Abstract
In his ‘Logic without Ontology’1, Professor Ernest Nagel denies that logic attempts to characterize either the way men really think about the world or the real world they think about. In fact, he says, logic is normative only: it seeks to prescribe standards, not to describe the real, and therefor it lacks all ontological implications.
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References
Ernest Nagel, ‘Logic Without Ontology’, in Naturalism and the Human Spirit (ed. by Yervant H. Krikorian), New York 1944; reprinted in E. Nagel, Logic Without Metaphysics, Glencoe, I11., 1957.
Compare ‘analgesic’, ‘anastigmatic’, ‘anesthetic’, etc. I owe the neologism to my colleague, Professor Erazim Kohak.
See, for instance, Rudolf Carnap, Philosophy and Logical Syntax, London 1935, pp. 58ff.
See W.V. Quine, Word and Object, New York and London 1960, pp. 270–276.
I have borrowed the term from Hao Wang. See his ‘The Categoricity Question of Certain Grand Logics’, Mathematische Zeitschrift 59 (1953) 47–56, where he apparently uses the expression in the above sense. Compare ‘grand duke’, ‘grand total’ and ‘grand larceny’. The latter suggests ‘petty logic’ as a label for small-bore calculi, but perhaps the suggestion is invidious. Wang’s own source may be the title of the unfinished Grand Logic of C. S. Peirce. See Peirce’s Collected Papers, vol. VIII, 278–280.
See W.V. Quine, ‘Two Dogmas of Empiricism’, in From a Logical Point of View, Cambridge, Mass., 1953, pp. 20–46. Also, Word and Object, pp. 273–276.
The rule permitting the inference of e.g. ‘α = 1’, ‘α = 2’, ‘α =y’ from ‘(x) α = x’. Its validity stems from the fact that what is true of everything must be true of anything.
For a fuller statement of which see my ‘Paradox and Logical Uncertainty’, Philosophical Forum 15 (1957) 25–40.
See Whitehead and Russell, Principia Mathematica, Cambridge, 1st ed., 1910–1913, especially ‘Introduction, Ch. II’ and the ‘Prefatory Statement of Symbolic Conventions’ beginning volume II.
See F. P. Ramsey, The Foundations of Mathematics and Other Logical Essays, New York and London 1931, pp. 20–29.
W.V. Quine, ‘New Foundations for Mathematical Logic’, American Mathematical Monthly 44 (1937) 70–80.
W.V. Quine, Mathematical Logic, revised edition, Cambridge, Mass., 1951.
See Paul Bernays, ‘A System of Axiomatic Set-Theory’. This appeared by installments in the Journal of Symbolic Logic 2, 6, 7, 8, 13, 19.
Particularly Zermelo’s. See Ernest Zermelo, ‘Untersuchungen über die Grundlagen der Mengenlehre I’, Mathematische Annalen 65 (1908) 261–281.
For an account of the best known ones, see Whitehead and Russell, op. cit., vol. I, pp. 60–65.
See W.V. Quine, ‘On What There Is’, in From a Logical Point of View, pp. 1–19.
In what follows, no distinction has been made between existence and being. Various such distinctions could, of course, be made but none seems relevant to present purposes.
See Mathematical Logic §§ 24, 26, 27.
Alfred Tarski, ‘Der Wahrheitsbegriff in den formalisierten Sprachen’, Studia Philosophica 1 (1936) 261–405.
In some systems, in particular the system ML of Mathematical Logic and the system NNF introduced in Section VI above, individuals, which would ordinarily be non-classes, turn out to be classes of a certain sort, namely those which are their own sole members. Such odd classes might be termed improper classes, all others proper. The ordinary distinction of class from individual would then become the distinction of proper class from individual. For simplicity, I have retained the standard terminology; the reader may correct to fit the case. For the systems admitting improper classes, however, definitions (25) and (26) should be replaced as follows: (25’) \(Ux = _{df} \sim (y)(y = x \equiv y\varepsilon x)\) (26’) \( Px{ = _{df}}(y)(y = x \equiv y\varepsilon x) \).
Sometimes termed ‘the principle of extensionality for classes’. In symbols \((x)\,(y)\,(x = y \equiv (z)(z\varepsilon x \equiv z\varepsilon y)).\)
Usually designated by the Greek ‘∧’, defined as ‘x~(x =x)’ or the like.
For details and references, see my half of ‘The Ontological Significance of the Lowenheim-Skolem Theorem’, in Academic Freedom, Logic and Religion (ed. by Morton White), Philadelphia 1953, pp. 39–55.
The standard logical analysis of relations ignores the feeling, evidently pretty common, that they are somehow more concrete than properties or classes, and interprets them as classes of sequences. Thus a dyadic relation z is identified with the class of exactly those two-membered sequences or ordered pairs {x, y} each of whose first members x bears z to its second member y.
Once the dyadic relation z is taken as a class of ordered pairs, to say “x bears z to y” means that the ordered pair {x, y} is a member of z. In symbols, z(x,y) {x, y}εz.
