Chapter 2, ‘Taking Plurals at Face Value’, continues at an introductory level.
Oddly, Florio and Linnebo give almost no examples of the full range of plural expressions which they think a formal logic of plurals might aim to regiment (compare, for example, the rich diet of examples given by Oliver and Smiley in §1.2 of their Plural Logic, ‘Plurals in Mathematics and Logic’). Rather F&L start by immediately sketching three singularist strategies for eliminating plurals, starting the with familiar option of trading in a plural term denoting many things for a singular term denoting the set of those things.
They will be returning to discuss these singularist strategies in detail later. But for now, in their §2.2, F&L introduce the rival idea that “plurals deserve to be understood in their own terms by allowing the use of plural expressions in our regimenting language”. §2.3 then announces “the” language of plural logic. But that’s evidently something of a misnomer. It is a plural formal language, but — for a start — it lacks any function expressions (and recall how central it is O&S’s project to have a workable theory account of function expressions which take plural arguments).
F&L leave it open whether one should “require a rigid distinction between the types of argument place of predicates. An argument place that is open to a singular argument could be reserved exclusively for such arguments. A similar restriction could be imposed on argument places open to plural arguments.” But why should we want such selection restrictions? O&S remark very early on (their p. 2) that — bastard cases aside — “every simple English predicate that can take singular terms as arguments can take plural ones as well.” Are they wrong? And if not, why should we want a formal language to behave differently?
F&L seem think that not having selection restrictions would depart from normal logical practice. They write
In the philosophical and logical tradition, it is widely assumed that if an expression can be replaced by another expression salva congruitate in one context, then it can be so replaced in all contexts. This assumption of “strict typing” is true of the language of first-order logic, as well as of standard presentations of second-order logic.
But that’s not quite accurate. For example, in a standard syntax of the kind F&L seem to assume for singular first-order logic, a name can be substituted salva congruitate for a variable when that variable is free, but not when it is quantified. (As it happens, I think this is a strike against allowing free variables! — but F&L aren’t in a position to say that.) Any anyway, there is a problem about such selection restrictions once we add descriptions and functional terms, or so Oliver and Smiley argue (Plural Logic, p. 218). If we allow ostensibly plural descriptions and multi-valued functions (and it would be odd if a plural logic didn’t) it won’t in general be decidable which resulting terms are indeed singular arguments and which are plural; so having singular/plural selection restrictions on argument places will make well-formedness undecidable. (If F&L don’t like that argument and/or have a different account of ‘singular’ vs ‘plural argument’, which they haven’t previously defined, then they need to tell us.)
Moving on, §2.4 presents what F&L call “The traditional theory of plural logic”. I’m not sure O&S, for example, would be too happy about that label for a rather diminished theory (still lacking function terms, for a start), but let that pass. This “traditional” theory is what you get by adding rules for the plural quantifiers which parallel the rules for the singular quantifiers, plus three other principles of which the important one for now is the unrestricted Comprehension principle: ∃xφ(x) → ∃xx∀x(x ≺ xx ↔ φ(x)) (if there are some φs, then there are some things such that an object is one of them iff it is φ).
Evidently unrestricted Comprehension gives us some big pluralities! Take φ(x) to be the predicate x = x, and we get that there are some things (i.e. all objects whatsover) such that any object at all is one of them. F&L flag up that there may be trouble waiting here, “because there is no properly circumscribed lot of ‘all objects whatsoever’.” Indeed! This is going to be a theme they return to.
§2.5 and §2.6 note that plural logic has been supposed to have considerable philosophical significance. On the one hand, it arguably is still pure logic and ontologically innocent: “plural variables do not range over a special domain but range in a special, plural way over the usual, first-order domain.”
And pressing this idea, perhaps (for example) we can sidestep some familiar issues if “quantification over proper classes might be eliminated in favor of plural quantification over sets”. On the other hand, a plural logic is expressively richer than standard first-order logic which only has singular quantification — it enables us, for example, to formulate categorical theories without non-standard interpretations. F&L signal scepticism, however, about these sorts of claims; again, we’ll hear more.
