Some Ideas on Logic

المخاطبات- العدد 14- أفريل 2015 AL-MUKHATABAT ISSN 1737-6432 Numéro-Issue 14 Avril-April SOME IDEAS ON LOGIC

Colin MCGINN (Auteur indépendant)

(1) Inverted Logic

Abstract. It is argued that classical logics are not the only genuinely logical systems. In addition to modal logic, deontic logic, epistemic logic, and other recognized systems, we must make room for a logic devoted to predicates. This can be done by treating predicates as logical constants and rendering every other expression by means of schematic letters. The result is an inverted logic in which what are logical constants in one system are schematic letters in another. Logic is really the study of any kind of entailment, no matter what the category of expression is.

ملخّص. لا يعتبر المنطق الكلاسيكي على نحو حقيقي النسق المنطقي الأوحد، فبالإضافة الى منطق الجهات و المنطق التوجيهي و المنطق الابستمي و بعض الأنساق الأخرى المعترف بها، يجدر بنا أن نخ ّصص مجالا لمنطق خاص بالمحمولات يقوم على معالجة هذه الأخيرة باعتبارها ثوابت منطقية و استصفاء ك ّل عبارة أخرى بواسطة أحرف التمثيل. و تكون النتيجة عندئذ منطقا معكّوسا يكون فيه ما يعتبر ثوابت منطقية في نسق ما أحرف تمثيل في نسق آخرا، فالمنطق هو حقا دراسة أ ّي ضرب من ضروب الإستلزام بغ ّض الطرف عن ماهية

مقولة التعبير.

Résumé. Les systèmes logiques classiques ne sont pas à vrai dire les seuls systèmes. A côté de la logique modale, la logique déontique, la logique épistémique et d’autres systèmes bien connus, nous devons laisser la place à une logique dévouée aux prédicats. Cela peut être réalisé en traitant les prédicats comme des constantes logiques et en rendant compte de toute autre expression au moyen de lettres schématiques. Le résultat est une logique inversée dans laquelle ce qui est une constante logique dans un système devient une lettre schématique dans un autre. La logique est vraiment l’étude de tout type d’inférence, sans préférence pour une catégorie d’expressions en particulier.

The standard logical systems of propositional calculus and predicate calculus include two sorts of symbols: logical constants and schematic letters. Thus we have the constants “and”, “or”, “not”, “if”, “for some x”, “for all x”; and the

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schematic letters “p”, “q”, “F”, “G”, “a”, “b”. A formula will contain a mixture of these symbols: for example, “p and q” and “for some x, (Fx)”. What such formulas say can be expressed informally as follows: “the conjunction of any two propositions” and “something is such that it has a property F”. When stating the entailments of formulas in these logical systems (i.e., propositional calculus and predicate calculus), we say things like: “No matter what the two propositions p and q you conjoin are, you will have each of p and q as an entailment”, or “No matter what F you choose is, if everything is F, then a particular thing is F”. The aim is to express the generality of the entailments that depend on the logical constants in these systems; this is achieved by employing a combination of schematic letters (“placeholders”) and expressions with a constant meaning (“interpretation”). Hence we say that any argument of a certain general form is valid, where the form is fixed by the logical constants and the schematic letters. Logical form is the residue left when ordinary interpreted expressions of certain categories are replaced by schematic symbols, leaving only the designated logical constants.

Different logical systems may treat different expressions as logical constants with characteristic entailments. Modal logic adds new symbols:  and , which represent the constants of “necessarily” and “possibly” respectively, to standard systems and investigates the entailments thereby generated. Similarly for epistemic logic, deontic logic, tense logic, indexical logic, mereology, and so on. It is a question whether the expressions treated as logical constants in these various systems have anything interesting in common—is there a well-defined notion of a logical constant that transcends what we are treating as a constant in various systems. Is it merely arbitrary what we call a logical constant? Could any expression be a logical constant in some system? It may seem that the answer to that last question, at least, must be no, since no existing system treats sentences and predicates as logical constants. But could we construct a logical system that treats predicates, say, as logical constants, with other expressions treated as schematic letters? That is, can we invert the roles of the two sorts of expression in standard logic? Can we coherently treat predicate expressions as logical constants and connectives and quantifiers as schematic letters? Can we thereby investigate the entailments of predicates, in particular, by generalizing over other semantic categories? If we can, the question of what counts as a logical constant becomes completely system-relative; at any rate, predicates will be seen to possess a “logic”, as much as connectives and quantifiers.

