Propositional Functions Research Articles

La distinction valeur générique/valeur non spécifique est-elle soutenable ?

The present article supports the combinatory (nonsemantic) distinction between the generic and nonspecific value of the indefinite noun phrase. The line of argument presented is the following : nonindividual concepts and terms taken out of context have a general value, which is their only inherent value, identical with their extension. Any other values associated with them appear in a given context, i.e. when they take on syntactic functions. The conclusion drawn is that these values are relational (combinatory). • Two types of syntactic functions are distinguished : attributive function (attributive use) or constitutive propositional function, and referential function (referential use) or function of binding the argument variable in the scope of another term. Therefore, the referential use of a term is identical with its restrictive function. The generic value of a general term is the result of its referential use (as the restrictor of an argument variable) in the scope of a general term, the latter being in attributive use. The nonspecific value of the same general term is generated by its attributive use. The only condition to be met is that the argument variable it has in its scope should be bound by a singular term incorporating a modal concept that excludes

Some Difficulties Concerning Russellian Intensional Logic

Although Russell never formalized the intensional logic implicit in Principles of Mathematics, Church has recently provided a reconstruction in the style of Principia Mathematica. A sign -for equality of propositions is added, synonymous with the (defined) identity sign of Principia but having the syntax of a connective. The resulting theory embodies a stringent criterion of identity of propositions and propositional functions (corresponding to Church's Alternative (0) for Fregean intensional logic (Church 1951, 1974)) and supposed protected from antimony by the ramified theory of types. In this paper I show that the resulting Russellian intensional logic allows the proof of a paradox about unit domains and that a principle embodying one of Russell's semantical ideas cannot be consistently added. These defects can be repaired by returning to a notion of propositional function closer to Russell's own. I outline the construction of models supporting this suggestion and explain the changes in the logic which are necessary or desirable given this semantical basis. I conclude with a discussion of an independent difficulty deriving from Russell's theory of descriptions and which seems to threaten the feasibility of the Russellian approach to intensional logic.

Simon Kochen and E. P. Specker. Logical structures arising in quantum theory. A reprint of XL 507(21). The logieo-algebraic approach to quantum mechanics, Volume I, Historicale evolution, edited by C. A. Hooker, The University of Western Ontario series in philosophy of science, vol. 5, D. Reidel Publishing Company, Dordrecht and Boston1975, pp. 263–276. - Simon Kochen and E. P. Specker.

Simon Kochen and E. P. Specker. Logical structures arising in quantum theory. A reprint of XL 507(21). The logieo-algebraic approach to quantum mechanics, Volume I, <b><i>Historicale evolution</i></b>, edited by C. A. Hooker, The University of Western Ontario series in philosophy of science, vol. 5, D. Reidel Publishing Company, Dordrecht and Boston1975, pp. 263–276. - Simon Kochen and E. P. Specker.

Pattern formation, logistics, and maximum path probability.

The concept of pattern formation, which to current researchers is a synonym for self-organization, carries the connotation of deductive logic together with the process of spontaneous inference. Defining a pattern as an equivalence relation on a set of thermodynamic objects, we establish that a large class of irreversible pattern-forming systems, evolving along idealized quasisteady paths, approaches the stable steady state as a mapping upon the formal deductive imperatives of a propositional function calculus. In the preamble the classical reversible thermodynamics of composite systems is analyzed as an externally manipulated system of space partitioning and classification based on ideal enclosures and diaphragms. The diaphragms have discrete classification

