LOGIC AND EXISTENCE 1Czeslaw Lejewski
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.
When we have to commit ourselves to asserting or rejecting propositions like
(1) electrons exist
(2) minds exist
(3) Pegasus exists,
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.
On page 150 of Quine's Mathematical Logic we read: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.
Let us translate this argumenLinto a symbolic language so that its logical structure may become more perceptible. I think that the following translation stands a fair chance of being acceptable to Quine:
(4) ((∃x)(x does not exist) ≡ (∃x)( x exists. ∼(x exists))
If instead of 'x does not exist' in the antecedent of (4) we write '∼(x exists)' then from (4) and from the circumstance that the consequent of (4), being a contradiction, is false, we get immediately(5) (x)(x exists).which in accordance with the law relating the existential quantifier to its universal counterpart is equivalent to(6) (x)(x exists).
This result seems to confirm our interpretation of the passage, which is in complete harmony with the opening paragraph of "On What There Is" as this paragraph runs as follows:A curious thing about the ontological problem is its simplicity. It can be put into three Anglo-Saxon monosyllables: 'What is there?' It can be answered moreover in a word -- 'Everything' -- and everyone will accept this answer as true.'3
But let us revert to page 150 of Mathematical Logic. The passage which we began to analyse just now continues thus: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.
This logical rule, which in Methods of Logic is referred to by Quine as the rule of universal instantiation, owes its validity to a certain logical law which with the aid of symbols can be expressed in the following way:(7) (x)(Fx) ⊃ Fy.
Now difficulties begin to appear and Quine sets them out as follows:But this rule of inference leads from the truth '(x)(x exists)' not only to the true conclusion 'Europe exists' but also to the controversial conclusion 'God exists' and the false conclusion 'Pegasus exists' if we accept 'Europe,' 'God,' and 'Pegasus' as primitive names in our language.4
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.
In that paper Quine discusses inferences which would be exemplified by the one whereby from(8) Pegasus does not existwe infer(9) (∃x)(x does not exist)in virtue of the rule which allows us to infer '(∃x)(Fx)' from 'Fy.'5 In "Notes on Existence and Necessity" and in Methods of Logic this rule is described as the rule of existential generalisation, and we may add at once that it derives its validity from the following logical law:(10) Fy ⊃ (∃x)(Fx).According to Quine (8) would be true but (9) would be false. Regarding the rule he observes the following:The idea behind such inference is that whatever is true of the object designated by a given substantive is true of something; and clearly the inference loses its justification when the substantive in question does not happen to designate.6
I think that we have come to a point where a brief summary of Quine's argument may not appear to be superfluous. We have two inferences:
(x)(x exists)we infer Pegasus existsby universal instantiation.
Inference IIFrom Pegasus does not existwe infer (∃x) (x does not exist)by existential generalisation.
According to Quine both these inferences are objectionable to our intuitions. In his opinion they lead from truths, '(x)(x exists)' and 'Pegasus does not exist,' to falsehoods, 'Pegasus exists' and '(∃x) (x does not exist).' In other words the logical laws (7) and (10), which we thought to be behind the rule of universal instantiation and the rule of existential generalisation, do not hold for every interpretation of 'F' and every substitution for the free variable. If we interpret 'F' as 'exists' and substitute 'Pegasus' for 'y,' (7) turns out to be false. Similarly, (10) turns out to be false for the same substitution if we interpret 'F' as 'does not exist.' But no difficulty arises if the noun-expressions substituted for 'y' in (7) and (10) are non-empty:(11) (x)(x exists) ⊃ Socrates existsare true propositions.
(12) Socrates does not exist ⊃ (∃x)(x does not exist)
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.
