§0. A familiar if obscure idea: an indicative conditional presents its consequent as holding in the actual world on the supposition that its antecedent so holds, whereas a subjunctive conditional merely presents its consequent as holding in a world, typically counterfactual, in which its antecedent holds. Consider this pair: (1) If you had starved as a child, you would have been as tall now as you actually are. (2) If you starved as a child, you are as tall now as you actually are. I do not know how tall you are; I doubt that you starved as a child. My empirical evidence tells strongly against (1), by indicating that childhood malnutrition tends to reduce adult height. In contrast, (2) is logically trivial. How things actually are includes your actual height, whether or not it also includes your childhood starvation; how things would have been if you had starved as a child does not include your actual height. A 1 similar contrast arises between metaphysical and epistemic modalities. As a claim of metaphysical possibility, (3) is obviously true, while as a claim of epistemic possibility (4) is obviously false: (3) You could have been less tall now than you actually are.

English contains singular terms, quantifiers and predicates (e.g. ‘it’, ‘something ’ and ‘... is an elephant’). But it also contains plural terms, quantifiers and predicates (e.g. ‘they’, ‘some things ’ and ‘... are scattered on the floor’). 1 Philosophers have become increasingly interested in plurals over the past couple of decades. The purpose of this paper is to explain why plurals might be thought to have philosophical importance, and why they have led to philosophical debate. 1

This paper is divided into four sections. The first two identify different logicist theses, and show that their truth-values can be conclusively established on minimal assumptions. Section 3 sets forth a notion of ‘content-recarving ’ as a possible constraint on logicist theses. Section 4—which is largely independent from the rest of the paper—is a discussion of ‘Neo-Logicism’. 1 Logicism 1.1 What is Logicism? Briefly, logicism is the view that mathematics is a part of logic. But this formulation is imprecise because it fails to distinguish between the following three claims: 1. Language-Logicism The language of mathematics consists of purely logical expressions.

Most of the time our quantifications generalise over a restricted domain. Thus in the last sentence, ‘most of the time ’ is arguably not a generalisation over all times in the history of the universe but is restricted to a sub-group of times, those at which humans exist and utter quantified phrases and sentences, say. Indeed the example illustrates the point that quantificational phrases often carry an explicit restriction with them: ‘some people’, ‘all dogs’. Even then, context usually restricts to a sub-domain of the class specified by the count noun. Although teenagers like to have fun by being, they mistakenly think, overly literal — ‘Everyone is tired, let’s get to bed’: ‘everyone: you mean every person in the entire universe?’ — competent language users have to be sensitive to context virtually all the time. But is it always the case that generalisation is over a restricted domain? On the face of it, to claim this is paradoxical. If we say: (1) For every generalisation and every domain D, if D is a domain

The aim of this chapter is to tighten our grip on some issues about quantification by analogy with corresponding issues about identity on which our grip is tighter. We start with the issues about identity. I In conversations between native speakers, words such as ‘same ’ and ‘identical ’ do not usually cause much difficulty. We take it for granted that others use them with the same sense as we do. If it is unclear whether numerical or qualitative identity is intended, a brief gloss such as ‘one thing not two ’ for the former or ‘exactly alike ’ for the latter removes the unclarity. In this paper, numerical identity is intended. A particularly conscientious and logically aware speaker might explain what ‘identical ’ means in her 1 mouth by saying: ‘Everything is identical with itself. If something is identical with something, then whatever applies to the former also applies to the latter. ’ It seems perverse to continue doubting whether ‘identical ’ in her mouth means identical (in our sense). Yet other interpretations are conceivable. For instance, she might have been speaking an odd idiolect in which ‘identical ’ means in love, under the misapprehension

0. Logic is sometimes conceived as metaphysically neutral, so that nothing controversial in metaphysics is logically valid. That conception devastates logic. Just about every putative principle of logic has been contested on metaphysical grounds. According to some, future contingencies violate the law of excluded middle; according to others, the set of all sets that are not members of themselves makes a contradiction true. Even the structural principle that chaining together valid arguments yields a valid argument has been rejected in response to sorites paradoxes. In each case, a deviant metaphysics corresponds to the deviant logic. Of course, if one is trying to persuade deviant metaphysicians of the error of their ways, one is unlikely to get far by relying on logical principles that they reject. But that obvious dialectical exigency stably marks out no realm of logic. Each logical principle has persuasive force in some dialectical contexts and not in others. We do better to admit that logic has metaphysically contentious implications, and embrace them ─ if we know what they are. 1 Logic and metaphysics are not mutually exclusive. They overlap in the logic and metaphysics of existence, identity and possibility, for instance. The exploration (but not