It is actually quite easy to construct systems of this kind. I will call such a logical system a “predicate logic”, contrasting it with what are better called

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“truth-functional logic” and “quantifier logic”. The question is what category of expressions are we constructing a logical system for—what entailments are we seeking to capture? Once we select such a category, we formulate the entailments by generalizing over the other semantic categories, by use of suitable schematic letters. So consider a system containing two interpretedpredicates, “is a vixen” and “is a lioness”–the logical constants of the system– along with associated schematic letters. There are two cases to consider: a truth- functional system and a quantifier system. To deal with the former, add a pair of symbols, “a” and “b”, which can stand for particular animals. Then we can form conjunctions, disjunctions, and negations of whole sentences, like “Not (a is a vixen and b is a lioness)”. But suppose we don’t use particular interpreted connectives; instead we use schematic letters that hold a place for any connective. For example, let us use “C” to stand for either conjunction or disjunction (a schematic letter for two-place truth-functional connectives). A formula of this language will then look like this: “a is a vixen C b is a lioness”, where “C” functions as a schematic letter replaceable by particular connectives. Now we ask about the entailments of such a formula. Since the predicate “is a vixen” entails “is a female fox” and “is a lioness” entails “is a female lion”, theentailments of the complex sentence follow simply from these two entailments; so we obtain, “a is a female fox C b is a female lion”. We may infer this formula from the previous formula, based on the entailments of the two predicates. We don’t need to worry about what C is. If C is conjunction, then the entailed sentence is a conjunction that follows logically from the first sentence (by virtue of the meaning of the predicates), while if C is a disjunction, then the disjunctive sentence also follows logically. No matter how we interpret the connective schematic letter, the inference goes through, because the entailments depend purely on the predicates involved. The connectives don’t interfere with these entailments, just as the identity of the predicate doesn’t affect the entailments due to the connectives in classical logic. We can generalize over connectives by replacing them with schematic letters, thereby focusing on the logical properties of the predicates in the formulas. Thus we treat the predicates as logical constants in this system of “predicate logic” and the connectives as mere placeholders. (We can do the same for one-place connectives like negation, but there is only one to consider, since standard logic contains only negation as a one-place connective—still, the distinction between constant and schematic letter applies also with respect to the class of one-place connectives.)

Now we move to languages with quantificational structure, so we will need an apparatus of individual variables to go with the quantifiers. A typical formula would be this: “For some x, x is a vixen” or “For all x, x is a lioness”. Now we

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introduce a schematic quantifier letter “Q”, to be read “any quantifier”, so that we can write, “Qx, x is a vixen”. The letter can be read disjunctively—either existential or universal quantification—as the schematic predicate letters of standard logic can be read as disjunctions of interpreted predicates (either “is white” or “is red” or “is a man”, etc). What are the entailments of a formula containing the letter “Q” in this system? There are none to speak of in virtue of that letter, since the entailments of quantifiers depend on the particular quantifier, not the general category of being a quantifier. But the predicates in the formula have their usual entailments. Thus we can say: anything of the form “Qx, x is a vixen” will entail “Qx, x is a female fox”, no matter what quantifier we substitute for “Q” is (existential or universal). We can also, of course, deduce “Qx, x is a fox” from “Qx, x is a vixen”. Again, we have generalized over quantifiers to tease out an entailment due solely to a specific predicate. It is easy to see that the same principles will apply once we start constructing complex formulas using connectives and embedded quantifiers. We will be able to write things like: “Qx, Qy (x is a vixen C y is a lioness C x is smaller than y”, which has such substitution instances as: “For all x, there is a y such that x is a vixen and y is a lioness and x is smaller than y”, which we obtain simply by substituting on the schematic letters “Q” and “C”. We can obviously make other substitutions–say, by inverting the initial quantifiers or using disjunction not conjunction. It is easily seen that, no matter what we replace the schematic letters by, we will derive the same consequences in virtue of the meaning of the constant predicates “vixen” and “lioness”. The validity of the inference does not depend on the choice of quantifier or connective, but solely on the meaning of the specific predicates.

