Quantifiers, Quantifiers and Quantifiers: Themes in logic, metaphysics and language

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The question of how to develop an appropriate model theory for the language of quantified propositional logic is no substitute for the more substantive question of whether there is an intelligible interpretation of propositional quantification on which it is not construed as objectual quantification over propositions or substitutional quantification over sentences. While Prior and Grover , for example, take propositional quantification to be importantly different from objectual and substitutional quantification, Richard suggests that propositional quantification should at the end of the day be treated as a species of objectual quantification, e.

Much of contemporary ontology builds on the assumption that existence is to be understood in terms of quantification: in a slogan, to exist is to be something. Ontology is largely concerned with the domain of the existential quantifier. By the link between quantification and existence, we mean the claim that to exist is to be identical to something.

This view can be traced back to Frege and Russell, who offered roughly the same broad analysis of quantification in terms of predication. The entry on existence provides a helpful overview and discussion of their account of existence against the background of the historical context in which they produced it. In what follows, we highlight a distinction between the Frege-Russell analysis of quantification in terms of predication, on the one hand, and the link between quantification and existence, on the other.

Frege a, b explicitly analyzed quantification in terms of predication. For Frege, first-level predicates express concepts under which objects fall. A quantifier expresses a second-level concept under which first-level concepts fall. In particular, he proposed to take a sentence like 3 below to predicate of a first-level concept that it has instances:.

Russell offered a similar account of quantification. The entry on propositional functions provides more detail on the role of propositional functions in the early development of modern logic. Both Frege and Russell combine their account of quantification in terms of predication with the substantive thesis that existence is to be understood in terms of quantification.

For Russell, existence is identified with a certain property of propositional functions, e. This may seem overkill; if true, it might seem that only first-level concepts exist, not the objects they instantiate. This point is perfectly compatible with the existence of a first-level concept under which all and only those objects that exist fall.

Take, for example, the concept being self-identical. Likewise for Russell. Note, however, that one could in principle retain their analysis and deny, for example, that first-level concepts and propositional functions, respectively, can only be saturated by objects that exist. One could take the view that some great philosophers who once existed, no longer exist. Socrates, for example, was a great philosopher who no longer exists. He can nevertheless instantiate the first-level concept admired by many philosophers.

Or consider the bookcase I would have built, had I finally assembled all the materials I purchased according to the assembly instructions that came with them. I have had concrete plans to build the bookcase for ages now; never mind the fact that I may never find the skill, time or energy to assemble it. My planned bookcase does not exist yet, and knowing myself, it might well never exist. But one might take the view that we can refer to it and that it instantiates many first-level concepts such as being a planned bookcase.

On a view like this, the assertion that there are planned bookcases that do not exist would remain true even if none of its instances exist. A view like this would be classified as a form of Meinongianism by Nelson , and it would be subject to some of the difficulties other forms of Meinongianism face. Suffice it to say, for present purposes, that it would, however, require a radical departure from a very influential approach to ontology advocated by Quine and endorsed by philosophers like Peter van Inwagen and David Lewis and their followers. The problem with this answer is that it is largely uninformative.

All parties agree that everything is something, but there is still plenty of room of disagreement as to what kinds of objects are there. Are there mereologically composite objects? Are there concrete possible worlds? Are there mathematical objects? However, philosophers are still intensely divided as to whether they do. He advised philosophers to look at the ontological commitments incurred by our best global theory of the word—best by ordinary scientific standards or principled extensions thereof—when appropriately regimented in the language of pure quantificational logic with identity.

More details are given in section 5 of the entry on Quine. One of the most familiar instances of such arguments is the Quine-Putnam argument for the existence of mathematical objects. Since mathematical objects are indispensable for scientific purposes, we should expect the quantifiers of our best global theory of the world to range over them. But once we settle the question of whether the quantifiers of our best global theory of the world range over mathematical objects, we have settled the ontological question of whether such objects exist.

