What is meant by the words ‘ontological structures of reality’? Does it mean what is the case, what isn’t thecase,what could be the case, and what cannot be the case in the case, of reality?
According to Quine, it is science that decides ‘what is’. However, this does not automatically mean that all scientists at all times will get this right. The philosopher’s task, therefore, is to
make explicit what had been tacit, and precise what had been vague.
This simply means that scientists are not, after all, philosophers. They too need to take on board logical and ontological research into what they say is the case. However, scientists are still on the front line when it comes to saying ‘what there is’. It is not the ontological and logical status of our everyday language that is considered by logicians and ontologists, but of the scientific theories that posit the objects that are supposed to provide the ‘furniture of the world’. In a sense, many scientists are not even concerned with the ‘kinds’ of the objects that they investigate. It is up to the ontologist and logician to determine what belongs to which kind, and even the status of the kinds themselves. And if we bring on board kinds, then perhaps we are also committed to ‘essences’. And here too we can say that scientists aren’t concerned with essences either. There is still much work to do when scientists offer up their various theories. Perhaps, for example, scientists are positing too many kinds or too many objects, or perhaps not enough.
Science, Ideal Languages and Reformulations
Quine’s re-workings of our language are not translations. And they are certainly not synonymous with its sentences. Quine says that they are analyses and explications that ‘supply lacks’, unlike what the ordinary language philosophers, say, claimed to do with everyday expressions.
Quine was quite a revisionary philosopher. Claims about changing everyday ontology are truly radical. Something that even philosophers find hard to do when they are away from their desks. Perhaps it is obvious that Quine was a revisionary rather than a descriptive philosopherbecause of his strong allegiance to science. It certainly isn’t the case that science simply describes what people say and what people think. If that were the idea, quantum mechanics would have never had entered the scientific picture. Science moves forward all the time. Of course Quine thought that everyday ontology should change over time. After all, what the scientists have told us about the world needs to be taken on board by not only philosophers, but also the man in the street. Only Wittgensteinians and neo-Aristotelians see philosophy as essentially descriptive. And that may be primarily the case because of their negative attitude to science. The usual riposte is that no matter what science comes up with, there are certain philosophical fundamentals and problems that will not be changed or solved by science. Indeed to some philosophers, like Strawson and Tom Nagel, such an attitude about the relation between philosophy and science is essentially an a priori belief. Science can never supersede philosophy because they each have different and incompatible subject matters.
Unlike Carnap, however, Quine would not have attempted to create a new ‘ideal language’ from scratch. Again, this may have something to do with his attitude to science. If a philosopher created a genuinely new ‘ideal language’, then evidently it would not take into account the various scientific ontologies that already exist, or any other aspects of science that are important to philosophy.A genuine ‘ideal language’ would be in a sense an a priori construction. And such philosophers who created such an ‘ideal language’ would want it to be taken up by individual scientists and science as a whole. Quine had the exact opposite view of the relation between science and philosophy. If anything, any changes in language or conceptual schemes must be determined by science and taken up by philosophers, not the other way around. To Quine, philosophy is part of science andscience is part of philosophy. It makes no sense to say which one primary – science or philosophy – because they are concerned with different areas of thought and research, even though those areas essentially belong to the same ultimately scientific or empirical domain.
Quine stresses the essentially revisionarynature of philosophy by arguing that what the philosopher says need not be ‘strictly synonymous’ with what the man in the street says. Indeed if there should be any synonymy, it should be a synonymy with what is said in science, not what is said in everyday life. As we’ve already said, if the philosopher offered strictly synonymous sentences, then they wouldn’t actually be doingphilosophy at all. A strict synonymy often gives us no new knowledge of any description. What philosophymust do is “reconstruct”, “explicate or “analyse” what is actually said in everyday language. For example, a particular process of explication may be the
disposing of certain concepts with the substitution with new concepts.