Regarding Quine's interpretation of the quantifiers, which is the one accepted by the majority of modern logicians, we have a very useful passage in the Methods ojf Logic. It reads as follows: If we think of the universe as limited to a finite set of objects a, b, . . . , h, we can expand existential quantifications into alternations and universal quantifications into conjunctions; '(∃x)Fx' and '(x)Fx' become respectively:Fa ∨ Fb ∨ . . . ∨ Fh,
Fa.Fb . . . . . Fh7
To have a still simpler though fictitious example let us think of the universe as limited to two objects a and b. Then the corresponding expansions would be:Fa ∨ Fb and Fa · Fb
Our language, which for reasons of simplicity needs not synonyms, may leave room for noun-expressions other than the singular names 'a' and 'b.' We may wish to have a noun-expression 'c' which would designate neither of the two objects, in other words which would be empty, and also a noun-expression 'd' which would designate either. Introducing noun-expre;ssions is a linguistic matter. It does not affect our assumed universe, which continues to consist of a and b only. The new noun-expressions can now be put to use. For instance we can form the following true proposition; 'c does not exist,' and on turning 'd' into a predicate-expression 'D' we can further assert that '(x)(Dx)' is true. To say that something exists means the same as to say that it belongs to the universe. Thus 'a exists' and 'b exists' are true propositions. From the expansion of the existential quantification we see that either of these two propositions implies '(∃x)(x exists).' But we have no ground to contend that 'c does not exist' implies '(∃x)(x does not exist).' Again since 'a does not; exist or b does not exist' is false, we conclude from the expansion that '(∃x)(x does not exist)' is also false. Hence '(x)(x exists)' is true. This is confirmed by the expansion of the universal quantification and in view of the circumstance that 'a exists and b exists' is true. We also learn from this expansion that '(x)(x exists)' does not imply 'c exists.' Within our fictitious universe the rule of universal instantiation and the rule of existential generalisation, fail if applied to 'c' or 'd.' They are vadid rules of inference if their application is restricted to reasoning with 'a' and 'b.'
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.
Our universe consisting of a and b remains the same but the quantifiers read differently. Under the present interpretation '(∃x)(Fx)' becomes:Fa ∨ Fb ∨ Fc ∨ FdCorrespondingly, '(x)(Fx)' is to mean:Fa·Fb·Fc·FdThus one can argue that '(∃x)(x does not exist)' is true because 'c does not exist,' which under the new interpretation is one of the components of the existential expansion, is true. Hence '(x)(x exists)' is false. This is confirmed by the corresponding universal expansion, which contains a false component, namely 'c exists.' Under this interpretation the rule of universal instantiation and the rule of existential generalisation are valid without any restrictions. They can be safely applied in reasoning with any noun-expressions: singular non-empty like 'a' and 'b,' empty like 'c,' or general non-empty like 'd.' The noun-expression 'd' need not be changed into the corresponding predicate-expression 'D.'
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.
The two interpretations of the quantifiers give rise to two different theories of quantification and we may well be expected to say a few words on the relation in which one theory stands to the other. In this respect the most important point is that whatever is said with the aid of the theory of restricted quantification, can be easily expressed in terms of the unrestricted qnantification provided we are allowed to use the notion of existence. A few examples set out below in the form of two lists will suffice to illustrate the procedure:
I. Expressions to be understood in the light of the restricted interpretation:(7) (x)(Fx) ⊃ Fy
(10) Fy ⊃ (∃x)(Fx)
(13) (x)(Fx) ≡ ∼(∃x)(∼Fx)
II. Corresponding translations to be understood in the light of the unrestricted interpretation:(7a) (x)(x exists ⊃ Fx) ⊃ Fy
(10a) Fy ⊃ (∃x)(x exists · Fx)
(13a) (x)(x exists ⊃ Fx) ≡ ∼(∃x)(x exists · ∼Fx)
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.
Thus, for instance, (6) and (9) translated for the purpose of the unrestricted interpretation become(14) (x)(x exists ⊃ x exists)and(15) (∃x)(x exists·x does not exist)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.
