by Corry Shores
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[The following is summary of Priest’s text, which is already written with maximum efficiency. Bracketed commentary and boldface are my own, unless otherwise noted. I do not have specialized training in this field, so please trust the original text over my summarization. I apologize for my typos and other unfortunate mistakes, because I have not finished proofreading, and I also have not finished learning all the basics of these logics.]
Summary of
Graham Priest
An Introduction to Non-Classical Logic: From If to Is
Part II:
Quantification and Identity
13.
Free Logics
13.5
Quantification and Existence
Brief summary:
(13.5.1) We might want a free logic where quantifiers range over all objects and not just existent ones. (13.5.2) Quantifiers ranging over the outer domain D are called the outer quantifiers, and they are written as ∃ and ∀. The quantifiers that range over the inner domain E are called inner quantifiers, and they are written as ∃E and ∀E. (13.5.3) We read ∀xA as ‘Every x is such that A’; ∀ExA as ‘Every existent x is such that A’; ∃xA as ‘Some x is such that A’ or as ‘Something is A’; and ∃ExA as ‘there exists an x such that A’ or as ‘there is an x such that A’. (13.5.4) We should not think that the existential quantifier of natural language necessarily implies existence. (13.5.5) There is an argument for reading the existential quantifier as “there exists”. The argument wants to avoid problems like the ontological argument, so it does not allow existence to be a predicate. Instead, it sees as the only other viable option for expressing existence as being the existential quantifier. Part of the thinking is that only things that are there can be predicated. But this is not a convincing argument, because there are many examples of predication of non-existing objects, like Zeus being worshipped. (13.5.6) If we wish, we can define inner quantifiers in terms of outer ones, which means that “in a free logic with outer quantifiers, we can dispense with inner quantifiers altogether,” namely, in the following way:
∃ExA ∃x(ℭx ∧ A)
∀ExA ∀x(ℭx ⊃ A)
(p297). However, “There is no way of defining outer quantifiers in terms of inner quantifiers” (297). (13.5.7) These new semantics make one problematic inference no longer problematic, namely, Ax(a) ⊨ ∃xA, now meaning that if something can be predicated, it is either an existent or non-existent object (previously it implied that any predicable thing must be existent). But it may not make the logical truth ∃x(A ∨ ¬A) unproblematic (it implies now that there must be at least a non-existent object, while before it implied there must be at least an existent object.)
[Free Logic Quantifiers Ranging Over Non-Existent Objects Too]
[Inner and Outer Quantifiers]
[Reading the Quantifiers]
[Not All Existential Quantification Should Imply Existence]
[An Unconvincing Argument for Reading the Existential Quantifier as “there exists”]
[Redefining the Inner Quantifiers]
[Deproblematizing Some Inferences]
Summary
[Free Logic Quantifiers Ranging Over Non-Existent Objects Too]
[We might want a free logic where quantifiers range over all objects and not just existent ones.]
[Recall from section 13.2.2 that we are examining free logics, where there is the main domain D of all objects and the “inner domain” E of existent objects, which is a subset of D, with the remainder in D being non-existing objects. As we saw in section 13.2.4, the semantics for free logics are mostly the same as in classical first-order logic except that the truth conditions for quantifiers concern only the domain of existents:
v(∀xA) = 1 iff for all d ∈ E, v(Ax(kd)) = 1 (otherwise it is 0)
v(∃xA) = 1 iff for some d ∈ E, v(Ax(kd)) = 1 (otherwise it is 0)
(p.291, section 13.2.4)
But we might instead want quantifiers that range over all objects.]
Free logics of the kind at which we have been looking contain names for non-existent objects, but they do not allow us to quantify over them. This may be thought somewhat arbitrary, especially given the semantics. Why not allow quantifiers to range over all objects? Thus, we might add another kind of quantifier whose truth conditions are exactly the same as those in classical logic, with domain of quantification D. In tableaux, these quantifiers would function, of course, just as do quantifiers in classical logic.
(295)
[Inner and Outer Quantifiers]
[Quantifiers ranging over the outer domain D are called the outer quantifiers, and they are written as ∃ and ∀. The quantifiers that range over the inner domain E are called inner quantifiers, and they are written as ∃E and ∀E.]
[As we mentioned above, in section 13.2.2 we saw that the domain of existent objects is called the “inner domain.” So the quantifiers that only range over the inner domain are called inner quantifiers. They are written as use ∃E and ∀E, while those ranging over the outer domain, called the outer quantifiers, are written just as ∃ and ∀.]
