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On the one hand, the logical predicates $\forall$ and $\exists$ are defined using the concept of a Domain of Discourse, which itself is defined as a set (at least according to wikipedia). On the other hand, axiomatic set theory defines the basic properties of set using these quantifiers (e.g. the Axiom of the Empty Set: $\exists x\forall y\,\neg(y\in x)$). This seems like a circular definition. If someone could please clarify this, I would be most greatful.

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  • $\begingroup$ I'm not a logician, so I don't know whether my understood is right. $\forall$, $\exists$ are only logic symbols, in the sense of formal language, namely, first-order logic. Systems such as ZFC is also first-order logic systems. $\endgroup$
    – Yai0Phah
    Feb 7 '14 at 12:19
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    $\begingroup$ But when you are considering the semantic of a logic system, you need a model for that. It seems me that logicians strictly distinguish the concept of a formal language and its models. $\endgroup$
    – Yai0Phah
    Feb 7 '14 at 12:23
  • $\begingroup$ For example, see Gödel's completeness theorem $\endgroup$
    – Yai0Phah
    Feb 7 '14 at 12:24
  • $\begingroup$ Strictly speaking, the logical quantifiers are not "defined" with appeal to axiomatic set theory. Only if you want to "formalize" the semantics of quantifiers in a mathematica theory, you need the resources of something like $ZFC$. You explain the usage and meaning of quantifiers with the concept of domain of discourse, but also to define what is an expression you must start with a set of symbols, and so on. So, there is really some sort of circularity, but this is unavoidable. This sort of "circularity" doesn not mean that the explanations and clarifications provided are useless. $\endgroup$ Feb 7 '14 at 12:24
  • $\begingroup$ @MauroALLEGRANZA So we do things in the ambient space, founded by, say, ZFC or other systems, but we still study a logic system such as ZFC, considered a formal system in the ambient space and our deduction is based on the formal rules of the logic system and the principles of the ambient space, right? $\endgroup$
    – Yai0Phah
    Feb 7 '14 at 12:30
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The logical predicates ∀ and ∃ are defined using the concept of a Domain of Discourse, which itself is defined as a set.

This much is true: to fix the content of e.g. $\forall xFx$, we need to know which objects the quantifier is ranging over -- i.e. which objects are such that each of them supposedly satisfies the predicate $F$.

But to understand $\forall xFx$ we don't have to assume that the objects which are being quantified over form a set. (If quantification required the objects we are quantifying over to form a set, that would be very bad news for set theory! -- for in a quantified claim of ZFC, the quantifiers are supposedly quantifying over all sets, yet according to ZFC itself the sets do not themselves form a set!)

It is a quirk of the history of logic that the formalized theories of logic inference that became canonical aimed to regiment singular reference and associated quantifiers and ignored plural reference and plural quantifiers (even though we use plural talk in informal maths all the time). So in formally regimenting our informal semantics for quantifiers we find ourselves substituting natural talk about the objects (plural) a quantifier runs over by talk about the domain of quantification (singular). But that's an artefact of our formalisation, not an insight!

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    $\begingroup$ I'm not sure that's entirely right. For instance, when I attempt to make absolutely general claims, I may not know which objects my quantifier ranges over (other than the uninformative ``all objects whatsoever"). Whether I manage to effect absolutely general quantification is, of course, a point of debate. Nevertheless, it seems that when I attempt it, (1) I don't know, in any informative sense, what the range of my quantifiers is; and yet (2) my claims still have content. $\endgroup$
    – GME
    Feb 7 '14 at 18:24
  • $\begingroup$ Yes, I agree with you entirely about natural language quantifiers. I was considering quantifiers in a regimented first-order language with a determinate range for its quantifiers. $\endgroup$ Feb 7 '14 at 18:35
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    $\begingroup$ Fair enough. If one builds in that the quantifiers have to have a range which is informatively specifiable (in some relevant sense), then sure. But, just to be clear, I don't think regimentation in a first-order language is enough to deal with the worry. After all, we can always regiment our metaphysics (for instance). In other words, there's no reason that $\exists$ couldn't be used to formulate purportedly absolutely general claims in a regimented language. $\endgroup$
    – GME
    Feb 7 '14 at 18:42
  • $\begingroup$ I think I'd rather put it this way. Suppose, doing metaphysics, we have (purported) natural language absolutely general quantifiers. Then can we regiment our metaphysics into a first-order language with the usual understanding of the semantics (pluralized to avoid unnecessary set-talk)? Yes, if there is a determinate totality of all that there is. No, if there isn't. $\endgroup$ Feb 7 '14 at 20:31
  • $\begingroup$ Oh, I see. If you put it that way, then I don't see that we can't regiment our metaphysics into a first-order language with a plural semantics (pluralities as domains and as interpretations of (monadic) predicates). I mean, take the plurality of absolutely everything. Perhaps it's not determinate, but prima facie (and, I'd argue, on reflection) that's a different question. $\endgroup$
    – GME
    Feb 7 '14 at 20:50
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We need not think of $\forall$ as defined in terms of a domain of quantification. Rather, we can take it to be a ${\it primitive}$ of our language. Definition has to stop somewhere, and the quantifiers seem like a pretty basic place to stop! Of course, a quantifier may ${\it have}$ a domain of quantification even if it isn't defined in terms of it. "is red", for instance, isn't defined in terms of the set of red things, although its extension (the set of things it applies to) is the set of red things.

Here's one way to see quantifiers aren't in general defined in terms of sets. It is perfectly coherent to think there are no sets (although it's not true) -- that is, to think $\neg\exists x(x$ is a set$)$. Perhaps one thinks that the only things which exist are physical. But if the quantifier $\exists$ were defined in terms of sets, it wouldn't be coherent to think this.

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The quantifiers are symbols in syntax, nothing more. The string $\exists x: \forall y: \neg (y \in x)$ is simply a string in a "game of symbols" (axiomatic set theory). The axioms and proof rules serve for manipulating strings (from the viewpoint of the metatheory).

Remaining within the object theory, we take a proof theoretic approach. In this case, we usually use a finitistic metatheory.

When we interpret the symbols of a theory in a metatheory that contains "sets" (e.g., the "domain of discourse"), then what we are doing is using as our metatheory set theory itself. It could be the same theory. So, the metatheory in this case involves infinitary reasoning. This approach is usually called model theory.

One way to think about this is that we have set theory as rules about symbols, and then we can ask the question "Can we encode this theory itself using values (sets) within itself?". Gödel did so to show that if we encode ZF within itself and try to prove it is consistent, then either we cannot prove it (in case ZF is consistent), or we can (and that shows ZF is inconsistent).

Kenneth Kunen gives a very good introduction to the subject in his book Foundations of mathematics.

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  • $\begingroup$ This is only one approach to the philosophy of mathematics—not the only one. Mindless manipulation of symbols according to rules must require a thinking mind already armed with understanding of the semantics for there to be any knowledge involved. Knowledge cannot exist in the absence of a mind. Your answer isn't necessarily wrong but it's very limited in scope (and it's a bit confusing). $\endgroup$
    – Wildcard
    Jan 17 '17 at 9:53
  • $\begingroup$ I do not know what "knowledge" means. There are well-formed formulae (strings), some are axioms, and there are some rules for manipulating strings to obtain other strings. Douglas Hofstadter discusses this well in GEB. Neither is "mind" defined anywhere within a formal theory. $\endgroup$ Jan 17 '17 at 10:00

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