The position is different if we choose to understand the quantifiers in the light of the unrestricted interpretation. (3), which is 'Pegasus exists,' and (8), which is 'Pegasus does not exist,' are meaningful propositions either of which contains a noun-expression, viz. 'Pegasus.' 'Thus (3) and (8) may be regarded as components of quantificational expansions. Now, (3) being false, the corresponding universal quantification, i.e. '(x)(x exists),' which ought to imply any component of its expansion, must also be false. On the other hand, (8) being a true proposition the corresponding existential quantification, i.e. '(∃x)(x does not exist),' which is implied by any component of its expansion, must also be true. Thus under the unrestricted interpretation of the quantifiers the two inferences [i.e., Inference I and Inference II above] cannot be used as counterexamples to disprove the validity of the rules of universal instantiation and existential generalisation in application to reasoning with empty noun-expressions. In Inference I both the premise and the conclusion are false, while in Inference II both the premise and the conclusion are true. There is nothing wrong with the inferences provided we adopt the unrestricted interpretation of the quantifiers. Furthermore, under the unrestricted interpretation the logical laws (7) and (10) turn out to be universally true. For every proposition of type 'Fa' where 'a' stands for any noun-expression, empty or non-empty, is now regarded as a component of the quantificational expansions and consequently is implied by the corresponding proposition of type '(x)(Fx)' and implies, in turn, the corresponding proposition of type '(∃x) (Fx).' (7) and (10) being universally true, the rule of universal instantiation and the rule of existential generalisation are universally valid as the principles that are behind them are no longer principles by courtesy.8
The unrestricted interpretation of the quantifiers seems to remove yet another difficulty from quantification theory. It has been argued by several authors that(16) (∃x)(Fx ∨ ∼Fx)and(17) (x)(Fx) ⊃ (∃x)(Fx),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.
On considering (16) and (17) we readily admit that these two formulae fail for the empty universe if we understand the quantifiers in accordance with the restricted interpretation. This becomes evident from(16a) (∃x)(x exists·(Fx ∨ ∼Fx))and(17a) (x)(x exists ⊃ Fx) ⊃ (∃x)(x exists·Fx),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.
It ought to be evident from what has already been said that under the unrestricted interpretation existential quantifications have no existential import. In fact it would be misleading to read '(∃x)(Fx)' as 'there exists an x such that Fx.' The non-committal 'for some x, Fx' seems to be more appropriate. Similarly the terms 'existential quantification' and 'existential quantifier' no longer apply and could be convenietly replaced by such expressions as 'particular quantification' and 'particular quantifier.' The rule of existential generalisation could perhaps be referred to as the rule of 'particular generalisation.'
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.
When we consider (7), (10), (13), (16), and (17) as understood in light of the restricted interpretation and compare them with their corresponding translations for the purpose of the unrestricted interpretation, i.e. with (7a), (10a), (13a), (16a), and (17a), we cannot fail to notice that the idea of the restricted quantification is not a simple one. The translations reveal that it can be analysed into two separate constituents: the idea of the unrestricted quantification on the one hand and the notion of existence on the other. In my opinion the most serious disadvantage of the theory of the restricted quantification is that by merging the idea of quantification with the notion of existence it has put logicians and philosophers on a wrong track in their endeavours to elucidate the problem of existence in logic. In what follows we shall adhere to the theory of unrestricted quantification and we shall attack the problem of existence in logic by determining the meaning of the constant term 'exist(s)' as used in the function 'x exist(s).'
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.
The meaning of 'exist(s)' can best be determined on the basis of the logic of noun-expressions constructed as a deductive system by Lesniewski in Warsaw in 1920 and called by him 'Ontology.'12 The original system of Lesniewski's Ontology is based on singular inclusion (a is b or in symbols a ∈ b) as the only primitive function. For various reasons, however, I prefer to continue my analysis of 'exist(s)' with reference to a system of Ontology based on ordinary inclusion, which I shall write in the following manner:(18) a ⊂ bI 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.
The functor of ordinary inclusion is a proposition forming functor for two arguments either of which is a noun-expression. If in (18) we substitute constant noun-expressions for the variables 'a' and 'b' then the result of the substitution will be true if and only if everything named, or designated, by the noun-expression substituted for 'a' is also named by the noun-expression substituted for 'b.' It may be of some historical interest to mention that the above semantic characterisation of inclusion be traced back to Hobbes who used it in order to determine the meaning of the copula 'est' in propositions such as 'homo est animal.'14
From the semantic characterisation of inclusion15 it is evident that the following propositions are true:man ⊂ animal
man ⊂ man
Socrates ⊂ man
Socrates ⊂ Socrates
Pegasus ⊂ animal
Pegasus ⊂ Socrates
Pegasus ⊂ Pegasus
The last three propositions are true because nothing is designated by 'Pegasus.' Thus whatever is designated by 'Pegasus,' is designated by anything you like. The corresponding proposition can be formulated in symbols as follows:(19) (a)(Pegasus ⊂ a)
If instead of 'Pegasus' in (19) we write the constant noun-expression 'Λ' which designates nothing, then we shall get the following ontological thesis:(20) (a) (Λ ⊂ a)'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).
