Posts Tagged ‘Plural Logic’

h1

Composition as identity, part I

December 11, 2008

I’ve been thinking a bit about the (somewhat radical) thesis that an object is literally identical with its parts. So, for example, these things, my parts, are identical to me. One nice thing about this is that you seem to get unrestricted composition for free: you get it from the plural comprehension schema.

However, its main drawback is it requires you to be able to make sense of many one identity. Lewis notes one problem with this, namely: my parts are many, whereas I am not. There are a couple of responses out there: Baxter takes this to be a failure of Leibniz’s law, and Sider has a language where plurals and and singular terms are intersubstituteable. Predicates are polymorphic and you can say truly that I’m both one, and many.

Both these views have crazy consequences (see Sider’s paper “Parthood” to see why.) So I’ve been trying to come up with a more natural way for the composition as identity theorist to go.

Note firstly that Alice, Bob and Fred are human iff Alice is human, Bob is human, and Fred is human. ‘Human’ is a distributive property. Consequently, the atoms that compose me are human iff each atom individually is human. They’re not, so the atoms that compose me aren’t human. However, there is a non-distributive property my atoms have, being human*, which some things have, roughly, if they compose a human. Thus I am human iff my atoms are human*.

So that’s the first step: every monadic predicate of the language, F, has a pluralised homonym, F*. For example, ‘one*’ is short for ‘many’: I am one, the atoms that compose me are one* (they’re many.) The second step: for every singular variable (or name), x,  there is a pluralised version, x*. I shall follow the tradition in plural logic, and use xx for x*. So, for example, ‘Andrew*’ is short for ‘Andrew’s parts’. Finally identity. We have one-one identity, =, many-one identity, *=, one-many identity, =*, and many-many identity, *=*. For n-place relations, well, you can work out your own notation, but it’s the same idea as identity.

We are now in a position to state Leibniz’s law. There are actually lots of versions, I’ll just state a couple

  • \phi^*(xx), xx ^*\!\!=y \vdash \phi(y)
  • \phi(x), x=^*yy \vdash \phi^*(yy)

(you must also add suitable identity axioms such as x =^* xx, xx ^*\!\!=x, etc…). So, for example, Fred is one, Fred is Fred’s parts (that is, Fred =* Fred*), therefore Fred’s parts are one* (Fred* are one*.) So, Fred’s parts are many. I’m human, I’m my parts, so my parts are human*. That’s the idea.

So much for identity. How do we get mereology out of this? Define x is a part of y, iff the xx’s are among the yy’s. Supposing \sqsubseteq is parthood, we have the following definition

  • xx ^*\!\!\sqsubseteq^* yy \leftrightarrow \forall z(z \prec xx \rightarrow z \prec yy)

where \prec is the ‘is one of’ relation from plural logic. Thus ^*\!\!\sqsubseteq^* is defineable in purely logical vocabulary, so if \sqsubseteq is truly a homonym parthood is logical. What’s more, unrestricted composition falls out from plural comprehension as desired.

But the other good thing about this formulation is that it avoids some of the crazy consequences Sider claims they get. For example, allegedly the principle: x is one of y_1, \ldots, y_n iff x=y_1 or … or x=y_n, fails. But his argument required moving between (in my language) ‘x is part of y’ and ‘x is part* of y’, rather than ‘x is a part* of yy’. Similarly he had to move between ‘x is-one-of xx’ and ‘x *is-one-of yy’ rather than ‘xx *is-one-of yy’ (his argument is just ungrammatical in this framework.)

Similarly, because he doesn’t pay attention to the difference between parthood, parthood*, *parthood and *parthood*, he gets all kinds of weird things coming out, e.g. ‘Tom, Dick and Harry carried the basket’ iff ‘Dom, Hick and Tarry carried the basket’, where Dom is the fusion of Dicks head and Toms body, Hick the fusion of Harry’s head and Dick’s bady, and Tarry the fusion of Toms head and Harry’s body. Following in the spirit of my rules, you can get from the LHS to ‘Tom*, Dick* and Harry* (carried the bucket)*’, where ‘(carried the bucket)*’ is a superplural predicate. But you can’t then swap bits from the plural terms ‘Tom*’, ‘Dick*’ and ‘Harry*’ and expect it to still satisfy (carried the basket)*.

