] contain no free propositional function variables, and hence it is sufficient to replace each one by an expression obtained from it by quantifying all the individual variables by means of universal quantifiers initially placed—thus, in particular, x = x is replaced by (x)[x=x]. Moreover the axiom x=y→[F(x)→F(y)] may be replaced by the following set of axioms:and similar axioms for each of the functions b1, b2, …, bk.

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.

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