As for philosophers determining what people “really have in mind when they say something” seems to be a task for the psychoanalysis, or, within philosophy, a Wittgensteinian, or an ‘ordinary language’ philosopher. Even in these cases I do not believe that it is a case of offering strict synonyms, descriptions, translations, orwhatever. And certainly they wouldn’t be the descriptions of “what we really have in mind”. No, even these descriptive philosophers surely must transcend pure description at least at certain point. I reiterate. Pure synonyms would not offer us any new information about the original sentence under scrutiny. More relevantly, such synonymous translations or interpretations would certain not be cases of philosophy in any known sense of that term.
The explication of everyday expressions, then, will “supply lacks”. These explications will be essentially attempts to rectify what is logically or philosophically wrong with these everyday sentences under scrutiny. Such rectifications or reformulations are actually discovered or uncovered through philosophical analysis and explication. I suppose that it has been the case that many revisionary philosophers have not reallyconcerned themselves with everyday expressions at all. They did not even offer us reconstructions of everyday sentences but simply left them to themselves.Instead, perhaps, such philosophers actually begin their philosophical enterprises by analysing and reformulating already-existing philosophical statements that may have themselves begun as reconstructions or translations of some kind. Some everyday statements are so ontologically or scientifically suspect that certain revisionary philosophers will not concern themselves with them at all, in any way whatsoever.
Quantification and Ontological Commitment
Many people do of course believe that quantificational logic implies a commitment to the bound variables having extensions. Indeed Follesdal didn’t see the point of a quantificational logic that didn’t have existential commitments of some kind. That was Quine’s main reason for developing quantificational logic. To codify and make sense of our ontological commitments. What does the backwards-E commit us to if it doesn’t commit us to the existents that it binds? What does Nolt mean by saying that the existential quantifier is only “a syntactic form”? I suppose that traditional logic is sometimes syntactic rather than material. Wasn’t this the reason why quantificational logic was introduced, by Frege and others, in the late 19th century? When we are not concerned with our or other people’s ontological commitments, can’t we then simply rely on traditional syntactic logic?
Quine’s ontology, at first, seems unflinchingly strict. There are so many philosophy constructs that do not have satisfactory ‘criteria of identification’: propositions, concepts, meanings, universals, God, the I, and so on. And no one agrees, for example, on what is the nature or reality of a proposition. And the same is true of meaning, etc. Quine also includes, against the fashions of his day, that possible worlds and possible world scenarios have no ‘criteria of identification’. He says that the possible fat man in the doorway is essentially indistinguishable from the possible bald man in the doorway. If they are the same possible person, then how would or how could we decide that? Essentially, because possible worlds are fictions, they have no defining criteria of identity. They are useful, sure, but they are non-actual. And just the same can be said about concepts, beliefs, propositional attitudes, meanings, universals, etc.
In a sense, Quine, by talking about ontological commitment, went against what many logicians and philosophers thought is the very essence of logic – its ontological neutrality. Unlike these philosophers and logicians, Quine thought that the applicationsof logic are of vital importance. More specifically, the use of logic in physics and, perhaps, in the other less hard sciences. This is not a surprise when we bear in mind the view that Quine held that philosophy essentially deals with the same issues as the hard sciences, if only on a broader and more general scale. There is no absolute distinction between the hard sciences and philosophy, according to Quine. Perhaps it should come as no surprise that he thought that philosophical logicians should have certainontologicalcommitments. Indeed such ontological commitments would be both defined and clarified care-of logic itself. In any case, if logic were truly and absolutely ontologically neutral, what would be the point of it? Even the most formal and abstract logical systems can still be applied to domains outside pure logic. Indeed that’s what’s so interesting about pure logic – that something so pure, formal and abstract can have so many applications to things that are far from being pure, formal and abstract.
We must be committed to the existence of those entities that we quantify over. We need to say why it is that we are committed to a specific kind of entity or to an actual individual of some kind. This commitment to various existents is shown in quantificational schema. We begin with a universal or existential quantifier. And follow them with variables. The existents that we are committed to become the ‘values of the variables’. The very act of using a quantified variable means that we are committed to at least one ‘value’ of the variable.