Lastly, a predicate, P, is distributive iff P(x_1, \ldots x_n) \Leftrightarrow P(x_1) \wedge \ldots \wedge P(x_n). Sider claims there are no distributive predicates if you’re a composition as identity theorist. But again, the argument seems to rely on being able to freely move between plural and singular terms, without moving between the corresponding plural and singular predicates.

Ok, so it seems to be a natural way to formulate the position. That said, I think the position is ultimately incoherent, so I’ll talk a bit about that in the next post…

Advertisements
h1

Is second order logic really first order?

November 6, 2008

Nowadays, I guess, a lot more people are sympathetic to the idea that second order logic is real logic than in Quine’s day due to the popularity of plural logic. However, this falls short of full second order logic by quite a long way due to the fact that it can’t quantify over relations. For example, you can’t state various facts about sizes or the axiom of choice.

In the first order case, the question seems to be more tractable. If we identify the logical vocabulary as those terms that are not sensitive to the particular identities of the individuals (i.e. whose extensions remain unchanged if you permute the domain) then we get the cardinality quantifiers and arbitrary unions of the cardinality quantifiers as logical terms. McGee confirms the intuition that these truly are logical by showing that the permutation invariant (first order) vocabulary are precisely those defineable from intuitively logical operations: negation, identity, arbitrary conjunctions, universal quantification with respect to an arbitrary block of variables. Admittedly, this language (L_{\infty, \infty}) is not a language that anyone can speak, but that is a deficiency on our part, and should not place constraints on logic. Thus, first order quantification seems to be ontologically innocent, even for quantifiers like ‘there are uncountably many F’s’.

Indeed, similar results hold if we allow second order quantifiers. They are also permutation invariant, and conversely, the permutation invariant second order quantifiers is precisely those that can be defined in the equivalent of L_{\infty, \infty} with arbitrary blocks of second order quantifiers too. But the difference here, it seems, is that it is not clear that second order quantification over relations is ontologically innocent. Sure, plural quantifiers are, but as soon as we leave the realm of monadic quantification there is less reason to think so (although some have suggested that you can get around it: e.g. Burgess, and Rayo and Linnebo.)

Anyway, I was wondering if it would be possible to reduce second order quantification to first order quantification in our infinitary language. If this were possible then we could happily use the second order quantifiers and safely know that the are not ontologically committing, because they are definable using first order vocabulary.

I think you can do it, but I’m not entirely sure so this might be wrong. Let \kappa be antizero – the size of everything. For each second order variable X of the language keep aside \kappa many variables: x_\alpha for \alpha \leq \kappa. Then define a translation schema as follows: [UPDATE: I reformulated it slightly so that it wasn’t quite so confusing.] For a subset of the domain, I, we define the translate of \phi with repsect to I as follows:

(Xx)^I \mapsto \bigvee_{\alpha \in I}x=x_\alpha

(\forall X \phi)^I \mapsto \forall x_1 \ldots x_\kappa(\forall y\bigvee_{\alpha \leq \kappa}x_\alpha=y \rightarrow \bigvee_{J \subseteq \kappa} (\phi)^J)

For the other connectives and quantifiers translation just commutes in the natural way. A couple of notes: this isn’t like L_{\infty, \infty} in that it must allow truly arbitrary disjunctions and quantifications (including proper class length conjunctions.) Secondly, it’s not really as simple a translation as it looks because in the first clause I left I “free”, to be later “bound” by an earlier application of the second translation clause. What this really means is that the length of the disjunction in the first clause is really determined by when it is called in the second clause. Lastly – that’s just monadic quantification, which we already had – but it seems it will extend nicely to polyadic second order quantifiers (this time we disjoin (x = x_\alpha \wedge y = y_\alpha) instead.)

h1

Is the axiom of choice a logical truth?