In order to determine the meaning of 'exist(s)' we shall need three definitions, which I write below in the form of equivalences:(21) (a)(ex(a) ≡ (∃b)(∼(a ⊂ b)))
(22) (a)(sol(a) ≡ (b,c,d)(∼(c ⊂ d) · (b ⊂ a) · (c ⊂ a) ⊃ (b ⊂ c)))
(23) (a)(ob(a) ≡ ex(a)·sol(a))16
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.
Thus from (21) we immediately get(24) ex(Λ) ≡ (∃a)(∼(Λ ⊂ a))Since from (20) we know that(25) ∼(∃a)(∼(Λ ⊂ a))we use (25) and (24) to show that(26) ∼(ex(Λ))From (26) we obtain, by particular generalization,(27) (∃a)(∼(ex(a)))which is equivalent to(28) ∼(a)(ex(a))From (23) and (26) we conclude that(29) ∼(ob(Λ))From (29) we get(30) (∃a)(∼(ob(a)))which is equivalent to(31) ∼(a)(ob(a))
We can now draw Pegasus into our deductions. From (19) we immediately obtain(32) ∼(∃a)(∼(Pegasus ⊂ a))and use it together with (21) to show that(33) ∼(ex(Pegasus))From (23) and (33) we derive(34) ∼(ob(Pegasus))
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).
Further evidence is supplied by (33) and (34) from which it follows that 'ex(Pegasus)' and 'ob(Pegasus)' are false just as 'Pegasus exists' is admittedly false. It remains then to be explained what is the difference in the meaning of 'ex' and 'ob.' We can find the required explanation if we consider the meaning of (22), which is a definition of 'sol.' The right hand side of this definition turns out to be true in two cases: if there is no such a thing as a or if there is only one a. Thus 'sol(a)' can be read as 'there is at most one a.' Now if we agree to read 'ex(a)' as 'a exists' or as 'a's exist' then in accordance with (23) 'ob(a)' will have to be read as 'there exists exactly one a,' which is equivalent to 'a is an object' or to 'a is an individual.' Thus the distinction between 'ex' and 'ob' roughly corresponds to the one made by Quine in his "Designation and Existence," where he talks about general existence, statements and singular existence statements.17
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.
2 Quine's most important contributions in this connection are the following: "A Logical Approach to the Ontological Problem," Journal of Unified Science, Chicago, 1940, 9. This paper was read at the Fifth International Congress for the Unity of Science, Cambridge (Mass.), 1939; "Designation and Existence," The Journal of Philosophy, New York, 1939, 36, also in Readings in Philosophical Analysis, edited by H. Feigl and W. Sellars, New York, 1949; "Notes on Existence and Necessity," The Journal of Philosophy, New York, 1943, 40; "On What There Is," The Review of Metaphysics, New Haven, 1948, 2, also reprinted in Proceedings of thi Aristotelian Society, Supplementary Volume XXV, London, 1951. [See above, pp. 146ff.]
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.
3 W. V. Quine, "On What There Is," The Review of Metaphysics, New Haven, 1948, 2, 21. [See above, p. 146.]
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.
10 See W. V. Quine, Methods of Logic, London, 1952, 96
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).
15 Strictly speaking the meaning of inclusion ought to have been determined axiomatically. I understand from Dr. B. Sobocinski that Lesniewski had an axiom for inclusion. It has never been published and I never saw it when I was studying with Lesniewski in Warsaw before the war. But I found some time ago that a system of Ontology can be built up on the basis of the following single axiom:
(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))))).
16 In Lesniewski's original system of Ontology the three functors are defined as follows:(a)(ex(a) ≡ (∃b)(b ∈ a))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.
(a)(sol(a) ≡ (b,c)((b ∈ a)·(c ∈ a) ⊃ (b ∈ c)))
(a)(ob(a) ≡ (∃b)(a ∈ A))
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.