Section Two: Technical Considerations
De Re and De Dicto Modality
In a well-known example, Quine is trying to get to the heart of necessity. It is necessarily the case that the number 9 is higher than the number 7. The number of planets that surround the sun is also 9. Does that mean that the number of planets is necessarily larger than the number 7? Of course not. The number of planets is indeed 9. That this is the case is not necessarily so. There could easily be 8 or 3 planets. However, if we are talking only about numbers, then in all situations 9 is greater than 7. The locution
The number of planets is larger than 7.
and
9 > 7
are clearly of a different logical order. One is a necessary statement. The other is simply a description. The latter is true at all possible worlds. And the former is only true at our world. The reference to planets must have a different content to that of the simple numerical expression. At another possible world the expression
9 = the number of planets
could be false, whereas
9 is greater than 7
is true at all possible worlds.
As Kripke puts is. If ‘9 = the number of planets’ is a true identity statement, then that identity must be a necessary identity statement. However, it is not necessary that the number of planets = 9 – it could have been otherwise. Also, Kripke says that even if we do not know a priori that Venus is identical to the Morning Star, that identity would still be necessary. It would be a posteriorinecessity. This means that we needed to find out that Venus and the Morning Star are one and the same object. That it is not only a priori statements that are necessary. In terms of ‘9 = the number of planets’, this statement is true, a posteriori and contingent, not necessary.
‘9 = the number of planets’ is in Quine’s scheme because he is an extensionalist. He doesn’t believe in possible worlds. However, he does not accept that that the number of planets necessarily is more than 7 because the number of planets does not necessarily equal 9. If we don’t except that the definite description equals 9 because of possible world scenarios, then we have no “clear meaning” what the description ‘the number of planets’ refers to. It could be 10 at possible world W1 and 1000000 at another possible world. In fact, if possible worlds are infinite in number, ‘the number of planets’ could equal any number. If this is the case, then the definite description ‘the number of planets’ quite evidently “lacks a clear meaning”. The number of planets can either have no de re meaning, or it has a different meaning at every possible world. However, Quine must accept, and I think he does accept,
∃ (9 > 7)
According to Marcus, the description ‘the number of planets’ cannot be a proper name precisely because it designates different numbers at different possible worlds. However, the name ‘Tony Blair’ does indeed designate the same object at all possible worlds. What, precisely, does that mean? It doesn’t mean that Tony Blair is called ‘Tony Blair’ at all possible worlds. However, it does mean that our name ‘Tony Blair’ designates the object Tony Blair at all possible worlds, whether or not Tony Blair is actually called ‘Tony Blair’ at all these possible worlds.
Kripke’s necessary a posteriori means:
If a and b are identical, then necessarily a = b at all possible worlds.
No mention of actual names is mentioned here. The ‘a’ and ‘b’ designate objects, not names. The person Tony Blair could be named ‘Frank Parsons’ at one possible world, just as Cicero was also named ‘Tully’ at our world.Therefore Tony Blair could not be two different entities at different worlds, though he could have two or more different names at other possible worlds.
Instead of looking at de re and de dicto modality in terms of individuals, which we have just done, we can bring in the issue of concepts as it applies to these issues. Take this statement:
The British monarch is necessarily the head of the British government.
The first de dicto statement means that part of the concept [British monarch] includes or contains the concept [head of the British government]. Of course it could be the other way around. The concept [head of the British government] contains or includes the concept [British monarch]. We could say that one is more of a definite description than the other. However, [British monarch] is still descriptive in that different individuals could fulfil that role at different possible worlds.
The de re reading applies an essential or necessary property to a res – to a person or object. It could be read this way:
MaÉ∃Ba.
Here the necessity operator is applied to the person who is a British monarch. It says that if this is the case, then he or she must be head of the British government. In the de dicto reading, the necessity operator is placed differently. Here we have:
∃ (MaÉBa) ¸ ∃ (BaÉMa)ˆ (Ma≡ Ba).
Here the necessity operator has as its scope the whole statement (‘narrow’ scope), including all the constituent concepts or predicates.It reads:
Necessarily, if a is M, then a is B and necessarily if a is B, then a is M. Therefore a is M if and only if a is B.