Sellars on Kantian intuitions

this is quite another matter. (B) Although Sellars does not explicitly say so, the demonstrative expression "this" as it occurs in the judgement 'this is a cube' seems to represent an empirical intuition in virtue of its "obvious connection" with 'this cube'. And it must be said that the demonstrative pronoun would seem to be a plausible choice to represent an empirical intuition for it expresses singularity and possibly immediacy and we can be said to subsume it under a concept by attaching it to a predicate ("This is a red circle"). It seems simply to represent an indeterminate, or at least undetermined, object which has submitted to the conditions of sensibility and therefore to be an adequate representation of an empirical intuition. There are important difficulties associated with this view, however, and the seemingly ingenuous demonstrative conceals more than might at first appear. A minimal prerequisite of an expression which might represent an empirical intuition is, if it refers at all, that it refer to at least and at most one object, for Kant insists throughout the Critique on the singularity of intuitional representation.7 And in order that this be achieved by a demonstrative expression, as I have suggested, certain conditions must be fulfilled. Hence, "this" may be said to refer to at least and at most one object if, it is added: "as uttered by speaker P at place L and at time T". But the only condition which Kant places upon intuitional representation is that, in order to be such, it must have submitted to the forms of sensibility. The introduction of the coordinates of speaker, place, and time is an introduction of conceptual determinations of the intuitional representation. The demonstrative pronoun must therefore be discounted as a possible candidate for the linguistic representation of an empirical intuition. (C) It may be further argued against the possibility of representing an empirical intuition linguistically that an intuition, however empirically indeterminate, is at least categorially determinate; that is to say, is a least something.8 If it is possi7 Cf. A3zo, B376-77, A7I3/B74I8 Az5o: "All our representations, it is true, are referred by the understanding to some object." SELLARS ON KANTIAN INTUITIONS 4 I 5 This content downloaded from 157.55.39.254 on Sun, 04 Sep 2016 06:31:21 UTC All use subject to http://about.jstor.org/terms ble to linguistically represent an intuition, the intuition will necessarily be represented as an object and will be seen, therefore, not merely to have submitted to the conditions of sensibility, but to the concept of an object in general. The expression "this", for example, represents an object, albeit empirically indeterminate, and cannot, therefore, represent an intuition for it already expresses a representation which has submitted to a minimal conceptual determination. Consider Sellars, however: "The view that before we can have representations of the form 'X is a cube' we must have representations of the form 'this cube' is a puzzling one." But this is only puzzling when the expression 'this cube' is said to represent an intuition, at least in the manner of referring to it, and I have been arguing against this. What ought to be said, a view to which Kant would subscribe, is that judgement is not possible without intuition and intuition cannot be represented apart from judgement. It is doubtful, in any case, that Kant would have accepted Sellars' formulation: "Before we can have judgement we must have intuition." Talk of having judgements and intuitions would certainly lead one to believe that we possess such "things". And the question here is whether an analytic device is being interpreted ontologically that is, whether the method of distinguishing elements is being mistaken for an interpretation which demands separating "things". In any case the point Sellars wishes to make here is that the expression 'this cube' is incomplete unless it forms part of a judgement such as 'this cube is a die'. It will be seen that this is a curious inversion of Frege's argument that the concept or propositional function 'X is a die' for example, is "incomplete" or "unsaturated" unless a value is found for the variable. And the trouble is that, as far as Sellars is concerned, there is no argument presented as to why it should be that a singular expression is incomplete or, indeed, precisely what this means.

Sellars on Kantian Intuitions

is quite another matter. (B) Although Sellars does not explicitly say so, the demonstrative expression this it occurs in the judgement 'this is a cube' seems to represent an empirical intuition in virtue of its obvious connection with 'this cube'. And it must be said that the demonstrative pronoun would seem to be a plausible choice to represent an empirical intuition for it expresses singularity and possibly immediacy and we can be said to subsume it under a concept by attaching it to a predicate (This is a red circle). It seems simply to represent an indeterminate, or at least undetermined, object which has submitted to the conditions of sensibility and therefore to be an adequate representation of an empirical There are important difficulties associated with view, however, and the seemingly ingenuous demonstrative conceals more than might at first appear. A minimal prerequisite of an expression which might represent an empirical intuition is, if it refers at all, that it refer to at least and at most one object, for Kant insists throughout the Critique on the singularity of intuitional representation.7 And in order that be achieved by a demonstrative expression, I have suggested, certain conditions must be fulfilled. Hence, this may be said to refer to at least and at most one object if, it is added: as uttered by speaker P at place L and at time T. But the only condition which Kant places upon intuitional representation is that, in order to be such, it must have submitted to the forms of sensibility. introduction of the coordinates of speaker, place, and time is an introduction of conceptual determinations of the intuitional representation. demonstrative pronoun must therefore be discounted a possible candidate for the linguistic representation of an empirical (C) It may be further argued against the possibility of representing an empirical intuition linguistically that an intuition, however empirically indeterminate, is at least categorially determinate; that is to say, is a least something.8 If it is possi7 Cf. A3zo, B376-77, A7I3/B74I8 Az5o: our representations, it is true, are referred by the understanding to some object. SELLARS ON KANTIAN INTUITIONS 4 I 5 This content downloaded from 157.55.39.254 on Sun, 04 Sep 2016 06:31:21 UTC All use subject to http://about.jstor.org/terms ble to linguistically represent an intuition, the intuition will necessarily be represented an object and will be seen, therefore, not merely to have submitted to the conditions of sensibility, but to the concept of an object in general. expression this, for example, represents an object, albeit empirically indeterminate, and cannot, therefore, represent an intuition for it already expresses a representation which has submitted to a minimal conceptual determination. Consider Sellars, however: The view that before we can have representations of the form 'X is a cube' we must have representations of the form 'this cube' is a puzzling one. But is only puzzling when the expression 'this cube' is said to represent an intuition, at least in the manner of referring to it, and I have been arguing against this. What ought to be said, a view to which Kant would subscribe, is that judgement is not possible without intuition and intuition cannot be represented apart from judgement. It is doubtful, in any case, that Kant would have accepted Sellars' formulation: Before we can have judgement we must have intuition. Talk of having judgements and intuitions would certainly lead one to believe that we possess such things. And the question here is whether an analytic device is being interpreted ontologically that is, whether the method of distinguishing elements is being mistaken for an interpretation which demands separating things. In any case the point Sellars wishes to make here is that the expression 'this cube' is unless it forms part of a judgement such 'this cube is a die'. It will be seen that is a curious inversion of Frege's argument that the concept or propositional function 'X is a die' for example, is incomplete or unsaturated unless a value is found for the variable. And the trouble is that, far Sellars is concerned, there is no argument presented to why it should be that a singular expression is or, indeed, precisely what means.