The chapter finishes with §2.7, promisingly titled ‘Our methodology’. One of the complaints (fairly or unfairly) about O&S’s book has been the lack of a clear and explicit methodology: what exactly are the rules of their regimentation game, which pushes them towards what some find to be a rather baroque story? Why insist (as they do) that our regimented language tracks ordinary language in allowing empty names while e.g. cheerfully going along with the material conditional with all its known shortcomings? (What exactly are the principles on which conventionally tidying the conditional is allowed, but not tidying away the empty names?) Disappointingly, despite its title, F&L’s very short section doesn’t do any better than O&S. “We aim to provide a representation of plural discourse that captures the logical features that are important in the given context of investigation.” Well, yes. But really, that settles nothing until the “context of investigation” is articulated.
To be continued.
1. I don’t think that F&L need to be addressing O&S at every step. But, like it or not, O&S’s Plural Logic is the most seriously worked discussion of both philosophical and formal aspects of plural logic that we have. So I was expecting more direct engagement by F&L.
2. O&S: “We think that marking predicates for number is like marking them for person. We would never regard ‘I am F ’, ‘you are F ’, ‘he/she/it is F ’ as featuring three different predicates, and we shall be giving reasons to hold that the same predicate occurs in ‘it is F ’ and ‘they are F ’. We are with the grammarians who say that these are different inflectional forms of the same lexical item.” There are arguments to be engaged with on this topic, which F&L seemingly ignore.
3. Good question! My “for example” wrongly suggested I had other examples up my sleeve! (But how about a syntax with an absurdity constant which can only appear stand-alone and not as a sub-wff; that would be perfectly workable and useful, but would restrict allowable substitutions.)
4. For O&S, the classification of terms as singular/plural is modal and semantic (“can denote at most one thing” vs “can denote zero, one, or many things”), and that’s what leads to undecidability issues. Of course, F&L could propose a different criterion for distinguishing singular from plural terms. But they don’t. (Or at least, a possibly inadequate search didn’t turn one up.)
1. It looks to me that whenever F&L discuss something also discussed by O&S, you might want them to engage — even if their divergence from O&S is just leaving something open rather than addressing a claim O&S made on their p 2:
See also item 4, where you want F&L to address something because it comes up when terms are classified as singular or plural in the way O&S do it.
2. Person and number look like very different sorts of distinction to me. Singular ‘I’, ‘he/she/it’, and ‘you’ are just different ways to refer to an individual, so of course the same predicate can be applied. That doesn’t mean it must be possible and make sense to apply the same predicate to 100 individuals at once, or to an infinite number.
3. I don’t think such examples invalidate F&L’s basic point: they just require that, for their claim to be strictly correct, F&L need to complicate the way they express it. (And if O&S can make a claim ‘bastard cases aside’, F&L might be allowed something similar.)
4. I can’t find my copy of O&S 2nd ed, so I’ve looked at the 1st where p 208 seems relevant. Anyway, I think F&L are most likely making an essentially ‘syntactic’ distinction. If so, then they don’t have an undecidability problem. Instead, their problem (if it does count as a problem) is that the ‘syntactic’ distinction diverges from O&S’s ‘semantic’ one. Suppose I define a function and declare its return type to be ‘plural’, meaning it can return zero or more things. It might turn out that it in fact will always return exactly 1 thing. For O&S, that would make it ‘semantically’ singular. Since whether it returns 1 thing or some other number might depend on some complicated calculation, it’s not necessarily possible to determine by analysis how many values it returns. So it’s not decidable. It would still be ‘syntactically’ plural, though, and since ‘zero or more things’ can just be 1 thing, it doesn’t violate that ‘contract’. Anyway, that’s how things seem to me, and (provisionally, at least), I don’t mind F&L diverging from O&S in that way.
1. I approach Florio and Linnebo’s book from a different direction. For me, Oliver and Smiley’s book was too tendentious and ideological and, consequently, was in some places quite irritating to try to read. So I’m pleased that Florio and Linnebo are presenting things in their own way, rather than addressing Oliver and Smiley at every step.
2. It seems to me that English makes a singular vs plural distinction (for instance, ‘is’ vs ‘are’), and so having a singular – plural distinction in the formal language would make sense.
3. Is this “for example” the only example, or are there others?
4. This is surprising:
It looks like the sort of thing programming language implementations work out when determining what type the value of an expression will have. Is the problem that a description or function is only *possibly* plural or multi-valued, rather than definitely?