Let us call such quantifiers outer quantifiers, as opposed to the quantifiers with domain E, which are inner quantifiers. If we use ∃ and ∀ for the outer quantifiers, then we need a different notation for inner quantifiers. For the rest of this section (only) I will use ∃E and ∀E for them (the superscript ‘E’ indicating existential loading).
(295)
[Reading the Quantifiers]
[We read ∀xA as ‘Every x is such that A’; ∀ExA as ‘Every existent x is such that A’; ∃xA as ‘Some x is such that A’ or as ‘Something is A’; and ∃ExA as ‘there exists an x such that A’ or as ‘there is an x such that A’.]
[One convention for reading ∃xA is “there exists an x such that A.” But now that this ∃ symbol ranges over the outer domain, that means it can include non-existent objects, which no longer is properly represented by the conventional reading. (For, the reading implies that the object exists, when in fact it may not.) However, this reading holds correctly for ∃ExA. Instead, we could read ∃xA as ‘Some x is such that A’ or ‘Something is A’. The outer universal quantifier can keep its conventional reading of “Every x is such that A’, but the inner universal one should now be ‘Every existent x is such that A’. To avoid further complications regarding a distinction between “exists and is (existence and being), and to [not] impute to non-existent objects some different – usually some second-class – kind of existence”, we should only use ‘there is an x such that A’ for inner quantification.]
Of course, if one proceeds in this fashion, one must precisely not read the outer particular quantifier, ∃xA, as ‘there exists an x such that A’. That is how one reads ∃ExA. ‘Some x is such that A’ will do nicely as a reading. Thus, ‘∃x x is a cat’ can be read as ‘Some x is such that x is a cat’, or more simply, ‘Something is a cat’. The outer universal quantifier, ∀xA, note, can still be read as ‘Every x is such that A’. It is the inner quantifier ∀ExA that now needs to have its standard reading changed to ‘Every existent x is such that A’. What of the locution ‘there is an x such that A’? Conceivably, one might use this for either inner or outer particular quantification: we can, after all, use words to mean whatever we wish, provided that it is clear to all concerned what we are doing. My own inclination, however, is to use it only for inner quantification. Doing otherwise invites us to draw a distinction | between exists and is (existence and being), and to impute to non-existent objects some different – usually some second-class – kind of existence.1 But if an object is non-existent, it is non-existent. End of story.
(295-296)
1. A view, incidentally, often attributed – fallaciously – to Meinong. It was an early view of Russell.
(296)
[Not All Existential Quantification Should Imply Existence]
[We should not think that the existential quantifier of natural language necessarily implies existence.]
[(ditto)]
The founding fathers of classical logic, Frege and Russell, certainly read the quantifier ∃ as ‘there exists’, and Quine famously took the quantifier to be definitional of existence, in his slogan: ‘to be (= to exist) is to be the value of a bound variable’. But it is not easy to find arguments that natural language quantifiers ought always to be understood as existentially loaded, and there are many places in English where this appears not to be the case. Suppose, for example, that I dreamed of an ugly monster last week, and I dreamed of it again last night. Then it would be quite natural to say that I dreamed about something last night which I dreamed about last week, even though that thing does not exist.
(296)
[An Unconvincing Argument for Reading the Existential Quantifier as “there exists”]
[There is an argument for reading the existential quantifier as “there exists”. The argument wants to avoid problems like the ontological argument, so it does not allow existence to be a predicate. Instead, it sees as the only other viable option for expressing existence as being the existential quantifier. Part of the thinking is that only things that are there can be predicated. But this is not a convincing argument, because there are many examples of predication of non-existing objects, like Zeus being worshipped.]
[I might have this next idea wrong, so please check the quotation below. We will follow some reasoning (not our own) that will arrive upon the conclusion that the existential quantifier should be read as “there exists”. We read it as such, because it is our only viable option for representing existence. This is because our other available option for expressing existence is to use the existence predicate, which according to this line of reasoning, is not a genuine predicate. It seems one motivation for rejecting the existence predicate is that it leads to the ontological argument, which we probably want to avoid. Priest says that at the basis of this line of thinking is that “to predicate anything of an object, it must be there, in some sense, to be available for predication”. But I am not certain yet how that works into the line of reasoning. For now I am guessing it is the following, but I will change this later. We begin with this intuition that we can only predicate things that are there (for how can something not there substantially have properties that would modify it? The modifications, attributes, accidents, whatever, require something there that they modify or feature in or whatever). (Here is where I start guessing badly). But, there are things that we want to predicate existence to which are not there (but I am not sure what yet. Maybe God, maybe abstract things like justice. I have no idea still.) Thus, we need another means to express the existence of such things. That leaves as our only available option to use the existential quantifier. But Priest notes that this line of thinking is not very compelling. We can think of many things that do not exist but which still receive predication, as for example Zeus being worshipped by Homer. (Note that some of this discussion can be found in Priest’s Logic: A Very Short Introduction, ch.4.])
