Formal Logical System Research Articles

A formal system of logic

The main purpose of this paper is to present a formal systemPin which we enjoy a smooth-running technique and which countenances a universe of classes which is symmetrical as between large and small. More exactly,Pis a system which differs from the inconsistent system of [1] only in the introduction of a rather natural new restrictive condition on the defining formulas of the elements (sets, membership-eligible classes). It will be proved that if the weaker system of [2] is consistent, thenPis also consistent.After the discovery of paradoxes, it may be recalled, Russell and Zermelo in the same year proposed two different ways of safeguarding logic against contradictions (see [3], [4]). Since then various simplifications and refinements of these systems have been made. However, in the resulting systems of Zermelo set theory, generation of classes still tends to be laborious and uncertain; and in the systems of Russell's theory of types, complications in the matter of reduplication of classes and meaningfulness of formulas remain. In [2], Quine introduced a system which seems to be free from all these complications. But later it was found out that in it there appears to be an unavoidable difficulty connected with mathematical induction. Indeed, we encounter the curious situation that although we can prove in it the existence of a class V of all classes, and we can also prove particular existence theorems for each of infinitely many classes, nobody has so far contrived to prove in it that V is an infinite class or that there exists an infinite class at all.

An Outline of Semantics for the Psychologist

It is possible that many of the difficulties of the contemporary psychologist are related to a continued reliance on antique systems of formal logic, ancient philosophy, or hand-me-down oriental or primitive mysticism. The purpose of this paper is to indicate ways in which such factors are influencing present day psychology, with the object of creating a greater awareness of the metaphysical bases of much of our current theory and practice. The primary consideration in a paper such as this is to define the function or functions of language in scientific investigation. Korzybski (3, pp. 21-24) holds that the only possible function that words can perform is in terms of their direct influence on the inter-relationships existing between organisms and events. He illustrates this point by asking a person to define a word, any word, and then asking him to define the term or terms used in the original definition, and the terms of each successive definition, until a point is reached at which definitions either become circular (such as defining thought in terms of mind and mind in terms of thought) or the response I know but cannot tell is given. We suggest that language is essentially referential in nature and contains no 'meaning' of itself. All efforts at verbal definition must eventually end with a group of terms which cannot be defined further. Psychologically, language is a behavior situation in which a person refers someone (or is referred) to some object or event either vocally or by gesture (2). Language behavior is an adjustment of an individual to another individual and the object referred to: it is not a set of symbols that act as carriers for mysterious meanings. Language from this point of view is seen as a potential tool of science, a tool which can be of aid if the priority of the or primary datum of the science is recognized, but as Korzybski indicates, ... verbal levels are only auxiliary, sometimes useful, but at present often harmful (3, p. 23). Harm results from the fact that popular language, philosophy, and formal logic are based on the universe as it was assumed to be when

The inconsistency of certain formal logics

A proof that certain systems of formal logic are inconsistent, in the sense that every formula expressible in them is provable, was published by Kleene and Rosser under the above title in 1935. For the case where the underlying system satisfies an additional condition—viz. the possession of an operator analogous to Schönfinkel's K—I gave a simpler derivation of this condition in a previous paper. That argument, like the original one of Kleene and Rosser, was a refinement of the Richard paradox. The object of the present paper is to show that, if we use other paradoxes, an inconsistency will result from a very much simpler argument and on much less restrictive hypotheses. The contradiction can no longer be called “the paradox of Kleene and Rosser,” because it is based on an entirely different principle; but, in deference to the work of the original discoverers of the inconsistency, the paper is given the same title as that which their paper bears. The central idea of the new derivation was suggested by some work of R. Carnap.This paper is based on the one above cited, which will be referred to as PKR. However, acquaintance with that paper will be presupposed only through 3.4, i.e., through the statement of the basic hypotheses and conventions for a combinatorially complete system—except that for the second (alternative) method of construction given below the substance of PKR through 9.6 is needed.

Schröder's anticipation of the simple theory of types

The attempt to exclude the well-known pardoxes of set theory from a system of formal logic, without at the same time rendering the system so weak as to be inadequate for the purposes of mathematics, may be made in various ways. Currently the most favored methods are the Zermelo axiomatic set theory and the simple theory of types. As is well known, the simple theory of types may be described in the following terms. A particular domain, to be called the domain of individuals is selected; this may be any domain within very wide limits, but in any particular system must be treated as fixed. Classes and relations (and functions, if the system provides for such) are then classified into a hierarchy of types. For simplicity in the description of this hierarchy it is convenient to think of relations (and functions) as defined in terms of classes following a suggestion first made by Wiener in 1914. On this basis, the first type is composed of the individuals, the second type of classes of individuals, the third type of classes of classes of individuals, and so on. The restriction imposed by the simple theory of types is that the members of a class which belongs to the type n +1 must all belong to the type n. There is no provision for classes which do not belong to a type: classes not obeying the restriction just stated are regarded as non? existent, and names for such classes are excluded from the system. And even to say or write a e b (that a is member of the class b) is excluded as not well-formed unless the variables or constants a, b show by their syntactical form that, for some n, they refer to types n and n + 1 respectively. (Of course, the foregoing statement cannot be made within the system based on the simple theory of types which is being described, but only within some stronger system, e.g., one employing transfinite types. For the purpose of the formal construction of the system it is not

Frederic Brenton Fitch. A system of formal logic without an analogue to the Curry W operator. The journal of symbolic logic, vol. 1 (1936), pp. 92–100.

Frederic Brenton Fitch. A system of formal logic without an analogue to the Curry W operator. The journal of symbolic logic, vol. 1 (1936), pp. 92–100.

Frederic Brenton Fitch. A system of formal logic without an analogue to the Curry W operator. The journal of symbolic logic, vol. 1 (1936), pp. 92–100.

Frederic Brenton Fitch. A system of formal logic without an analogue to the Curry W operator. The journal of symbolic logic, vol. 1 (1936), pp. 92–100.

Oskar Becker. Eudoxos-Studien IV. Das Prinzip des ausgeschlossenen Dritten in der griechischen Mathematik. Quellen und Studien zur Geschichte der Mathematik Astronomie und Physik, Abteilung B, vol. 3 (1936), pp. 370–388.

Oskar Becker. Eudoxos-Studien IV. Das Prinzip des ausgeschlossenen Dritten in der griechischen Mathematik. Quellen und Studien zur Geschichte der Mathematik Astronomie und Physik, Abteilung B, vol. 3 (1936), pp. 370–388.

A system of formal logic without an analogue to the curry W operator

The system of formal logic to be presented in this paper has the following main properties:(1). The Mengenlehre paradoxes do not arise in it.(2). It seems to be free from other contradictions.(3). It does not resort to a theory of types.(4). It uses negation and material implication and does not weaken the principle of excluded middle.(5). It contains no analogue to the Curry W operator; consequently, given a propositional function ϕ, a propositional function Ψ cannot always be found such that, for all x, ϕ(x, x) and Ψ(x) are equivalent propositions.(6). It employs a universal quantifier, G, such that (x)f(x), (x)f(x, x), (x)f(x, x, x), …, would be respectively expressed thus, G(f, f), G(f, f, f), G(f, f, f, f), ….(7). It employs Schönfinkel's functional notation, so that f(x1, x2, …, xn) will be written as (( … ((fx1)x2) …)xn) and abbreviated to (fx1x2… xn) and, when no ambiguity results, to fx1x2…xn.(8). A theory of integers and real numbers cannot, apparently, be deduced from it.(9). A form of the axiom of choice is provable in it.