October 5, 2008

I actually think there are a bunch of related statements which we might think of as expressing choice principles. The most striking contrast is probably the set theoretic statement of choice, and the choice principle as it is stated in second order logic: \forall R(\forall x \exists y Rxy \rightarrow \exists f \forall x Rxf(x)). I want to argue that the second principle is a purely logical principle, unlike the first, despite the fact that the question of whether or not the latter is a logical truth seems to depend on the (ordinary) truth of the former.

Let’s start off with the set theoretic principle. I believe this is non logical. Note, however, that this is not because of the Gödel Cohen arguments – I think set-choice is a logical consequence of the second order ZF axioms, given SOL-choice. It is rather because the ZF axioms themselves are non logical. For example consider a model with three elements such that: a \in b \in c – clearly c is a set of nonempty sets, but there isn’t a choice function for it because there aren’t any functions at all (that would require a set of set of set of sets.) Simply put: membership is not a logical constant, and so admits choice refuting interpretations. Note, I don’t mean to downplay the importance of the Gödel Cohen arguments; forcing and inner model theory are important tools in the epistemology of mathematics. Set-choice and CH may not be logically independent of the ZF axioms, but they do show us that, for all we are currently in a position to know, CH might be a logical consequence of second order ZF. It provides a method for showing epistemic independence and epistemic consistency, despite falling short of logical independence and consistency.

It might then be surprising to say that the second order choice principle is a logical truth. For following the Tarskian definition of logical truth for second order languages, i.e. truth in every set model, it follows that SOL-choice is a logical truth just in case set-choice is an ordinary truth (“true as a matter of fact”.) For example, if our metatheory was ZF+AD, SOL-choice would be neither a logical truth nor a logical falsehood!

I think this is to put the cart before the horse. Once the logical constants are a part of our metalanguage, then it is possible to do model theory in such a way that the non-logical fragment doesn’t affect the definitions of validity – indeed the non-logical component can be reserved purely for the syntax (see particularly, Rayo/Uzquiano/Williamson (RUW) style model theory.) So much the worse for Tarskian model theory.

But why think that SOL-choice is a logical truth or a logical falsehood, rather than neither? I guess I have three reasons for thinking this. Firstly, SOL-choice is stateable in almost purely logical vocabulary: Plural logic plus a pairing operation. While it is possible for it to fail under non-standard interpretations of the pairing function, it is enough to provide well orderings of many sets of interest: e.g. the plural theory of the real numbers gives us enough machinery for pairing, so well orderings under this encoding of pairs is possible. Secondly, SOL-choice is stateable in purely logical vocabulary. If you treat the binary quantifier “there are just as many F’s as G’s” as a logical quantifier, then you can state cardinal comparability in Plural logic+”there are just as many F’s as G’s” (which is certainly equivalent to choice in the ZF metatheory, I’m not sure what you need for this in the RUW setting.) I argued here that “there are just as many F’s as G’s” is a logical quantifier.

Lastly, imagine that we interpreted the second order quantifiers as ranging completely unrestrictedly over all pluralities there are. Suppose we still think that SOL-choice is not logically true or false. I.e. SOL-choice and it’s negation is logically consistent in the strong sense (not just that there are refuting Henkin models – that you can’t prove a contradiction from standard axioms.) Then there is a model in which SOL-choice is true, and a model in which it is false. But since our domain is everything, and the quantifiers in both models range over every plurality there is, the second order quantifier in the choice-satisfying model ranges over a choice function, which the second order quantifiers in the choice-refuting model must have missed. This is a contradiction, because we assumed that the quantifiers ranged over every plurality there is. Basically, choice-refuting models are missing things out. If there’s a choice interpretation and a ~choice interpretation for our unrestricted plural quantifiers, the choice model quantifiers range over more pluralities, in which case the ~choice model wasn’t really unrestricted after all. It seems then, that if SOL-choice is logically consistent, then it is logically true! (Note: this is kind of similar to the Sider argument against relativism about mereology. If there is an interpretation of our unrestricted quantifier that includes mereological fusions, and one that doesn’t, then the latter wasn’t really unrestricted after all.)