The entry on indispensability arguments in mathematics includes extensive discussion of this and related indispensability arguments in mathematics. Similar arguments have been deployed outside mathematics to argue for the existence of possible worlds, mereologically complex objects, and the like. This brings us to a family of applications for different styles of quantification explored above. It is not uncommon to respond to indispensability arguments for the existence of objects of a certain kind by means of a paraphrase strategy, which if successful, would show how to make do without the ontological commitment to objects of the offending kind.

Oftentimes, the paraphrase will involve a different style of quantification, whether an alternative to classical quantification or an extension thereof. For example, Gottlieb attempted to respond to the Quine-Putnam indispensability argument for the existence of numbers by dispensing with objectual quantification in favor of substitutional quantification in arithmetic. Or take the question, for example, whether there are composite objects of various kinds.

This observation applies to ontological inquiry more generally. When a nominalist asserts that there are no mathematical objects, she does not intend the thesis to be qualified by a restriction to a domain of concrete objects; otherwise, the thesis would be devoid of interest. But unrestricted quantification is not a very common phenomenon outside highly theoretical contexts such as logic and ontology.

Take a typical use of a quantifier expression in English as exemplified in a typical utterance of the sentence:. In particular, it would be inappropriate for another participant in the conversation to point out that the Moon is not on sale.

Metaphysics as Logic

The Moon is not an exception to the statement made by my utterance of 11 because the Moon does not lie in the domain of quantification associated to my use of the quantifier. But the fact that unrestricted quantification is relatively uncommon is no reason to doubt it is attainable in certain contexts. Unfortunately, many philosophers have recently doubted that genuinely unrestricted quantification is even coherent, much less attainable.

First, they face the question of what to make of the prospects of ontological inquiry without unrestricted generality. How should we formulate substantive ontological positions such as nominalism, if we cannot hope to quantify over all objects at once? The second challenge for the skeptics is to state their own position. To the extent to which the thesis that we cannot quantify over everything appears to entail that there is something over which cannot quantify, skeptics seem to find themselves in a bind by inadvertently quantifying over what, by their own lights, lies beyond a legitimate domain of quantification.

At the core of the problem lies the assumption that the set-theoretic paradoxes cast doubt upon the existence of a comprehensive domain of all objects. What they reveal, according to Dummett , , is the existence of indefinitely extensible concepts like set , ordinal , and object. The set of all non-self-membered sets in the initial domain cannot, on pain of contradiction, be in that domain, which means that it must lie in a more comprehensive domain of all sets.

If there is no domain of all sets, there is, the thought continues, no hope for a domain of all objects. For another line of attack against the coherence of unrestricted quantification, one may ask what exactly is a domain of quantification supposed to be. Skeptics often take a page from modern model theory and think of a domain of quantification as a set—or at least as a set-like object, which contains as members all objects over which the putatively unrestricted quantifier is supposed to range.

They take for granted that for a speaker to be able to quantify over some objects, they must all be members of some set-like object, which constitutes a domain of quantification. Cartwright called this thesis the All-in-One Principle. Now, if there is unrestricted quantification, then the domain of quantification associated to it cannot be a set-like object. The All-in-One principle is not beyond doubt.

One application of plural quantification, which is explored by Cartwright , is to abandon the All-in-One principle and to understand talk of a domain of quantification not as singular talk of collections but rather as plural talk of its members. To speak of a domain of certain objects is just to speak of the objects themselves—or to speak of a first-level concept under which they all fall; and to claim of a given object that it lies in the domain is to claim of the object that it is one of them.

Alternatively, one may opt for the expressive resources of second-order quantification with the second-order quantifiers taken to range over Fregean concepts. Williamson outlines a conception of a domain of quantification as a Fregean concept under which certain objects may fall. To speak of a domain of all objects on this view is to speak of a Fregean concept under which all objects—without restriction—fall, and to claim of a given object that it lies in the domain is merely to claim that the object falls under the relevant Fregean concept.

The combination of quantification and modality traces back to Barcan , and Carnap Quine had anticipated the issue in Quine , and he pressed the criticism again in Quine , The interaction of quantification and modality turns out to raise difficult philosophical problems. As we will see, some of these difficulties have, in fact, inclined some philosophers to rethink the scope of pure quantificational logic. The problem is that certain theorems of quantified modal logic suggest that everything is necessarily something. The subject matter of ontology is necessary: nothing could have failed to be something.

