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Published in British Journal for the Philosophy of Science, Vol. V (1954), pp. 104-119. Reprinted in Gary Iseminger (ed.), Logic and Philosophy: Selected Readings, 1968.
I have given my essay this title because it roughly indicates the foundaries of the topic to be discussed and at the same time hints at the method that will be adopted in my analysis. The problem of existence will interest me only to the extent to which it enters the province of logical enquiry and I shall try to disentangle it a little by departing from the generally accepted interpretation of the quantifiers and by bringing in other concepts related to that of existence.
our hesitation can be traced to twofold causes. In the first place we may not be willing to give our judgment because we are not quite certain what we mean by 'electrons' or 'minds,' or we may not understand the word 'Pegasus.' In the second place we may be confused as regards the meaning of the term 'exist(s).' It is the latter cause of our embarrassment that calls for closer attention. Let the physicist, the psychologist, and the mythologist deal with the meaning of the words 'electrons,' 'minds,' and 'Pegasus' respectively. The logician's task, as I understand it, will be to establish the meaning of the constant term 'exist(s)' as it occurs in the function 'x exist(s)' where 'x' is a variable for which any noun-expression can be substituted.
I hope that it will be a permissible simplification to say that in recent times the discussion over the logical side of the problem of existence centres around what Professor Quine has written on the subject.2 In presenting the views of this author I shall have to use a little more quotation than is customary as the whole matter is of exceptional subtlety.
To say that something does not exist, or that there is something which is not, is I clearly a contradiction in terms; hence' (x)(x exists)' mustbe true.
Moreover we should certainly expect leave to put any primitive name of our language for the 'x' of any matrix '. . . x . . . ,' and to infer the resulting singular statement from '(x)(. . . x . . .)'; it is difficult to contemplate any alternative logical rule for reasoning with names.
From the whole passage quoted from page 150 of Mathematical Logic we seem to be entitled to conclude that for Quine (6), i.e. '(x)(x exists),' is a truth while (3), i.e. 'Pegasus exists,' is a falsehood. Regarding the rule which allows us to infer 'Fy' from '(x)(Fx),' Quine cautiously remarks that "it is difficult to contemplate any alternative logical rule for reasoning with names." He is more outspoken in his "Notes on Existence and Necessity," to which we now turn.
From (x)(x exists) we infer Pegasus exists by universal instantiation.
Inference II From Pegasus does not exist we infer (∃x) (x does not exist) by existential generalisation.
A remedy that might suggest itself to an unscrupulous mind would be to ban the use of empty noun-expressions and consider them as meaningless. Quine is right in not following this course. One may disagree as to the truth-value of the proposition 'Pegasus exists' but one would have to have attained an exceptionally high degree of sophistication to contend that the expression was meaningless. Quine does think that empty noun-expressions are meaningless just because they do not designate anything. He allows for the use of such words as 'Pegasus,' 'Cerberus,' 'centaur,' etc. under certain restrictions and tries to distinguish between logical laws which prove to be true for any noun-expressions, empty or non-empty, and those which hold for non-empty noun-expressions only. It follows from his remarks that before we can safely use certain laws established by logic, we have to find out whether the noun-expressions we may like to employ, are empty or not. This, however, seems to be a purely empirical question. Furthermore, all the restrictions which according to Quine must be observed whenever we reason with empty noun-expressions, will have to be observed also in ihe case of noun-expressions of which we do not know whether they are empty or not.
This state of affairs does not seem to be very satisfactory. The idea that some of our rules of inference should depend on empirical information, which may or may not be forthcoming, is so foreign to the character of logical enquiry that a thorough re-examination of the two infeences may prove to be worth our while. Let us then try to find out on what grounds (6) is asserted as true while (9) is rejected as false, and let us find out also on what grounds the rules of universal instantiation and existential generalisation are regarded as inapplicable to reasoning with empty names. In seeking answers to the above questions we shall turn to the interpretation of the quantifiers.
But this is not the only possible interpretation of the quantifiers. With the aid of the same fictitious example I shall now present an interpretation which as far as I can judge, is in harmony with the one adopted for instance by Lesniewski of the Warsaw School.
These two different interpretations of the quantifiers, which in what follows, will be referred to as the restricted interpretation and the unrestricted interpretation respectively, can now be generalised to apply to a universe with any number of objects. In the case of a finite universe we have finite expansions which are equivalent to their respective quantifications. If, however, we think of the universe as consisting of an infinite or unknown number of objects then we cannot have equivalences for the simple reason that we can never form complete expansions of our quantifications. Consequently we abandon equivalence in favour of implications. We say that an existential quantification is implied by any component of its infinite or unknown expansion and that a universal quantification implies any component of its infinite or unknown expansion. Now the expansions will vary depending on how we choose to interpret the quantifiers. Under the restricted interpretation every component of an expansion contains a noun-expression which designates only one of the objects belonging to the universe. Under the unrestricted interpretation every component of an expansion contains an expression of which we can only say that it is a meaningful noun-expression. It may designate only one of the objects belonging to the universe, it may designate more than one, or it may designate nothing at all.