It could be said that the biconditional in and of itself entails necessity.
In specifically technical terms, the following examples may be useful.
The ‘x’ in ‘$x’ occurs free in the scope of
$x ∃Fx
Therefore the statement is de re. On the other hand, no variable or constant is free in the scope of ‘∃’ in the formula
∃ $x Fx
Therefore it is a dedicto statement. The de re statement reads:
There is at least one x, such as that x is necessarily F.
The de dicto statement reads:
Necessarily there is at least one x, such that that x is F.
The de re statement ascribes a necessary property to an object. Whereas the de dicto statement ascribes necessity to the formula itself. Necessarily at least one object is F. The de dicto statement is saying that there is necessarily at least one x such as that x is F. Whereas the de re statement says that there is at least one object, but not necessarily one object, such that the object is necessarily F. In the former case, the necessity operator works on the object, whereas in the latter case the operator works on a property of the object. In terms of possible worlds, the de re statement says that all x’s have the property F at all possible worlds. And if a = x, then a has the property F at all possible worlds. The de dicto statement, on the other hand, says that there is at least one xsuch that that x is F. However, x’s aren’t necessarily F at all possible worlds. x, or an x, on the de dicto reading, need not be F in all possible worlds.
Variables Bound by Quantifiers
For Quine
∀(x)
does not mean ‘for every x’. It means ‘for every x which is a…’ The ‘x’ has to belong to a domain of discourse and perhaps a class or set. It never applies to literally every thing or entity. In that case, the domain of x’s is the class of all men. And if a ‘y’ appears in the formula, then the domain of y will be different to that of the domain of x, unless an identity relation has already been set up previously (such as ‘y= x’).
This means that classes or sets, that is, extensions, are of vital importance to quantificational logic. The domain or universe is never “unrestricted”. If it were unrestricted, then the quantificational formula would not even make sense because it would refer to the entire universe or, alternatively, the set of allsets.
We can highlight the nature of bound variables with a short take on an argument that works against whatearlier has been said about Quine’s ontological commitments
*) From the logical truth ‘(∀x) (x = x)’ we obtain ‘Pegasus = Pegasus’ by the law of universal instantiation. Applying the rule of existential generalisation next, we derive the statement ‘(∃x) (x = Pegasus)’. A factual falsity has been inferred from a logical truth. (271)
*) The second solution consists of modifying the laws of first-order logic in such a way that it becomes free of existence assumptions with respect to singular terms. Hintikka (1959) produced a free logic by submitting the application of the rule of existential generalization f (a/x) | (∃x) fx to a condition: the truth of the premise (∃x) (x = a) which states that a exists. The third solution consists of treating denotationless singular terms as denoting nonexistent objects and taking bound variables as ranging over objects which are either existent or nonexistent…A variant of the third approach can be found in a version of first-order logic which operates with two pairs of quantifiers, viz (1) ∀ª and ∃ª which bind variables ranging over existent (‘actual’) individuals and (2) ∀and ∃ which bind variables ranging over possible individuals. (Gochet, 271)
So here at least quantificational logic can talk about non-existent entities. In this case, we simply put a small ‘a’ above the quantifier. For a start, it seems to go against the very essence of quantificational first-order logic that it can talk about non-existents. That is, the scientifically-minded quantificational logicians would have seen such a logic as making sense of the claims of science and how they compare with other kinds of discourse. However, Gochet at first does talk about first-order logic, and not quantificational first-order logic. However, if there is a difference, it doesn’t change the case of Hintikka in that he used quantificational logic to also talk about non-existents. (a/x) | (∃x) fx What seems to be being said in that formula, is that for all or some x’s that are named a, we have a proof that there is at least one x, such that this x is f. And it would follow that if an x is f, then so too is an a. (∃x) (x = a). This addition to Hintikka’s scheme says that there is at least one x, such that this x is a or is even given the name symbolised by ‘a’.