Foreword

In this paper, we will try to shed new light on the ‘Ezafe’ construction in Kurdish.1 Drawing on historical facts and tracing the Ezafe back to its origins on the one hand, and the unified relation of adjectives and possessors to the head nouns, namely modification, on the other hand, we will argue for the existence of the predicate phrase which Ezafe realizes its head above the NP. The occurrence of the Ezafe will be attributed to the presence of such a functional head which serves to turn a property expression, assigned to the modifier constituent XP, into a propositional function which in turn needs to be saturated by an argument, that is the head noun. Extending the syntactic realization of predication from clausal domain to nominal domain is part of a broader project, which aims at further assimilation of the structure of DPs to clauses.

Some reflections on predestination, providence and divine foreknowledge

It might appear to be beyond question, for Christian theism, both that God is omniscient and that omniscience includes knowledge of future truth. For it seems obvious that if P is true, then an omniscient being knows that P. P, in this propositional function, is entirely general, and must therefore include propositions of the form: ‘it will be the case that X’. If, truly, it will be the case that X, then God knows that truth.

Characterising Self-deception

In the introduction to the second volume of his collected papers Ryle tells us that though he himself was 'not qualified to be a real logician' he had realised that 'advances in Logic would and should result in the re-shaping of the questions, answers and especially the arguments of philosophers'. His view was that 'brokers were needed . . . to facilitate transactions between Logic and the philosophy of mind, between Logic and the theory of Sense/Nonsense and even between Logic and the should-be theory of pedagogy' (pp. vii-viii). Two obvious places in which we can see his brokership in action are his celebrated articles 'Systematically Misleading Expressions' (I932) and 'Categories' (I938). The first is largely an exploration of Russell's idea of an incomplete symbol and the second an application of his notion of a propositional function. It is no less clear that Russell's inventions are operating in the article 'Imaginary Objects' (I933). They generate there a view about characters in literature which, despite the fact that Russell's theory of descriptions is not now much in favour, seems still to be widely held. In Ryle's view, when a novelist creates one of his characters what in effect he does is to compose a complex description and then maintain that someone of whom it is a description did, or had done to him, whatever in the novel he is supposed to have done, or had done to him.