Quantified modal logic is the system that combines pure quantificational logic with propositional modal logic. A weak propositional modal logic called K logic includes as axioms all tautologies and all instances of schema K:. The rules of inference of the system include modus ponens and what is often known as a rule of necessitation:.

The combination of propositional modal logic and pure quantificational logic is often called quantified modal logic and is discussed in section 13 of the entry on modal logic and the entry on actualism. This combination immediately delivers what we may call the necessity of being :. Successive applications of the rule of necessitation and universal generalization are all it takes to complete the proof.

Take the Griffith Observatory, for example. It is a consequence of the theorem that it is necessarily something. The Griffith Observatory would have been something even if no funds had been available for construction and nothing had ever been built in the site it occupies. And what is true of the Griffith Observatory is true of us: each of us is necessarily something.

When we combine this result with the link between quantification and existence, we conclude that everything exists necessarily, which flies in the face of the assumption that we exist only contingently. The proof is given, for example, in one of the supplements to the entry on actualism. But this provides another route to the necessity of being: since necessarily, everything is something, everything is necessarily something.

In the presence of a further plausible modal principle, B, we may even prove the Barcan Formula. The Brouwerian principle, B, states that whatever is the case is necessarily, possibly the case:. The entry on actualism includes a proof of BF. But this formula is deemed unacceptable by many philosophers. Consider my plans to build a bookcase. Even if I never manage to build one, it is certainly possible for me to build one. So, it is possible for there to be some bookcase I have built.

By the Barcan Formula, it seems to follow that something is possibly a bookcase I build. In fact, to deny this is one way to come to terms with the Barcan Formula. What is true of the combination of pure quantificational logic with propositional modal logic is mutatis mutandis true of the combination of pure quantificational logic and propositional tense logic.

Certain theorems of quantified tense logic suggest that everything has always been, is, and will always be something. The domain of ontology is immutable: nothing ever fails to be something. In place of K, we have distribution axioms:. The combination of propositional tense logic and pure quantificational logic immediately delivers the eternality of being:.

The Griffith Observatory was something even before it was built in —and will always be something—even billions of years from now. And what is true of the Griffith Observatory is true of us: each of us has always been, is, and will be something—even before our birth and long after our death. We infer, for example, that there were dinosaurs only if there are past dinosaurs. If to exist is to be something, then past dinosaurs exist. But whether this commits us to flesh-and-blood dinosaurs is of course a different matter, since being a past dinosaur need not require being a dinosaur.

The interaction between pure quantificational logic and tense and modality suggests the domain of ontology is immutable and necessary. But we can only draw these conclusions by making use of some auxiliary assumptions, some of which are far from obvious. The highly counterintuitive character of the conclusions has consequently led many philosophers to place some of the auxiliary assumptions under close scrutiny. Other philosophers have followed argument where it leads, and they have sought to reconcile the eternal and necessary character of the domain of ontology with the apparent temporary and contingent character of existence.

In fact, Williamson set out to provide a separate battery of arguments for the necessity and immutability of ontology. One auxiliary assumption one may question is the traditional link between quantification and existence. Even if there is no change in the domain of quantification, you may nevertheless think that existence is only temporary. Socrates did not exist either before BCE or after BCE, and the mere fact that he is, has and will always be something is not reason to attribute existence to him.

But the Meinongian thesis is not compulsory for more plausible developments of the view that not everything exists, even though everything is something. The moral of the temporal versions of BF and CBF is only that unrestricted quantification ranges over an immutable and necessary domain of objects, whether or not they enjoy temporary and contingent existence.

For more on different variants of Meinongianism, the reader may consult the entry on existence. There is, in the second place, the assumption that the axioms of pure quantificational logic remain true when we expand the language to include other sentential operators, whether temporal, modal or otherwise. This is, for example, the path taken in Kripke , where Kripke proposes to weaken the axiom of universal instantiation in line with the first version of inclusive quantificational logic discussed earlier.