Extensions and Intensions
Just as proper names have ‘senses’, in the Fregean scheme, so do extensions have intensions (or meanings). For example, the intension of the set of people who are 21 or older may make the extension of this set come out differently in different contexts, times or at other possible worlds. The intension in our case is that the extension is true of our world today. But a different intension will determine that the extension has a different set of members in other contexts or at other possible worlds. The intension therefore helps pick out the set that belongs to the extension, just as a definite description helps pick out the referent of a proper name.
The set of adults at another possible world may be different, or will be different, to the set of adults at our world. The intension determines where, so to speak, the extension is pointing. Is it pointing at our world now, or our world in the past, or, perhaps, to the set of adults at another possible world?
To finish this section. There is a well-known problem with extensionalism. “Creature with a kidney” and “creature with a heart” both have the same extension – viz., human beings. However, what if the “creature with a kidney” is a fox and the “creature with a heart” is a human being again? Is the extension still the same? In that case the extension of these two phrases would include both foxes and humans. Is there a problem here? There could be “an extension of all extensions”. This would be “the infinite extension”. The extension of “creatures with a nose” would include many species.
Quantifying Over Predicates
Quine argues that predicates have no independent existence, unlike what Frege calls “saturated objects”.Quine’s position is not unlike Aristotle’s who believed that universals could only be found in their instantiations in particular objects. Such universals do not exist apart from their instances in particulars. Because of this, Quine believes that “predicates are not names”. They are “parties” to something else. In Gochet’s terms, to “primary substances” and “complete saturated objects”.The following formula is unacceptable to Quine. That formula is:
For at least one politician or red thing, and for every object that is an x, there is an xthat is a politician or a red thing.
Being a politician and being red seem to be different modes of being. Surely the class ofpoliticians has an extension of concrete individuals. Is the class of red things different from the class ofreds? In the first instance, red is a predicable property; it belongs to various things. In the next case, we are not talking about red as a predicable property, but as it is in itself – the class of reds. In the case of the class of reds, it seems more like they are universals in that they do not simply exist when theyareinstantiated.
We can quantify over properties or predicates. This is an example from Strawson:
In ‘There is something that Betty is and Sally is not’ we appeared to be quantifying in a higher-type region… (87)
We are saying that there is something, and that something is prettiness, that can be attributed toBetty but not Sally. We are quantifying over the predicate “prettiness”, even if it is the subject-position, or the property prettiness. We are not simply quantifying over Betty and Sally, we arequantifying over an attribute of Betty. If we were quantifying over Betty and Sally themselves, then theywould take the subject -position of the quantification. Our prime object of reference, or the object in thedomain of quantification, is prettiness not Betty and Sally. Indeed we could ask if Sally and Betty would be anything if they had no properties or predicates attributable to them.
Why are objects seen as theprime reference of quantification in Quine? This is almost a commitment to an anti-Bundle Theory in ontology. The object itself is of vital importance, not the properties of objects or, even, the properties that are the sole constituents of objects. Indeed how can we pick out, identify, or refer to an object without the object’s properties? We could even say that objects belong to a set of properties, rather thanproperties belong to particular objects. And if we do accept the Bundle Theory of objects, then this way of putting things would be even more acceptable. However, we could also accept substratums orbare particulars and still accept properties rather than objects as the prime targets of existential anduniversal quantification. It can be said that Quine is tacitly committed to a non-bundle view of objects. And that is why he refuses to quantify over properties or predicates. If objects were nothing butcollections of properties, then this emphasis on objects would seem to make little sense. Without rejecting bare particulars or substratums we could still ask Quine why he emphasises objects ratherthan attributes or properties.
Instead of saying that something is “thus-and-so” (Strawson, 88), we could say that the something is nothing but thus-and-so. There is nothing else for an object or entity to be other than thus-and-so. I mean that an object is quantified over or referred to exclusively in terms of its properties and notin terms of some fictional bare particular or substratum. However, when we use proper names to refer to objects, we do not thereby identify or even pick out that object exclusively in terms of applying a name to that object. If we accept non-contentful proper names, such names say nothing about the objectnamed and therefore don’t individuate, describe or even pick out that object without the help ofreferences to properties, etc.