Representing and locating deduction rules in a semantic network

A semantic network is defined with its arcs and nodes separated into various sets. Arcs are partitioned into descending, ascending, and auxiliary arcs. Nodes are partitioned into base, variable, assertion, pattern and auxiliary nodes. Nodes can be temporary or permanent.Some pattern and assertion nodes, called rule nodes, represent propositional functions of the nodes they dominate. Rule nodes may bind the variables they dominate with any one of a set of binding relations representing quantifiers. A rule node which dominates variables all of which are bound is a constant deduction rule.Deduction rules may be viewed as pattern-invoked procedures. The type of propositional function determines the procedure, the variables bound by the rule are the local variables, and the quantifier determines the type of binding.A binding is defined as a list of variables associated with the nodes they are bound to. A binding can be used like a substitution, except it is seldom actually applied. Instead, a pattern node and a binding for it are used as a pair.A match routine is defined which is given a source node and a binding and finds target nodes, target bindings and more fully specified source bindings. Target nodes that are patterns provide entrees into relevant rules.

The ${\bf Q}$-consistency of ${\cal F}_{22}$.

In his [CSC],1 Curry proved the consistency of a system, which he there defines and calls 922, and which is closely related to the system 9$λ of [CLg.Il] §15C.2 This is essentially a type-free intuitionistic predicate calculus without conjunction, alternation, or negation but with quantification over propositions and propositional functions. However, Curry's consistency proof is rather weak, since it only proves that every theorem of the system belongs to a class of obs (terms) which are defined to be canonical (called canobs) and since the canobs are those obs which are to be interpreted as propositions this proof leaves open the possibility that every ob which is to be interpreted as a proposition is a theorem of the

The work of Kurt Gödel

I first heard the name of Kurt Gödel when, as a graduate student at Princeton in the fall of 1931, I attended a colloquium at which John von Neumann was the speaker, von Neumann could have spoken on work of his own; but instead he gave an exposition of Gödel's results of formally undecidable propositions [1931].Today I shall begin with Gödel's paper [1930] onThe completeness of the axioms of the functional calculus of logic, or of what we now often call “the first-order predicate calculus”, using “predicate” as synonymous with “propositional function”.Alonzo Church wrote ([1944, p. 62] and [1956, pp. 288–289]), “the first explicit formulation of the functional calculus of first order as an independent logistic system is perhaps in the first edition of Hilbert and Ackermann'sGrundzüge der theoretischen Logik(1928).” Clearly, this formalism is not complete in the sense that each closed formula or its negation is provable. (Aclosed formula, orsentence, is a formula without free occurrences of variables.) But Hilbert and Ackermann observe, “Whether the system of axioms is complete at least in the sense that all the logical formulas which are correct for each domain of individuals can actually be derived from them is still an unsolved question.” [1928, p. 68].This question Gödel answered in the affirmative in his Ph.D. thesis (Vienna, 1930), of which the paper under discussion is a rewritten version.I shall not describe Gödel's proof. Perhaps no theorem in modern logic has been proved more often than Gödel's completeness theorem for the first-order predicate calculus. It stands at the focus of a complex of fundamental theorems, which different scholars have approached from various directions (e.g. Kleene [1967, Chapter VI]).

Response to Arnaud

Philosophers, and no doubt workers in other areas where problems are hard and theories soft, are prone to embark on ambitious and not really soundly motivated programs of reduction. An important modern example of this, I think, is the attempt to reduce relational to notional belief. There are frames of mind in which it seems that notional belief is basic and that relational belief is unintelligible unless it can be reduced to notional belief. One becomes enchanted with a picture of the mind in the light of which one can see how a mind can grasp, in the attitude of belief, a complete sentence or complete proposition, but one cannot see how a mind can grasp in any attitude a mixture made up partly of non-linguistic objects and partly of a predicate or propositional function. One wants to ask: how can a mind have one foot on extensions and the other on intensions? One may be put into this frame of mind by reflection on the idea that the sentence is the unit of meaning. Or one may be picturing a person's beliefs at a time as being like a map. Thus Ramsey: A belief of the primary sort is a map of neighbouring space by which we steer. It remains such a map however much we complicate it or fill in details ([2]: 238). Our beliefs are a map; but a map of France which had Paris and Vezelay and Mont Blanc on it would be France. Here I mean that the actual cities and the actual mountain, not symbols for them, are on the map, for that is what would be analogous to relational belief. Yet, when we let ourselves put pictures and preconceptions aside, take our language as it comes, and ask about the logical forms of belief sentences, we find that sentences which are most naturally represented as involving relational belief occur as frequently and make as much sense as those most naturally represented as involving notional belief. Moreover, with regard to its degree, the relation is, in Kenny's term, variably polyadic: Jones believes that someone loves someone, Jones believes of Ann that someone loves her, Jones believes of Ann and Stan that he loves her. This variable polyadicity of 'believes' formally parallels that of 'true of'.