Thus we have inverted the usual procedure of standard logic by treating different expressions as logical constants and schematic letters. What is going on here can be seen by considering an ordinary sentence like, “Every vixen is smaller than every lioness and some lionesses are nimbler than some vixens”. Here we have quantifiers, connectives, and predicates combined to produce a sentence. That sentence has various entailments in virtue of expressions in each semantic category. We can decide to focus on certain of these entailments by making logical generalizations. We do this by treating some expressions as logical constants and replacing others by placeholders. Thus we say that any conjunction of propositions entails each conjunct, or that if everything has a given property then each thing has that property. But we can also say that if anything is true of vixens it must be true of female foxes—whether involving quantifiers or connectives. We can express this in a logical system by generalizing over other types of expressions while keeping the predicate fixed.

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If we are interested in the logic of zoological taxonomic terms specifically, we can construct a system that focuses on their entailments alone, by replacing everything else with schematic letters. Another way of putting it is this: standard logic looks at the analytic entailments of connective concepts and quantifier concepts; an inverted logic looks at the analytic entailments of predicate concepts. To take a very simple example, we can investigate the logical properties of a sentence like “a is a vixen”. There are no connectives or quantifiers in this sentence, so the logic of connectives and quantifiers need not be invoked; but we can spell out the logical properties of the contained predicate, noting that it entails “a is a female fox” and “a is a fox”. Here we will say that it doesn’t matter what object you refer to in a simple sentence like this, the predicate entailment still holds. We might also note that the sentence entails “Something is a vixen” and is entailed by “Everything is a vixen”, focusing on the subject term and generalizing with respect to the predicate. Either procedure is acceptable and simply depends on what kind of entailment we want to highlight. Every word in a sentence has entailments of some sort, so every word has a logic associated with it. Standard propositional calculus and predicate calculus single out certain words for logical attention, but we can single out other words and not be accused of arbitrariness or error, as in the inverted logic. We can therefore add predicates to the list of words that can coherently be treated as logical constants, such as modal, epistemic, deontic, or mereological words.

It is true that the predicates I have selected do not occur as often in speech or writing as other words, so that their logic is not as pervasive as that of other words (e.g. “and” or “not”). But that is a merely statistical fact, with no bearing on logical questions. Should we say that epistemic logic isn’t really logic because “know” does not occur as often as “and” and “not”? Doesn’t it depend on the kind of speaker you and the kind of subject matter that most occupies you? You may have an intense and exclusive interest in simple subject-predicate propositions about knowledge, not caring to conjoin or disjoin such propositions or insert quantifiers into them. It is the logic of epistemic concepts that preoccupies you, not the logic of conjunction, disjunction, and quantification. Then it will be natural to construct logical systems based around the concept of knowledge, ignoring systems that focus on truth-functional compounding and quantifiers. Similarly, you may be gripped by the logic of zoological nouns like “vixen” and “lioness”, because of your frequent interactions with such animals as foxes and lions, being quite indifferent about those other constructions. You may find connectives and quantifiers logically boring. Then you will put your efforts into fashioning logical systems that

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formalize predicate entailments, maybe expanding your logical interests beyond your initial zoological preoccupation. You will leave connective and quantifier logic to those with different predilections. And it is noteworthy that such logics sprang up among mathematicians interested in formalizing the sentences of mathematics, in which quantifiers play a central role. But zoologists with little interest in general laws might have different logical interests: they may want to know how words for animal groups are logically related. They might then naturally develop a side interest in words for kin relations: “bachelor”, “spinster”, “widow”, “brother”, and so on. They thrill to the logical proposition that a brother is a male sibling, or that no one can be both a widow and a wife, or that husbands cannot avoid being married. What fascinate these logicians are the logical relations between classes of predicates; and such words occur most frequently in their speech. To them our favored logical systems, with their chosen logical constants, may appear perverse—inversions of the natural and universal logical order. They wonder why we are so interested in such bland logical material (what is so fascinating about “or”?). It might even be that they worked out our logical systems long ago, finding them trivial as well as boring; they find “predicate logic” systems far more intellectually challenging. This is what they teach in university logic courses, not our preferred systems (everybody would get an easy A in our systems).

Inverted logic is thus really a species of logic. Formally, it works in the same way orthodox logic works, with selected logical constants and general schematic letters. If this is right, then the question of whether the standard logical constants have any special status becomes particularly pressing. Maybe the right thing to say is that every word has a meaning that carries certain characteristic entailments, so that every word can be treated as a logical constant relative to other words, depending on interests. The word “and” is no more absolutely a logical constant than is the word “vixen”. Every word can be either a logical constant or not a logical constant, depending on the system. Logic is simply the theory of entailment, and entailment is a trait of every word. Any system that formally captures entailments deserves to be called a logical system.

