Frederic B. Fitch

An Extension of Basic Logic

A logical calculus Κ was defined in a previous paper and then shown in a subsequent paper to contain within itself a representation of every constructively definable subclass of expressions of a certain infinite class U of expressions, where Κ itself is one such subclass. (The former paper will be referred to as BL and the latter paper as RC.) The calculus Κ was called a “basic calculus” and its theorems were thought of as expressing the asserted propositions of a “basic logic,” that is, of a logic within which is definable every constructively definable system of logic and indeed every constructively definable class or relation. The notion of constructive definability was essentially equated with the notion of recursive enumerability.

Natural deduction rules for English

A system of natural deduction rules is proposed for an idealized form of English. The rules presuppose a sharp distinction between proper names and such expressions as ‘the c’, ‘a (an) c’, ‘some c’, ‘any c’, and ‘every c’, where ‘c’ represents a common noun. These latter expressions are called quantifiers, and other expressions of the form ‘that c’ or ‘that c itself’, are called quantified terms. Introduction and elimination rules are presented for any, every, some, a (an), and the, and also for any which, every which, and so on, as well as rules for some other concepts. One outcome of these rules is that ‘Every man loves some woman’ is implied by, but does not imply, ‘Some woman is loved by every man’, since the latter is taken to mean the same as ‘Some woman is loved by all men’. Also, ‘Jack knows which woman came’ is implied by ‘Some woman is known by Jack to have come’, but not by ‘Jack knows that some woman came’.

A Basic Logic

This paper is concerned with finding a fairly simple system of logic which is “basic” in the sense that every system of logic is definable in it. If a “system”is regarded as being a class of propositions, rather than a class of sentences, then every class of propositions which is a system should be definable in the basic logic. The system of logic proposed in this paper will not be proved to be a basic logic, but strong evidence that it is basic will be given at the end of the paper. Evidence will also be given that the class of propositions which are not provable in the system is not definable in any system of logic. It will be established that the decision problem is unsolvable for the system. Notable characteristics of the system are its lack of negation and universal quantification, and its similarity to systems proposed by Church, Curry, and Rosser.By a “system” will be meant a class of propositions the membership of which can be specified by ordinary recursive methods, so that if the membership of a given proposition in the class can be established at all, such membership can be established in a finite number of steps. (This is not the same as demanding that a criterion must exist for determining, in a finite number of steps, whether or not some given proposition is a member of the class. Such a demand would require a solution of the decision problem for every system.)

A Complete and Consistent Modal Set Theory

1.1. The aim of this paper is the construction of a demonstrably consistent system of set theory that (1) contains roughly the same amount of mathematics as the writers system K′ [3], including a theory of continuous functions of real numbers, and (2) provides a way for expressing in the object language various propositions which, in the case of K′, could be expressed only in the metalanguage, for example, general propositions about all real numbers. It was not originally intended that the desired system should be a modal logic, but the modal character of the system appears to be a natural outgrowth of the way it is constructed. A detailed treatment of the natural, rational, and real numbers is left for a subsequent paper.

A correlation between modal reduction principles and properties of relations

The aim of this paper is to generalize Kripke’s method of constructing models of systems of modal logic.1 A procedure is given for constructing models of inGnitely many different systems of modal logic. Among these systems are intlnitely many different systems of deontic logic, that is, systems in which [7p 2 Qp is a theorem for every p, but in which p 3 C)p is not a theorem for every p. In deontic systems the interpretation of q p is that p is obligatory (instead of necessary) and the interpretation of Op is thatp is permitted (instead of possible). By a reduction principle in modal logic will be meant any principle which can be expressed as q p3 op or as 0 . . . Opz F,F,...F,,p, where there are zero or more diamonds on the left and zero or more Fi on the right, and each Fi is either a diamond or a square. By a standard system of modal logic will be meant one which is obtained from two-valued propositional calculus by closing the class of sentences with respect to application of the necessity operator and by using zero or more reduction principles (with‘o’defined as ‘ 0 ‘), together with the principle, c] [p I q] 3 =) [Up1 q q], and the rule that if the necessity operator is prefixed to an axiom, the result is an axiom. (It would follow, for such a system, that if the necessity operator is prefixed to a theorem, the result is a theorem.) In order to construct a model of a standard system of modal logic by Kripke’s method, it is required that appropriate properties (e.g., perhaps transitivity, symmetry, etc.) be assigned to a relation R between ‘universes’ or ‘worlds’, where aRfi would assert that fi is a ‘possible universe’ relatively to the universe a, and where Op is a member of (true in) a universe a provided that p is a member of some universe to which a bears the relation R, and where Up is a member of a universe CI provided thatp is a member of every universe to which a bears the relation R. Formulas expressing the required properties of R (such as symmetry, etc.) corresponding to any given standard system of modal logic can be constructed in accordancewitha correspondence which exists betweensuchformulas and the reduction principles used in the given modal system, as will now be shown.2