It is possible to define relations which are in a sense heterogeneous with respect to type. For example, writing ‘[x]’ and ‘[x, y]’ for ‘the class whose sole member is x’ and ‘the class whose sole members are x and y’ respectively, the Wiener-Kuratowski definition essentially defines ‘{x, y}’ as ‘[[x], [x,y]]’. See Mathematical Logic, §36 for details. Where x and y are of like type, ‘{x, y}’ so defined is meaningful under type-theory restrictions: any class of such ordered-pairs {x,y} will be a homogeneous relation. Where x is of one type lower than y a new version of ordered pair is wanted: ‘{x, y}nn+1 = df [[[x]], [[x], y]]’ will serve to define it. Any class of such ordered pairs may be taken as a heterogeneous relation joining one object to an object one type higher. The difficulty here is that {x, y}nn+1 = {[x], y}, so that the heterogeneous relation of x to y is merely the homogeneous relation of [x] to y, and the system, instead of discussing two different relations, merely contains two different names for the one relation. Similarly with variants and elaborations of such schemes: the net result is to multiply terminology, yet still leave amid the ranks of the missing connections which the platonist would deem present.
See J.B. Rosser, ‘Definition by Induction in Quine’s “New Foundations for Mathematical Logic”’, Journal of Symbolic Logic 4 (1939) 80–81.
See E. Specker, ‘The Axiom of Choice in Quine’s “New Foundations for Mathematical Logic”’, Proceedings of the Academy of Science, U.S.A. 39 (1953) 972–975.
I have formulated the definitions and axioms to follow in terms of schemata using these schematic letters: ‘a’, ‘b’, ‘c’, ‘d’, whose instances are variables ‘w’, ‘x’, ‘y’, ‘z’, ‘w’, - etc.; ‘f’, ‘g’, ‘h’, whose instances are variables and abstracts; ‘p’ ‘q’, ‘r’, ‘s’, and ‘t’, whose instances are sentences. (‘Variables’ here means ‘variables to NNF’, analogously for ‘abstracts’ and ‘sentences’.) An instance of a schema is any expression formed by uniformly replacing all the schema’s schematic letters with instances of them. Two expressions are corresponding instances of two schemata if they are instances formed therefrom by the same replacements.
I have taken the chi, on the suggestion of my colleague, Mr. Walter Emge, from the Greek ‘ϰωρίζομαι’ meaning ‘to be separate from’.
Where x or y is an individual,’(xϰy)’ is translated as ‘x and y are distinct’; the effect in such cases is to make individuals improper classes. See note 20.
Any order will serve as alphabetic; for instance, that in which the variables are listed at the beginning of note 29.
The term was first used in this sense by Quine but the idea goes back to Russell. I have simply adapted Quine’s definition to fit NNF’s primitives. See From a Logical Point of View, p. 90f.
The reduction of NNF’s primitives to two was suggested by the similar reduction in Quine’s ‘Logic Based on Inclusion and Abstraction’, Journal of Symbolic Logic 2 (1937) 145–152. I have preferred exclusion to inclusion because the former simplifies the structure of alternate denial. Of NNF’s axioms and inferential rules, most have appeared in the literature in one status or another. (44) and (50) hark back to Nicod; see his ‘A Reduction in the Number of the Primitive Propositions of Logic’, Proceedings of the Cambridge Philosophical Society 19 (1916) 32–40. (45) and (46) are commonplace theorems of quantification-theory, and (51) is the familiar rule of universal generalization. (49) is simply a weakened form of the Principle of Abstraction. In developing NNF I have been aided considerably by the suggestions of three of my students - Victor Van Neste, Jr., Robert May dole, and Owen Gallagher. I am also grateful to Prof. Quine for pointing out to me the inadequacy of an earlier set of axioms. The distinction of normal from abnormal abstracts, which dominates NNF’s treatment of the logical paradoxes, is implicit in the ancillary system L2 of Halperin’s ‘A set of Axioms for logic’ [Journal of Symbolic Logic, 9 (1944) 1–19], which doubtless suggested it to me. Although differing in their axioms, rules of inference, and primitive notation, NNF and L2 are equivalent systems in the sense that the primitive expansion of any theorem of either forms the definitional abbreviation of some theorem of the other.
The proof may be sketched as follows. Where the instance of ‘c’ is as in (48), the latter yields (A) \(b\varepsilon \hat ap\, \supset \,(\exists c)(a)(a\varepsilon c \equiv p).\) By substitution \((a)(a\varepsilon c \equiv p)\, \supset \,\hat a(a\varepsilon c) = \hat ap,\) which is to say \((a)(a\varepsilon c \equiv p)\, \supset \,c = \hat ap.\) This successively implies \((a)(a\varepsilon c \equiv p) \supset (c = \hat ap\& (a)(a\varepsilon c \equiv p))\), \((\exists c)(a)(a\varepsilon c \equiv p) \supset (\exists c)(c = \hat ap\& (a)(a\varepsilon c \equiv p))\) and (B) \((\exists c)(a)(a\varepsilon c \equiv p)\, \supset \,(a)(a\varepsilon \hat ap \equiv p).\) From (A) and (B) together comes \(b\varepsilon \hat ap\, \supset \,(a)(a\varepsilon \hat ap \equiv p),\) thence \((\exists b)(b\varepsilon \hat ap)\, \supset \,(a)(a\varepsilon \hat ap \equiv p).\). From this the desired result follows by transposition since only the empty class has no members.
More accurately, ‘proper classes’. See note 20: the distinction made there is crucial here.
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© 1969 D. Reidel Publishing Company, Dordrecht, Holland
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Berry, G. (1969). Logic with Platonism. In: Davidson, D., Hintikka, J. (eds) Words and Objections. Synthese Library, vol 21. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1709-1_15
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