(2) Logic without Propositions (or Sentences)

Résumé. Ce papier fait valoir que la logique ne doit pas être conçue comme l’étude des relations logiques entre les propositions. La logique est plutôt l’étude de la structure logique de la réalité objective, telle qu’elle existe en dehors des propositions. Néanmoins, il peut y avoir une logique des propositions, si cela est dérivé en premier lieu d’une logique de la

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réalité. Le réalisme des universaux joue un rôle clé dans l’argument. L’objectif général est de mettre à nu le fondement métaphysique de la logique.

ملخّص. تحاول هذه الورقة إثبات أ ّنه لا يمكن اعتبار المنطق دراسة للعلاقات المنطقية بين القضايا، إذ هو بالأحرى دراسة البنية المنطقية للواقع الموضوعي كما يوجد خارج القضايا، و مع ذلك فإّنه بالإمكان وجود منطّق للقضايا (مع أّنه في ذاته مست ّل من منطق سابق للوقائع). تلعب النزعة الواقعية في الكليات دورا مركزيا في هذه المحاججة، و هدفنا العام هو بيان الأسس الميتافيزيقة للمنطق.

Abstract. This paper argues that logic should not be conceived as the study of the logical relations of propositions. Rather, logic is the study of the logical structure of objective reality, as it exists outside of propositions. Nevertheless, there can be a logic of propositions, though this is derivative from a prior logic of reality. Realism about universals plays a key role in the argument. The general aim is to provide the metaphysical basis of logic.

The way logic has been presented for a hundred years or so is as a theory of the logical relations between propositions. Propositions have entailments and figure as the premises and conclusions of arguments. Not much is said about the nature of propositions in the standard explanations of logic, but we are to assume that they correspond to the meaning of sentences—declarative sentences. So logic deals with representational entities—things that stand for states of affairs in the world. It does not deal with states of affairs themselves— with objects and properties. Sometimes talk of propositions is “eschewed” (Quine) and sentences are made the subject matter of logic, construed as marks and sounds, or some such. Then we hear what is called “propositional logic” described as “sentential logic”. If we wanted to go one stage further in the direction of concreteness, we could re-describe propositional logic as “statement logic” or “utterance logic”, where these are conceived as actual speech acts. Thus we would investigate the logical relations between speech acts. It is the same for what is called “predicate logic”: logic investigates the logical relations between predicates, especially as they interact with quantifier expressions. We are still investigating sentences, but we analyze them into predicates and quantifiers. If we don’t like the talk of predicates (bits oflanguage), we could re-name this branch of logic “concept logic”: then proposition logic and concept logic would both deal with what is expressed by language, while sentential logic and predicate logic address themselves to

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linguistic expressions. No matter how we formulate it, logic is conceived to operate at the level of representational entities, with logical relations defined over these entities. Logic is essentially concerned with the discursive. So conceived, modern logic is “the logic of (discursive) representations”. Consequence, consistency, and contradiction are all regarded as relations between sentence-like discursive entities (propositions, sentences, statements, assertions, utterances, speech acts). Premises and conclusions of arguments are precisely such entities. The laws of logic are the laws of the logical relations between these entities.

But there are two points about logical laws that call this representational conception into question. The first is that we presumably want logical laws to apply to worlds in which there are no representations. Suppose that no representational beings had ever evolved in the universe, so that there are neither sentences nor propositions (I will ignore Fregean Platonism about propositions)–there is no language and no thought. Then logical relations defined over representations will not exist in that universe; there will be no logical laws of this kind. But will there be no logical laws of any kind? Surely not: the universe will still be governed by the laws of logic, as they are traditionally conceived. Contradictions will still be impossible, by the laws of logic: but they will not be defined over anything propositional. Logical laws like this are no more language-dependent than natural laws, such as the law of gravity. We can state logical and natural laws by means of propositions, but the laws themselves don’t concern propositions. The laws can exist without the existence of any statement of them. So logical laws are not inherently propositional: they can hold in a world in which there are no propositions (a fortiori for sentences and speech acts). The universe would be subject to the laws of logic even if no thinking beings ever came into existence.