Others have fallen back into alternative forms of free logic, but there appears to be no consensus as to which one is the best alternative to pure quantificational logic. It may be helpful to note, however, that some of the relevant free logics merely restrict the axiom of universal instantiation by means of an existence predicate and do nothing to block the derivation of other allegedly problematic principles such as the necessity of identity, which is the thesis that everything is necessarily self-identical.

Since, presumably, an object can only be necessarily self-identical if it exists necessarily, opponents of CBF will also want to have some resources to block the necessity of identity. The derivability of the necessity of identity from the axioms of identity and the rule of necessitation may perhaps invite one to consider another option. Maybe the culprit is not pure quantificational logic, but rather the indiscriminate use of the rule of necessitation. The rationale for necessitation is that every provable sentence should be necessarily true, but it is not clear how this thought is supposed to generalize to open formulas, which, strictly speaking, are not true or false.

Open formulas are merely true or false under an assignment of values to their free variables. One could avoid the problems by barring closed terms from the language, but such a radical exclusion seems ad hoc and artificial. So, if one chooses to restrict necessitation, one must provide a different rationale for it.

Deutsch provides an example of this strategy. The fourth and final option to consider is to take the derivability of CBF in tense and modal logic at face value, and embrace the conclusion that existence is indeed immutable and necessary. The task for each approach is to explain our initial reluctance to embrace them in the first place.

Take the apparent resistance to accept the claim that Socrates will always be something despite the fact that he died in BCE. The permanentist may respond that if we are initially disinclined to accept this claim, it is only because we mistakenly think that because a person, for example, is a concrete object, a past or a future person must be concrete as well. Socrates, which is a past person, was not a person either before BCE or after BCE; indeed, Socrates is now not a person, nor will he be one in the future.

Likewise, a merely possible person need not be a person either and neither the Barcan nor the Converse Barcan Formula threaten the temporary and contingent character of concreteness. Since I could have found the time, skill and energy to build a bookcase, some object is possibly a bookcase built by me. But the necessitist will be at pains to add that to be a possible bookcase built by me is not to be a bookcase built by me, much less a bookcase. But there are other points of contact.

Likewise, all versions of necessitism seem committed to the actual existence of many more nonsets than can form a set. Classical Quantificational Logic 1. Departures from Classical Quantificational Logic 2.

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Extensions of Classical Quantificational Logic 3. Quantification and Ontology 4. Classical Quantificational Logic What is now a commonplace treatment of quantification began with Frege , where the German philosopher and mathematician, Gottlob Frege, devised a formal language equipped with quantifier symbols, which bound different styles of variables. Departures from Classical Quantificational Logic In what follows, we look at three rival accounts of quantification in modern logic.

The permutation principle, however, becomes redundant in the presence of axioms for identity. Extensions of Classical Quantificational Logic Each departure from classical quantificational logic we have considered originated from an objection to either axioms of pure quantificational logic or the Tarskian definition of satisfaction in a model by an assignment of objects to the variables of the language.

But an influential argument against the identification of plural quantification and singular quantification over sets traces back to Boolos , which claims that the following two sentences differ in truth value when we let the domain of quantification include all sets there are: There is a set of all, and only, non-self-membered sets.

Some sets are all, and only, non-self-membered sets. Quantification and Ontology Much of contemporary ontology builds on the assumption that existence is to be understood in terms of quantification: in a slogan, to exist is to be something. In particular, he proposed to take a sentence like 3 below to predicate of a first-level concept that it has instances: 3 There is at least one square root of 4.

Take a typical use of a quantifier expression in English as exemplified in a typical utterance of the sentence: 11 Everything is on sale. Bibliography Adams, R. Bacon, A. Barcan, R. Bernays, P. Boolos, G. Burgess, J. Carnap, R. Cartwright, R. Dedekind, R. Braunshweig: Vieweg. Deutsch, H. Dorr, C. Dummett, M. Dunn, J. Enderton, H. Fine, K. Perry, J. Frege, G. Black eds. Glanzberg, M. Gottlieb, D. Grover, D. Hand, M.

Jacquette ed. Hawthorne, J. Hellman, G. Higginbotham, J.

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