Now we can ask if properties themselves can be seen as objects or even quasi-objects. In fact, if objects are made up exclusively of properties, then it is necessarily the case that we must treat properties as objects because there are no propertyless objects or even objects simpliciter without properties. It may of course be the case that properties are frowned upon by Quine and others because they are deemed asabstract non-spatio-temporal entities. We need not take them that way if, for example, we accept the trope theory of properties. If we don’t take them as multiply instantiated universals, butrather as particulars themselves. The predicate “red” only applies to particular examples of redthat are connected to one another by the psychological act of noting instances of resemblance betweendifferent examples of the property red. This is not the place, however, to defend the trope theory ofpredicates, properties or attributes. It follows, therefore, that we need not even see tropes of properties as necessarily non-spatio-temporal.
Predicate-Extensions and Quantification
In his ‘On What There Is’, Quine tried to show us that the simple use of names does not mean that we believe weare necessarilyreferring to entities. Names or words can have uses that don’t entail a commitment to anyreferents of such things. Not all words have a reference. Indeed not all names have a referent. Similarly, the useof predicates was traditionally seen as areference to universals. The predicate ‘is red’ referred to theuniversal red. Again, we can use such predicates without also believing in universals. Instead Quine would argue that the predicate or property red has an extension – the class of allred things.
Words or names do not automatically commit us to the entities that are seen to be the referents of words and names. What does commit us, on the other hand, is the use of ‘bound variables’. Some would say that the use of bound variables specifically shows us, and other people, that we arecommitted to the entities that we arereferring to. When the variable ‘x’ is bound, it means that this variable has as its reference or extension at least one entity of a specific kind or nature.
Think of the quantifier in
Some dogs are white.
When we assert this statement, we also implicitly assert the statement
There is something that is both a dog and white.
Quine’s version, or alternative, makes the original statement clearer and more exact. When we say that somedogs are white, we are effectively saying that there is something that isboth a dog and whiteat the same time. The subject term ‘some dogs’ is replaced by the subject-phrase ‘there is something’. This is broader and less specific than the original subject-term. However, it is specified and clarified by the predicate ‘that is both a dog and white’, unlike the original predicate ‘are white’.
However, Quine is not committed to the universals whitenessand doghood. He would need another way to explain his use of predicates that does not rely on the existence of universals. Perhaps there is an extensional set of allwhite objectsand an extensional set ofall dogs. And if we fuse these extensions together, we have theextensionalset of objects that are both white and dogs. However, the predicate ‘is a dog’ is not of the same logical category asthe predicate ‘are white’. In the case of this predicate, Quine would need to commit himself to an extension that included properties rather than objects in its extension. Properties do not exist without theobjectsthat have these properties. Therefore Quine’s property-extension would need to be the set of allwhite objects.
Quantifying Over Belief States
There is more to a proper name that its reference. Both ‘Tully’ and ‘Cicero’ refer to the same object. However, Ralph believes that Cicero denounced Catiline. But he does not believe the same about Tully, even though Tully and Cicero are the same man. Of course this problem deals with the psychological states of a subject, not the identity of Tully and Cicero. Quine is arguing that we must not quantify over belief states. Quantification must be extensional. If we are to preserve ‘the substitutivity of identity’, then quantification must not include propositional attitude contexts. In other words, we must not quantify over beliefs. The only thing that matters to the extensional logician is that ‘Tully’ and ‘Cicero’ refer to the same object. Why can’t we quantify over belief states?
In this passage, it is clear what we can and cannot quantify over. We cannot quantify over ‘quotation’. This means reports of other people’s belief states. On a Wittgensteinian reading, we cannot take account of private beliefs states. And propositional attitudes are, of course, private, except, of course, when they are behaviourally expressed. We can quantify over the
physical constitution and behaviour of organisms.