The second point is that the traditional way of formulating logical laws does not make them about propositions or sentences. Thus: “Everything is identical to itself”; “Nothing can both have a property and lack it”; “Everything either has a given property or lacks it”. In stating these logical laws no mention is made of propositions or sentences; the subject matter consists entirely of objects and their properties. There is thus no need to invoke propositions when stating logical laws; and such laws can clearly hold in a world without representations—you just need objects and properties, with logical relations defined with respect to them. Then are there two kinds of logical law—laws of propositions and laws of objects and properties? That seems unappealing: one would like a uniform account of what a logical law is. The same goes for non-

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standard logics, like modal or deontic logic: they also hold in worlds that contain no propositions (or sentences). If we add to the traditional three laws of logic so as to include further logical truths, such as Leibniz’s law of identity or the logical relations between colors, then again we have logical laws that are not defined over propositions—they concern the logical nature of the identity relation and of color themselves. They deal with logical necessities that are not formulated by reference to propositional entailments: it is a logical truth about identity, say, that (in addition to being reflexive, transitive, and symmetrical) if a is identical to b, then a and b have all properties in common. Again, we talk here only of objects and properties (or relations), not of propositions about them. These are de re (referential) necessities, not de dicto (non referential) necessities. Identity itself entails that identical objects are indiscernible, not propositions about identity; just as having a particular property itself entails not having the negation of that property, not propositions about the property. Logical facts obtain independently of discursive entities like propositions or sentences.

Rather than accepting that there are two kinds of logical laws, it would be better to demonstrate some kind of relationship of dependence between them. It seems too much simply to deny that propositions enter into logical relations, since that would be to condemn standard logic as completely misguided, based on an outright falsehood. Instead, we could try to see its entailments as derivative from deeper logical laws that are not inherently propositional: thus propositions have “derived logicality”. But how do we set about doing that? I propose that we re-conceptualize the matter along the following lines. Suppose we accept an ontology consisting of particulars and universals (objects and properties); then we can distinguish the following three areas of investigation: (i) which particulars instantiate which universals, (ii) what the nomological relationships are between universals, and (iii) what the logical laws governing universals are. That is, there are three sorts of fact about universals: first, which objects fall under them, how many, and so on; second, what laws of nature apply to universals (e.g. the laws of motion); third, what logical characteristics universals have. Each of these questions is about universals themselves, not about propositions or concepts. We are interested here in the third question, but it is worth observing how it relates to the other two questions, which are clearly not at all concerned with propositions or sentences. And the answer we would give will reflect the nature of the question: we will refer only to universals and their inherent logical relations (though of course we will be using propositions or sentences to do so).

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These relations, I suggest, will be of four basic kinds: identity, exclusion, consequence, and combination. Logic is then fundamentally about these four basic logical relations—with proposition-centered logic depending on the more basic logical facts. The logical relation of identity is captured in the law of identity for universals (also objects): every universal is identical to itself, and to no other universal. Then we will detail the logical properties of identity, noting also that identity is a necessary relation. None of this concerns propositions or sentences about identity, though there will be consequences for identity statements of familiar kinds. By exclusion I mean the way one universal excludes others from being instantiated in the same object—any which is incompatible with the first. Thus being square will exclude being not square, say by being triangular or circular. Every universal necessarily (logically) excludes other universals—that is a logical law. This is a de re necessity, not a truth about concepts or predicates: it could obtain without there being any concepts or predicates. By consequence I mean the way one universal can be sufficient for another: it is sufficient for being an animal that something is a cat, sufficient for being a man that someone is a bachelor, sufficient for having a successor that something is a number. One universal necessitates another, and perhaps another in turn. Logic (in a broad sense) traces out these consequence relations. By combination I mean logical properties of collections of universals: for example, if an object x instantiates a collection of universals U, then x instantiates each member of U; and if an object x instantiates a given universal F, then x instantiates F or any other collection of universals (these laws are intended to correspond to the standard rules of conjunction elimination and disjunction introduction). The idea here is that we can move from facts about collections of universals to facts about specific universals, and from facts about specific universals to facts about collections of them. Intuitively: if x instantiates F and G, then x instantiates F; and if x instantiates F, then x instantiates F or G. Here we logically link objects with universals considered as members of collections. Objects can be in the intersection of two universals (F and G) and be in the union of two universals (F or G).