A historically influential argument for reading ∃ as ‘there exists’ is based on the claim that existence is not a genuine predicate (in some sense of ‘genuine’). If this is right, then it would seem that the only mechanism we have for expressing existence is the quantifier. (Of course, since even free logics with only inner quantifiers use an existence predicate, this is just as much an objection to these.) At root, the basis for this claim is the thought that to predicate anything of an object, it must be there, in some sense, to be available for predication. Maybe there is some sense in this thought, but identifying being there with existing is simply question-begging against someone who takes it that non-existent objects can have properties. And natural language would seem to have obvious counter-examples to the claim that an object must exist for one to be able to predicate something of it. Sherlock Holmes can be thought of without existing, and Zeus can be worshipped without existing.2
(296)
2. A more sophisticated argument against the claim that existence is a genuine predicate is to the effect that, if it were, the Ontological Argument for the existence of God – and of pretty much anything else – would be sound. But this does not follow. To run the Argument one needs not only an existence predicate; one needs also the principle that an object characterised in a certain way has its characterising properties (the Characterisation Principle). No one can accept this, whether or not existence is a predicate.
(296)
[Redefining the Inner Quantifiers]
[If we wish, we can define inner quantifiers in terms of outer ones, which means that “in a free logic with outer quantifiers, we can dispense with inner quantifiers altogether,” namely, in the following way: ∃ExA ∃x(ℭx ∧ A) ; ∀ExA ∀x(ℭx ⊃ A) (page 297). However, “There is no way of defining outer quantifiers in terms of inner quantifiers” (297).]
[(ditto)]
Two final comments. First, note that inner quantifiers can be defined in terms of outer quantifiers and the existence predicate. It is easy to check that the following pairs of sentences have the same truth values:
∃ExA ∃x(ℭx ∧ A)
∀ExA ∀x(ℭx ⊃ A)
Thus, in a free logic with outer quantifiers, we can dispense with inner quantifiers altogether. There is no way of defining outer quantifiers in terms of inner quantifiers.
(297)
[Deproblematizing Some Inferences]
[These new semantics make one problematic inference no longer problematic, namely, Ax(a) ⊨ ∃xA, now meaning that if something can be predicated, it is either an existent or non-existent object (previously it implied that any predicable thing must be existent). But it may not make the logical truth ∃x(A ∨ ¬A) unproblematic (it implies now that there must be at least a non-existent object, while before it implied there must be at least an existent object.)]
[In section 12.6.3, we encountered the following problem. There we saw that that Ax(a) ⊨ ∃xA is valid, meaning that anything that can be predicated must exist. But now in our current semantics, ∃xA does not imply the existence of the particularly quantified thing, so the validity of this inference should no longer seem problematic to us. The next point I do not follow well, but I will walk through it as best I can at the moment. In section 12.6.2, we noted that in classical first-order semantics, ∃x(A ∨ ¬A) is a logical truth. This means that for some predicate, something must exist that either takes or does not take that predicate. We found that problematic, because we might want to leave room for the case that nothing at all exists. Or at least, we can say that it does not seem like a logical truth that there cannot be nothing, even though that is what is implied by the logical truth ∃x(A ∨ ¬A). Priest now seems to be saying that even in our updated semantics, this validity of this formula does not seem so entirely palatable yet. It does mean that now we understand it as leaving open the possibility that nothing exists. (For, the x that is either A or the negation of A can be a non-existing object.) (The final point here I do not grasp, but I will just throw out a temporary guess to change later: However, we might still not find this logical truth palatable, for example, if we think that there are neither existent nor non-existent objects, or if we allow for truth-value gaps.)]
Second, if one interprets the quantifiers as outer quantifiers, the inference of 12.6.3, from Ax(a) to ∃xA, seems quite unproblematic. The fate of the inference of 12.6.2 is less clear. One cannot now object to the logical truth of ∃x(A ∨ ¬A) on the ground that it makes the existence of something a logical truth. It is less obvious that the logical truth of ‘something satisfies either A or ¬A’ is objectionable.
(297)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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