This would of course mean the linguistic expressions of persons, but not the beliefs as they are supposed to be within the private realm of mind. We can say that ‘X said…’, but not ‘X believes…’ This attitude to beliefs is thoroughly behaviourist. Quine himself has classed himself as a semantic behaviourist. Only the expression of beliefs are relevant, not the beliefs themselves. Indeed Quine went further and argued that to say ‘the expression of belief’ creates a false dichotomy between belief and expression. Instead Quine thinks the expression itself is the belief. It is naturalistically acceptable to talk of behavioural expressions, not beliefs or thoughts that somehow come before the expressions and are therefore abstract in nature. Such things, according to Quine, would have no individuation-conditions and must therefore be suspect entities. His rejection of propositional attitudes, therefore, may make Quine an eliminative materialist in the manner of Paul Churchland. Quine believes that there are no such things as propositional attitudes. And, therefore, no such things as beliefs. That is, entities that come before expressions but still, somehow, contain all the information that is contained in the expression itself.
Quantifying Over Sets or Classes
Sets are different from universals. Universals do not depend on their “instances”, whereas sets do depend on their members. Without members there are no sets, even when a set’s members are other sets that have individuals as members. On the other hand, we can have, according to the literature, uninstantiated universals. A universal round square that is not exemplified by any existing particular or individual. Perhaps Pegasus too is a uninstantiated universal, though it will have a different mode of being from that of the round square universal.
What are unsaturated entities? Relations and properties are examples. And relations and properties are universals in that they are predicable of numerous entities. Predicates take universals as values. For example, the universal wise is predicable of wise people. The universal bigger than is predicable of those entities that are bigger than other entities. And finally the universal black is predicable of the class of blackentities.
However, if we do use empirical statements instead, such statements will refer to entities of various descriptions. In predicate logic it is possible to refer to only concrete individuals. In such a logic we will be attributing properties to actual concrete existents. In set-theory, however, we will quantify over classes, which means that we are quantifying over things that are abstract rather than concrete.
Whenever we use quantificational logic, whenever we quantify over something, it is currently thought that what we quantify over must exist. Indeed quantificational logic clarifies and expresses our ontological commitments. In fact both Frege and Russell essentially defined the notion of existence by using quantificational logic. Such a logic effectively told us what existence is and what actually exists. More technically, they said that to say that something exists is to say, “some predicate is satisfied”. The word ‘predicate’ here is used in its wider sense to refer to any word, not just the attributive part of a subject-predicate statement. They were saying that, say, politicians, or John, or redness, exist if these predicates are actually satisfied. And their satisfaction or non-satisfaction will be shown within quantificational schemata. They will, say, show us if politicians exist and if they are all red, etc.
Set-theory is a different ball game altogether. In this case we will be quantifying over abstract objects, classes, rather than concrete individuals and their properties. Since the purpose of quantificational logic was originally to clarify and express our ontological commitments, and, also in many cases to take Occam’s razor to such commitments, then it is strange that we should start quantifying over classes. It depends on what we take classes to be. For example, at one point Russell thought that classes were nothing over and above their members taken as a collection. Classes were just a convenient way of lumping together such collections and given their members the same name, etc. Of course it is also the case that classes can be seen as something over and above their members. Indeed they have to be so thought if we take into account the null class or classes with only one member. It is also probably the case that infinite classes or classes with uncountable members cannot be characterised wholly in terms of their members, and for obvious reasons. We are still left with the same question:
What are classes if they are something over and above their members?
By accepting classes as something over and above their members we are clearly increasing the number of entities we must quantify over. Ayer gives us a good example of the situation with classes.
A man makes a left shoe; and so brings an entity into existence. He then makes a right shoe; and therefore brings another entity into existence. Now there are two new entities in existence. However, don’t the two shoes also make a pair? And isn’t a pair of shoes different from each individual shoe simply added together? If it the pair is a new entity, it is an abstract entity. It is a class. It can’t be just the addition of two shoes because a pair implies that there is something more to the shoes that just their addition. If they belong to a pair, then they belong to a class. A pair is a class with two members. This is the class of the shoemaker’s right and left shoes.