All these logical laws are stated over objects and properties. The claim then is that this is the metaphysical basis of logical laws as they are stated over propositions. It is in virtue of the former laws that the latter laws hold. It is fairly obvious how this goes: we just need to make a step of semantic ascent. Thus: if being F necessitates being G, then “x is F” entails “x is G”–and similarly forexclusion. The logical laws of “and” and “or” fall out of logical laws concerning objects and properties, as just outlined. The law of existential generalization is based on the fact that if a particular object instantiates a universal then something

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does; and the law of universal instantiation is based on the fact that if everything instantiates a given universal then any particular thing does. In the case of “not”, used as a sentence operator, we can take negation as applicable to universals themselves, so that not-F is itself a universal. Then we can interpret “not-not F” as meaning “the negative of the negative of F”, i.e. F. Alternatively, we can construe negation as equivalent to “belongs to the complement of F” (e.g. “x is not red”). What we are doing is simply taking negation to apply to properties, not concepts or words; and similarly for conjunction and disjunction. An object can have the property of being F and G, or the property of being F or G, or the property of being not-F. All the standard so-called sentence operators have a more fundamental interpretation as operations on universals, forming complex universals from simpler ones. There are then logical relations between these universals, and hence logical laws. This allows such laws to obtain in worlds that lack language or anything representational. It makes them de re not de dicto—about reality not our description of it.

We could express all this by speaking of states of affairs, but I think we get the basic ontology right by sticking to talk of objects and properties (particulars and universals)—these being what states of affairs are all about. Objects and properties have logical laws governing them, on this conception, as they have natural laws governing them, and as they form particular facts about the distribution of properties in the universe. None of these facts depends on propositions or concepts or words. Of course, we can formulate propositions about these laws and facts, but they are not themselves constituted by anything internal to propositions. A logical principle stated at the level of propositions is thus derivative from the more basic level of the logic of universals. Predicates entail other predicates because the universals they denote or express themselves necessitate other universals—this being an entirely non-linguistic matter. So- called predicate logic is really property logic, seen through the prism of language. Strictly speaking, predicates don’t have logical relations, except derivatively on properties. If there were no properties obeying logical laws, then there would be no predicate logic. If there were no universals that inherently exclude each other, then there would be no law of non-contradiction at the level of propositions or sentences. Words cannot inherently exclude one another, and neither can concepts, construed independently of properties (as, say, dispositions to assent, or bits of syntax in the language of thought). The things that stand in logical relations at the most fundamental level are objects and properties; any other logical relations are transmitted upward from that basis. It is meaning that transmits logic from its original home in the world to language. If we try to view meaning as cut off from objects and properties, then

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we lose logic defined at the discursive level. Objects and properties are “logic- makers” as well as “truth-makers”—they are ultimately where logical truth and truth in general come from. Nothing is true but reality makes it so, as Quine said—even logical truth.

Frege described logic as “the laws of truth”, thus locating it at the level of truth bearers (“thoughts” in his terminology). But this very formulation points to a different conception, since truth turns on the condition of the world beyond representation—and likewise for logical relations. Just as a proposition is true in virtue of the way the world is, so its entailments hold in virtue of the way the world is—specifically, the logical relations between universals. The truth- makers are also the logic-makers. The laws of logic are not fundamentally laws of truth but laws of what make truths true—that is, the logical matrix in which universals are embedded: identity, exclusion, consequence, and combination. Logic does not exist independently of the world, as if confined to the level of propositions—as if it reflected the structure of human thought—but rather is immanent in the world, part of what constitutes it. It is not that we impose logic on the world, having first found it in thought; rather, logic imposes itself on thought, having its origin in the world beyond thought. The propositional calculus and the predicate calculus, as they exist today, are really encodings of a mind-independent logical reality, which exists outside of sentences and propositions; they are not the primary bearers of logical relations (the same goes for modal logic, etc).

This way of looking at things clearly depends on a robust ontology of properties or universals—they cannot be identified with predicates or even concepts in the mind, or else the contrast I am insisting on would collapse. The logic of universals would simply be the logic of predicates or concepts. Perhaps this kind of nominalism or psychologism about universals is part of the motivation for the view of logic I am rejecting; but I take it such views should not be accepted uncritically, and indeed are very implausible—for how then could objects have properties in a world lacking words or human concepts? Once we accept the reality of universals, fully and unapologetically, the approach I am defending begins to look attractive, indeed unavoidable.

This incidentally implies that the usual separation between first-order logic and second-order logic is philosophically misguided (though technically correct): we are essentially concerned with properties and their relations even at the level of first-order logic, because we need to interpret the predicates as denoting universals that form the basis of logical laws. Particulars and

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universals are the foundation of the whole logical edifice, even when we are not quantifying explicitly over the latter. Universals are ontologically basic and enter into all our thought: they are the original ground of logical laws, even when dealing with first-order logic.