A Method for Avoiding the Curry Paradox

The Curry paradox1 may be formulated in the following way:2 Let ‘Y q ’ serve as an abbreviation for ‘\({}^\backprime \hat x\left[ {\left[ {x \in x} \right] \supset q} \right]^\prime .\) ’ Then, using the method of subordinate proofs, we have:

Recursive Functions in Basic Logic

1.1 The system K * of basic logic, as presented in a previous paper, will be shown to be formalizable in an alternative way according to which the rule [ E ], is replaced by the rule [ F ], 1.2. General recursive functions will be shown to be definable in K * in a way that retains functional notation, so that the equation, will be formalized in K * by the formula, where ‘ f ’, ‘ p 1 ’, … ‘ p n ’ respectively denote φ, x 1 , …, x n , and where ‘≐’ plays the role of numerical equality. Partial recursive functions may be handled in a similar way. The rule [ E ] is not required for dealing with primitive recursive functions by this method. 1.3. An operator ‘ G ’ will be defined such that ‘[ Ga ≐ p ]’ is a theorem of K * if and only if ‘ p ’ denotes the Godel number of ‘ a ’. 1.4. In reformulating K * we assume ‘ o 0 ’, ‘ o 1 ,’ ‘ o 2 ’, …, have been so chosen that we can determine effectively whether or not a given U-expression is the m th member of the above series. The revised rules for K * are then as follows. (Double-arrow forms of these rules are derivable, except in the case of rule [V].)

A Demonstrably Consistent Mathematics--Part I

This paper continues the exposition of a demonstrably consistent foundation for mathematics begun in An extension of basic logic (hereafter referred to as EBL) and in The Heine-Borel theorem in extended basic logic (hereafter referred to as HB). Still earlier papers related to these are A basic logic (referred to as BL) and Representations of calculi (referred to as RC). The main conventions and results of all the above papers will be presupposed in what follows. The numbering of sections and paragraphs will be a continuation of that of EBL and HB.

The Heine-Borel Theorem in Extended Basic Logic

A demonstrably consistent theory of real numbers has been outlined by the writer in An extension of basic logic 1 (hereafter referred to as EBL). This theory deals with non-negative real numbers, but it could be easily modified to deal with negative real numbers also. It was shown that the theory was adequate for proving a form of the fundamental theorem on least upper bounds and greatest lower bounds. More precisely, the following results were obtained in the terminology of EBL: If С is a class of U -reals and is completely represented in Κ′ and if some U -real is an upper bound of С, then there is a U -real which is a least upper bound of С. If D is a class of ( U -reals and is completely represented in Κ′, then there is a U -real which is a greatest lower bound of D .

Closure and Quine's *101

The purpose of this paper is to suggest two alternatives to Quines definition of closure. These new definitions have two advantages over Quines definition, and they probably are the simplest definitions having both advantages. The two advantages are: (1). Principle *101 becomes superfluous and may be dropped from Quines set of principles for quantification. (In the case of my second definition, however, the dropping of *101 must be balanced by a slight change in *104.) (2). Closure is made independent of the alphabetical order of variables. The second of these advantages turns on the fact that the “alphabetical order” possessed by variables in virtue of their respective positions in the alphabet (or arbitrarily assigned to them) is a mere convention and not of genuine logical significance. It seems therefore desirable to consider some alternatives to Quines definition of closure, since according to his definition the closure of a given formula will be one statement or another, depending upon whether or not one letter of the alphabet is alphabetically prior to a certain other letter. It is interesting that the removal of this minor artificiality also enables us to dispense with *101. According to Quine, the closure of a formula containing n free variables is obtained by prefixing to it in alphabetical order the n universal quantifiers formed from these variables by enclosing each in a pair of parentheses. (If n = 0 the formula is its own closure and is a “statement” rather than a “matrix.”) Thus the statement ‘( x )( y )( z )( x ϵ y ▪ y ϵ z )’ would be the closure of ‘ x ϵ y ▪ y ϵ z ’, but ‘( z )( x )( y )( x ϵ y ▪ y ϵ z )’ would not be its closure. Now there is no reason why ‘( z )( x )( y )( x ϵ y ▪ y ϵ z )’ or ( y )( z )( x )( x ϵ y ▪ y ϵ z ) and so on, could not just as well be regarded as “the” closure of ‘( x ϵ y ▪ y ϵ z ’ as ‘( x )( y )( z )( x ϵ y ▪ y ϵ z )’. I therefore propose to allow to each formula not merely one closure, but as many closures as can be obtained by permuting in various ways the n prefixed universal quantifiers. In this way alphabetical order becomes irrelevant and no preference is given to one order of prefixed quantifiers in contrast to other orders which seem equally good. This constitutes my first redefinition of closure.