The Construction of Logical Space


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Agustín Rayo

It is sufficiently obvious, in any particular case, that a value of a function does not presuppose the function.

If this were not the case, no function could be apprehended at all, since the number of values true and false of a function is necessarily indefinite and there are necessarily possible arguments with which we are not acquainted. Russell —13, 39— To classify propositions and even propositional functions as instances of the same logical phenomenon as definite descriptions requires a considerable broadening of the notion.

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If some things that are said by Russell in IV and V of his Introduction to the second edition may be taken as an indication of what is intended, it is probable that the contextual definitions would not stand scrutiny. Many passages in [ Russell ] and [ Whitehead and Russell —13 ] may be understood as saying or as having the consequence that the values of propositional functions are sentences.

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And since the passages in question seem to involve confusions of use and mention or kindred confusions that may be merely careless, it is not certain that they are to be regarded as precise statements of a semantics. Church , n. Gregory Landini has proposed that there is indeed a coherent semantics for propositions and propositional functions in PM, which treats functions and propositions as linguistic entities. Propositions and propositional functions are unlike definite descriptions and classes in that there are no explicit definitions of them in PM.

Early critical discussions of constructions, such as Wisdom , stressed the contrast between logically proper names, which do refer, and constructions, which were thus seen as ontologically innocent.

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Beginning with The Problems of Philosophy in , Russell turned repeatedly to the problem of matter. As has been described by Omar Nasim , Russell was stepping into an ongoing discussion of the relation of sense data to matter that was being carried on by T. Nunn , Samuel Alexander , G. Stout , and G. Moore , among others. In The Problems of Philosophy Russell Russell argues that the belief in the existence of matter is a well supported hypothesis that explains our experiences.

Russell saw an analogy between the case of simply hypothesizing the existence of numbers with certain properties, those described by axioms, and hypothesizing the existence of matter. Russell followed the progression through the exact sciences from logic through arithmetic, and then real numbers and then to infinite cardinals.

This rational Dynamics , would involve justifying many of the fundamental principles of physics with pure mathematics alone, from definitions that yield the geometry of space and time and the formal properties of its occupants, quantities of matter and energy. Influenced by William James, Russell came to defend a neutral monism by which matter and minds were both to be constructed from sensibilia, but in different ways.

Russell saw some support for this in the theory of relativity, and the fundamental importance of frames of reference in the new physics. Russell had been motivated by the logicist project of finding definitions and elementary premises from which mathematical statements could be proved. Stebbing and Wisdom were concerned, rather, with relating the notion of construction to philosophical analysis of ordinary language. We may sense patches of color next to each other in our visual field, but what that tells us about the causes of those sense data, about matter, is only revealed by the structure of those relationships.

Thus the color of a patch in our visual field tells us nothing about the intrinsic properties of the table that causes that experience. Instead it is the structural properties of our experiences, such as their relative order in time, and which are between which others in the visual field, that gives us a clue as to the structural relationships of time and space within the material world that causes the experience. See the entry on structural realism. That project was transformed, Demopoulos and Friedman argue, by replacing inference from the given in experience to the cause of that experience with an inference to the rather impoverished, structural, reality of the causes of those experiences.

Logical Constructions

According to Demopoulos and Friedman, Newman shows that there must be more to scientific theories than trivial statements to the effect that matter has some structural properties isomorphic to those of our sense data. The notion of logical construction had a great impact on the future course of analytic philosophy. One line of influence was via the notion of a contextual definition, or paraphrase, intended to minimize ontological commitment and to be a model of philosophical analysis. The distinction between the surface appearance of definite descriptions, as singular terms, and the fully interpreted sentences from which they seem to disappear was seen as a model for making problematic notions disappear upon analysis.

Wisdom proposed this application of logical construction in the spirit of Wittgenstein. A more technical strand in analytic philosophy was influenced by the construction of matter. The supreme maxim in scientific philosophizing is this: Whenever possible, logical constructions are to be substituted for inferred entities. Carnap , 6. More generally, the use of set theoretic constructions became widespread among philosophers, and continues in the construction of set theoretic models, both in the sense of logic where they model formal theories and to provide descriptions of truth conditions for sentences about entities.


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A pattern repeatedly illustrated in recent sections is that of the defective noun that proves undeserving of objects and is dismissed as an irreferential fragment of a few containing phrases. But some times the defective noun fares oppositely: its utility is found to turn on the admission of denoted objects as values of the variables of quantification.


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  • In such a case our job is to devise interpretations for it in the term positions where, in its defectiveness, it had not used to occur. Quine , In other words, that ordered pairs are distinguished by having unique first and second elements.

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    Quine then continues:. The problem is a neat one, for we have in 1 a single explicit standard by which to judge whether a version is suitable. Honest Toil 2. Logical Analysis and Logical Construction 3. Natural Numbers 4. Definite Descriptions 5. Classes 6.

    Series, Ordinal Numbers and Real Numbers 7. Mathematical Functions 8. Propositions and Propositional Functions 9. The Construction of Matter Whitehead with the solution to the problem of the relation of sense data to physics that he adopted in I have been made aware of the importance of this problem by my friend and collaborator Dr Whitehead, to whom are due almost all the differences between the views advocated here and those suggested in The Problems of Philosophy.

    Russell , Russell was referring to logical constructions in this frequently quoted passage from his Introduction to Mathematical Philosophy. Russell , 71 He charges that we need a demonstration that there are any objects which satisfy those axioms. Russell , In what follows these various features of logical constructions will be disentangled. Giuseppe Peano Peano , 94 had stated axioms for elementary arithmetic, which were later formulated by Russell , 8 as: 0 is a number. The successor of any number is a number. No two numbers have the same successor. Whitehead and Russell —13, Mathematical or descriptive functions are thus explicitly included among the incomplete symbols of Principia Mathematica.

    Whitehead and Russell —13, 43 Russell goes on to explain the multiple relation theory of judgment, which finds the place of propositions in this world of objects and qualities standing in relations. As a result Russell concludes that: It will be seen that, according to the above account, a judgment does not have a single object, namely a proposition, but has several interrelated objects. VII, — English translation in Russell , 3—38 and Russell , — In Russell , 39—56 and Russell , — In van Heijenoort , —82 and Russell , — Whitehead and B.

    Reprinted Oxford: Oxford University Press. Pears ed. Also in Russell , — and Russell , — Russell , — Marsh ed. Lackey ed. Here it is sufficient to recall that "men" as used in "men are mortal" refers to each individual man indirectly through the property of being human; hence, its significance does not depend upon acquaintance with each individual man. If the table is a class of appearances, then, in saying "This table is brown" there is a reference to each member of the class although the speaker is acquainted only with one member, namely, the appearance which is sensibly present to him, which Russell calls a 'sense-datum'.

    This line of reflexion leads to the conclusion that the table is not the sort of object which could be directly presented, so that the table could not be referred to demonstratively.


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    Russell, as was pointed out in Chapter IX, seems to assume that to say that a symbol is incomplete is to say that it does not name a constituent of the proposition in whose verbal expression it occurs, so that, from the fact that the table could not be named, he would derive the conclusion that the table is a logical construction. The second line of reflexion arises in the search for an indubitable datum.

    We may, in any given case, be mistaken in believing that we are seeming a table, for we may be victims of illusions or hallucinations.

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