Frege opposed psychologism about logic—the idea that logical laws have to do with the mind (apart from being apprehended by the mind). To this end he fashioned his ontology of objective “thoughts”—a clear oxymoron. These thoughts were taken to exist independently of the mind and to precede the existence of the mind. I won’t argue against this position here, merely noting itsextravagance; but I will say that I agree with the motivation behind it—we don’t want logic to depend on human constructions, whether psychological or linguistic. When logic is conceived as the deductive science of propositions there is a distinct danger of psychologism, but the way to avoid it is not to objectify propositions; rather, we should locate logic at a deeper level—in the world beyond thought. There is nothing at all psychological about universals, for a realist about universals—they exist quite independently of minds. They are the building blocks of reality, since there is no particular that precedes universals—there are no property-free objects. Thus psychologism is avoided by locating logical laws in the non-psychological world of objects and universals, not (pace Frege) in a supposed realm of objective transcendent “thoughts”.

It is a consequence of the position advocated here that some knowledge of extra-mental reality is a priori: for we know the laws of logic a priori, and yet these laws characterize the world beyond the mind. Again, this consequence may be part of the motivation for a propositional view of logical laws, because then we can confine a priori knowledge to the contents of the mind (with language reckoned to the mind). If we think of logical necessity as analytic, and construe analyticity as arising from language and concepts, then we will be inclined to suppose that logical laws arise from the inner nature of mental representations or words. But again, such views must not be accepted uncritically or assumed without acknowledgment—and upon examination they are very problematic. I won’t undertake a criticism here, merely noting that we need to take seriously the possibility that some a priori knowledge just is knowledge of the structure of extra-mental and extra-linguistic reality. We know from our grasp of the nature of universals that they have certain kinds of exclusion and consequence relations—however jarring that may sound to certain kinds of empiricist or positivist assumptions. We have a priori knowledge of logical laws, and these laws characterize the objective nature of

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independent reality. This is just the way things are, like it or not. Sorry!

Finally, does the notion of logical form rest on a mistake? Philosophers and logicians have been apt to speak of the logical form of propositions or sentences, but an astute follower of the logical realism defended here might protest that this is a category mistake, since logical form properly belongs to states of affairs not to sentences or propositions about states of affairs. I agree with the spirit of this protest, and admire its extremism, but I think it goes a bit too far. We can agree that universals themselves exhibit logical form, in the sense that they are arranged in a logically determined totality, as defined by identity, exclusion, consequence, and combination. But there is nothing to stop us from supposing that this form is reflected in the structure of propositions themselves. The subject-predicate form, say, is a reflection of the object- property form: two complementary elements in a relation of mutual entanglement (predication and instantiation, respectively). Nor is there any objection to selecting a class of expressions designated as logical constants, and then defining a notion of logical form on that basis (though this may be more arbitrary than has been recognized). What is mistaken is the composite idea that logical relations depend on logical form and that logical form is an intrinsic feature of propositions, considered independently of reality. That is just the dogma of logical representationalism (to give it a name) stated another way. Logical relations, to repeat, cannot be defined purely over representations, as a matter of their very nature: so they cannot result from the logical form of representations. Any logical form that propositions have must be derivative from a more basic logical reality—the logical form inherent in the underlying universals.

If the position of this paper is correct, we should stop talking of propositional and predicate logic (though we may still speak of the propositional and predicate calculus—this being a type of notation). For that gives the metaphysically misleading impression that logic is grounded in propositions or predicates, not in the logical order of the world itself. We have different symbolic systems for representing (a fragment of) natural languages, but logical reality itself has nothing essentially to do with these systems. Logical reality is external to such systems, being essentially not a matter of symbols at all (so “symbolic logic” is misleading too). Logical laws per se exist in the world outside of all representation, and it is the job of our systems of representation to reflect their nature as best we can. They may do so without claiming to be constitutive of logical laws. The laws of logic stand outside of any notation for

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less perspicuously.

(3) Love of Logic

Résumé. L’auteur cherche à répondre à une question négligée: d’où vient notre amour pour la logique? On fait valoir que la logique représente l’esprit humain dans sa correspondance essentielle avec la réalité logique objective, ce qui affirme que nous sommes des êtres rationnels.