Three contributions to the two-valued propositional calculus

Three chapters contain the results independent of each other. In the first chapter I present a set of axioms for the propositional calculus which are shorter than the ones known so far, in the second one I give a method of defining all ternary connectives, in the third one, I prove that the probability of propositional functions is preserved under reversible substitutions.

�ber die logische Struktur eines Nervensystems

This paper describes an attempt to apply the concepts of modern logic toward encompassing one sort of nervous system function, sensory perception, within a general theoretical framework. Starting from the biological role of a sense organ to receive factual situations of the environment, one may treat the individual receptor as an elementary proposition, and a nerve network as a molecular proposition. Taking the known neurophysiology of the system into consideration, one recognizes that the logical functions contained in these propositional forms are those of a many-valued logic. The neuronal principles of lateral and linear inhibition provided ground for two basic propositional functions from which a variety of further functions can be constructed. The elements of construction are due to the anatomical principles of nerve cell connections: linear, lateral, and recurrent.

An extension of the algebra of sets

We shall explain the aim which leads us in the construction of an extended system of the algebra of sets1. The symbol 1. {*:?(*)} denoting the set of these and only these elements of domain of the variable x which satisfy the propositional condition (propositional function or form) ?9 (x)" is in com? mon use nowadays, so that it is adopted in school courses of mathematics in many countries, and in Poland as well. This condition will be said to define the set 1. However, if we admit propositional conditions which are meaningless for some values of their variables then we encounter some difficulties connected with the ex? pression 1. The formulae 2. {x : 9 (*)} = {x : 9 (*)}' 3. {x : 9 (s) v <|/ 0)} = {x : 9 (*)} u {x : ty (*)} do not hold for such conditions2. E.g. if we assume x to run over the set of all real numbers and the set to be the universe of discourse then we obtain

The modular logic as a calculus of logical schemata

Several logical systems, with an intent of applying them to the experimental sen? tences of quantum mechanics, have been built. The two best known ones are: the logic of quantum mechanics defined by Birkhof f and v. Neumann [2] and known also as the modular logic and the calculus of partial propositional functions defined by Specker and Kochen [3]. The modular logic, though initiated 33 years ago is still subject of research and discussions. The paper [2] is one of the commonest ones quoted in the literature on mathematics.

Proof of axiomatizability of full many-valued systems of calculus of propositions

By a matrix3 we shall mean every ordering which ascribes a value of propositional function to an arbitrary set of values of its arguments. A system of calculus of propo? sitions will be said to have a matrix presentation if there are given matrices of all its primitive terms and the meaningful propositions which are numbered into the system are exactly these ones which for arbitrary sets of values of their arguments take dis? tinguished values only, according to matrices of functors which appear in the propo? sitions. Or ? to make it short ? the propositions numbered into the system are these and only these which are satisfied by appropriate matrices. If the primitive terms of a given system of calculus of propositions allow to define

On sets and propositional functions in grammar

On sets and propositional functions in grammar

The theoretical implications of multivariate analysis in the behavioral sciences.

We can start with a specific methodological impass: the method used by a field anthropologist and the method used by a sociologist versed in and proud of high-power statistics. The former looks for qualitative configuration of a case study, while the latter looks for quantitative generalization of sample cases. The same impass exists between psychiatrists and psychologists. Of course, historians abhor the barbarous term “case study,” let alone statistical generalizations. However, in the deceivingly simple propositional function, X = f(y, z…), both the qualitative configuration and quantitative generalization are taken care of. Anthropologists, psychiatrists, and historians must strive for generalizations. Those arrived at by sociologists and psychologists must be applicable to individual cases. Thus the formula, arrived at through multivariate analysis, can be used to formulate individual cases such that X1 = f(y, z…), X2 = f(y2, Z2…)…Xn = f(yn, zn…) with subscripts 1, 2,…n representing coefficients of the variables appropriate to individual cases. To generalize is not to explain away individuals. Rather, in sciences, individuals can be formulated in terms of the determinable variables of the general formula. Since the conventional wisdom suspects sciences usually on the grounds of concern for the uniqueness of individual persons, things, or events, the possible scientific formulation of individuals is stressed to offer consolation and relieve anxiety.