It is not difficult to check that under their respective interpretations the corresponding expressions in the two lists yield the same truth value for the same substitutions performed on the free variables regardless of how we choose to interpret the predicate letters. The general rule for translating expressions is simple: expressions of type '(x)(Fx)' and '(∃x)(Fx)' become expressions of type '(x)(x exists ⊃ Fx)' and '(∃x)(x exists·Fx)' respectively; other expressions remain unchanged.
respectively. These translations fully account for the assertion of (6) and the rejection of (9) under the restricted interpretation of the quantifiers. Similarly (7a) and (10a) make it clear why (7) and (10) do not turn out true for all interpretations of 'F' and for all substitutions for the free variable. For if we interpret 'F' as 'exists' and substitute say 'Pegasus' for 'y,' the antecedent of (7a) becomes a tautology but at the same time the consequent turns out to be false. And again, if we interpret 'F' as 'does not exist' and substitute 'Pegasus' for 'y' then the antecedent of (10a) comes out true but the consequent must be rejected as a contradiction. Thus for certain interpretations of 'F' (7) and (10) turn out to be false if we substitute an empty noun-expression for the variable. Consequently the rules of universal instantiation and existential generalisation, which derive their validity from (7) and (10), can no longer be applied without restrictions.
which are valid if the universe is not empty, fail for the empty universe as their truth depends on there being something.9 When discussing these laws Quine tries to dismiss the case of the empty universe as relatively pointless and reminds us that in arguments worthy of quantification theory the universe is known or confidently believed to be nonempty.10 This contention, however, does not quite remove our uneasiness particularly as (16) and (17), not unlike (7) and (10), are demonstrable in quantification theory.
which are the corresponding translations of (16) and (17) to be understood in the light of the unrestricted interpretation. If there exists nothing then (16a) and the consequent of (17a) are obviously false while the antecedent of (17a) is obviously true. Under the unrestricted interpretation, however, (16) and (17) come out to be true irrespective of whether the universe is empty or non-empty. For (16) is implied by any component of type 'Fa ∨ ∼Fa' where 'a' stands for a noun-expression. In particular it is implied by a component. 'Fa ∨ ∼Fa' in which 'a' stands for an empty noun-expression. Such a component is true for all choices of universe and so is (16). In the case of (17) we argue as follows: if we asume that the antecedent of (17) is true then a proposition of type 'Fa' where 'a' stands for an empty noun-expression must also be true in harmony with the unrestricted interpretation of the universal quantifier. Now any such proposition implies the proposition of type '(∃x)(Fx),' which again must be true. Thus in the establishing of the truth value of (16) and (17) the problem of whether the universe is empty or non-empty is altogether irrelevant on condition, of course, that we adopt the unrestricted interpretation of the quantifiers.
Finally, the unrestricted interpretation in comparison with the restricted one appears to me to be a nearer approximation to ordinary usage. Somehow we do not believe that everything exists and we do not see a contradiction in saying that something does not exist. It is only from logicians who favour the restricted interpretation that we learn that things are the other way round. We may further add that the unrestricted interpretation of the quantifiers is in complete harmony with the formal quantification theory. I do not know of any formulae which are demonstrable in the formal quantification theory and which, under the unrestricted interpretation, are not applicable to reasoning with empty noun-expressions or do not hold for universes of some specific size.
From the logical point of view there are two satisfactory methods of determining the meaning of a constant term. The one consists in setting forth a set of axioms for the term in question. The other is adopted whenever we give a definition of the term in question with the aid of other terms whose meaning has already been determined axiomatically.11 We may add at once that we shall employ the latter method.
I shall read it 'all a is b' or 'all a's are b's.' I prefer doing this because ordinary inclusion seems to be more intuitive to an English speaking reader than Lesniewski's singular inclusion. Thus for instance ordinaryl inclusion has recently been used by Woodger in his "Science without Properties"13 for the purpose of constructing a language whose general tendency approximates the tendencies embodied in Ontology.
'Λ' can be defined in terms of inclusion but for the sake of simplicity I prefer to introduce it as an undefined term with the aid of (20).
For the present we need not trouble ourselves with the question how to read the newly defined functors. We can proceed straight on to the consequences which can be deduced from (20) and the three definitions.
Now (27), (28), (30), (31), (33), and (34) show that the functors 'ex' and 'ob' are very close approximations of 'exist(s).' We remember that under the unrestricted interpretation of the quantifiers '(x)(x exists)' is false and so is '(a)(ex(a))' and '(a)(ob(a))' as is evident from (28) and (31). Under the same interpretation '(∃x) (x does not exist)' comes out true and so does '(∃a)(∼(ex(a)))' and '(∃a)(∼(ob(a)))' as is evident from (27) and (30).
We have already remarked that under the restricted interpretation every component of quantificational expansions contains a noun-expression which designates only one of the objects belonging to the universe. It therefore follows that the function 'x exists' as used by us when we discussed the two interpretations of the quantifiers, means in fact the same as 'there exists exactly one x' which is the rendering of the symbolic 'ob(x).' The functor 'ex,' on the other hand, appears to be a very close approximation of the 'exist(s)' as used in ordinary language as it forms true propositions with noun-expressions which may designate more objects than one.
I wish to conclude with a brief summary of the results. The aim of the paper was to analyse rather than criticise. I started by examining two inferences which appeared to disprove the validity of the rules of universal instantiation and existential generalisation in application to reasoning with empty noun-expressions. Then I distinguished two different interpretations of the quantifiers and argued that under what I called the unrestricted interpretation the two inferences were correct. Further arguments in favour of the unrestricted interpretation of the quantifiers were brought in, and in particular it was found that by adopting the unrestricted interpretation it was possible to separate the notion of existence from the idea of quantification. With the aid of the functor of inclusion two functors were defined of which one expressed the notion of existence as underlying the theory of restricted quantification while the other approximated the term 'exist(s)' as used in ordinary language.
It may be useful to supplement this summary by indicating some aspects of the problem of existence which have not been included in the discussion. I analysed the theory of quantification so far as it was applied in connection with variables for which noun-expressions could be substituted and my enquiry into the meaning of 'exist(s)' was limited to cases where this functor was used with noun-expressions designating concrete objects or with noun-expressions that were empty. It remains to explore, among other things, in what sense the quantifiers can be used to bind predicate variables and what we mean when we say that colours exist or that numbers exist. These are far more difficult problems, which may call for a separate paper or rather for a number of separate papers.
1 The first draft of this paper was presented to a post-graduate seminar at the School of Economics on 12th November 1953, and was also read and criticised by Professors T. Lukasiewicz. K. R. Popper, W.V. Quine, and J. H. Woodger, from whose generous comments I have benefited much.
This list would have to be supplemented with the titles of several, more technical papers, published by Quine in The Journal of Symbolic Logic, and also with some passages from his Mathematical Logic, Cambridge (Mass.), 1947 (second printing), and Methods of Logic, London, 1952.
4 W. V. Quine, Mathematical Logic, Cambridge (Mass.), 1947, 150.
5 See W. V. Quine, "Notes on Existence and Necessity," The Journal of Philosophy, New York, 1943, 40. The inference has been rephrased so that it may conform with the example taken from Mathematical Logic. The original propositions are 'There is no such thing as Pegasus' and '∃x (there is no such thing as x),' respectively. See op. cit. 116.
6 See W. V. Quine, op. cit. 116.
7 See W. V. Quine, Methods of Logic, London, 1952, 88.
8 Sec W. V. Quine, "Notes on Existence and Necessity," The Journal of Philosophy, New York, 1943, 40, 118.
9 For details see W. V. Quine, From a Logical Point of View, Cambridge (Mass.), 1953, 160 sq.
11 See J. Lukasiewicz "The Principle of Individuation," Proceedings of the Aristotelian Society, Sup. Vol. 27, London, 1953, 77 sq.
12 See S. Lesniewski, 'Über die Grundlagen der Ontologie,' Comptes rendus des seances de la Societe des Sciences et des Lettres de Varsovie, Classe III, 1930, 23. For a brief account of Ontology see L. Ajdukiewicz, 'On the Notion of Existence,' Studia Philosophica, Posnaniae, 1951, 8 sq., or J. Lukasiewicz, 'The Principle of Individuation.' Proceedings of the Aristotelian Society, Supplementary Volume XXVII, London, 1953, 77 sq.
13 See this Journal, 1951, 2.
14 Hobbes wrote: 'Ut qui dicit homo est animal intellegi ita vult ac si dixisset "si quem recte hominem dicimus eundem etiam animal recte dicimus".' See 'Leviathan,' Opera Philosophica, iii, 497 (Molesworth); see also 'De Corpore,' Opera Philosophica i, 27 (Molesworth).
(a,b)((a ⊂ b) ≡ (c,d)(∼(c ⊂ d)·(c ⊂ a) ⊃ (∃)e,f)(∼(e ⊂ f)·(e ⊂ c)·(e ⊂ b)· (g,h,i)(∼(h ⊂ i)·(g ⊂ e)·(h ⊂ e) ⊃ (g ⊂ h))))).
See T. Kotarbinski, Elementy teorji poznania, logiki formalnej i metodologji nauk (Elements of Epistemology, Formal Logic, and Methodology), Lwow, 1929, 235 sq.; see also K. Ajdukiewicz, 'On the Notion of Existence,' Studia Philosophica, Posnaniae, 1951, 4, 8, and J. Lukasiewicz, "The Principle of Individuation," Proceedings of the Aristotelian Society, Supplementary Volume XXVII, London, 1953, 79 sq.
17 See W. V. Quine, "Designation and Existence," in Readings in Philosophical Analysis, edited by H. Feigl and W. Sellars, New York, 1949, 44 sq.
Transcribed into hypertext by Andrew Chrucky by request from William J. Greenberg, July 24, 2004.

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