ملخّص. يسعى المؤّلف للإجابة عن سؤال مهمل : من أين ينبع حّبنا للمنطق ؟ إّنه من الواضح أ ّن المنطق يص ّور الذهن البشري في مطابقته المركزية للواقع المنطقي الموضوعي م ّما يثبت أ ّننا كائنات عقلان ّية.

Abstract. The author seeks to answer a neglected question: whence our love of logic? It is argued that logic depicts the human mind in its essential correspondence with objective logical reality, thus demonstrating that we are rational beings.

Why do we love the predicate calculus?1 Because it is a diagram of thought as it reflects the logical order of the world. It depicts thought in its essential relation to logical reality. So it does not depict thought as a psychologist might; it depicts thought as it conforms to the objective logical order. That order is not itself a matter of psychology: it pre-exists the human mind. It would not be wrong to say that the predicate calculus depicts this logical order, since it records objective logical truths. But that is not why we love it: we love it because it depicts us as logical, as bound to the logical order. Predicate calculus is the proof that we are a rational species; it is not merely a means to formulate proofs concerning logical reality. It diagrams the logical scaffolding of our thought—it pictures human reason (physics does not do that). Its symbols and structures remind us of our inherent rationality (res cogitans), and thus enhance our self-love. We love logic because it confirms our elevated self-image, and rightly so (“proper narcissism”). If we had never invented logic, we would be able to doubt our power of reason; but logic assures us that we swim in the

1 I choose the predicate calculus as an example; other logical systems can be loved too, e.g. modal logic. I am raising the question of our logical affections because it is never raised in the philosophy of logic—yet it is surely a familiar fact of human psychology. We have emotions about logic, positive ones: we find logic adorable, beautiful, “sexy”. I am asking why we have such emotions.

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medium of reason–that we occupy logical space as well as physical and temporal space. The formulas of predicate calculus are the precise expression of our nature as logical beings. As we gaze at great paintings and recognize our status as aesthetic beings, so we gaze at logical formulas and recognize our status as rational beings—that is, as beings that reflect the logical order of the world. Love of logic is bound up with love of ourselves: but this love of ourselves is a rational love, being the love of rationality.1

(4) Is

Résumé. La question est de savoir si « est » possède deux significations (comme le mot anglais « Bank »). On fait valoir que contrairement aux idées reçues il n’y a pas une bonne raison de discerner une ambiguïté entre le “est” de la prédication et le “est” de l’identité. Il est assez facile d’offrir des paraphrases des propositions pertinentes selon lesquelles « est » a un sens unifié. L’analyse montre que le « est » signifie toujours la prédication.

ملخّص. السؤال المطروح هو ماذا لو كان لكلمة “يوجد” معنيان (على غرار كلمة “بنك” في الانجليزية) ، ّفمن الواضح أ ّنه دون اعتبار غاية البحث عن الصواب ، فإ ّنه ما من سبب وجيه لرفع اللبس فإ ّن الوجود في كلمة “يوجد” غير ذلك الحادث بين كونها آداة للحمل أو آداة لمبدأ الذاتية أو الهوّية بالمعنى المنطقي. إ ّنه من السهل تماما تقديم جمل تفسيرية مت ّممة للجمل التي تكون فيها لكلمة “يوجد” دلالة واحدة. و قد أظهر التحليل أ ّن المعنى المقترن دائما

بكلمة “يوجد” هو عمل ّية الحمل.

Abstract. The question is whether “is” has two meanings (like “bank”). It is argued that contrary to received wisdom there is no good reason todiscern any ambiguity in “is”, as between the “is” of predication and the “is” of identity. It is quite easy to offer paraphrases of the relevant sentences according to which “is” has a unitary meaning. Analysis reveals that “is” always means predication.

The standard view, enunciated by Russell, is that “is” is ambiguous between the “is” of predication and the “is” of identity (we might also add the “is” of composition, as in “this state is bronze”). Thus we have, “the cup is red” and

1 Our love of logic is thus not like our love of other subjects, such as geology or astronomy, which do not depict us. Logic is special in that it depicts us as reflective of the objective order of logical relations—it is a kind of ideal psychology. It tells us how we think (in one sense), as well as what is logically correct.

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“Hesperus is Phosphorus”, where the two occurrences of “is” have different meanings. To claim that “is” has the same meaning in both occurrences would produce absurd consequences. If the “is” in “the cup is red” expressed identity, then the sentence would mean that the cup is identical to redness, which is false and absurd. If the “is” of “Hesperus is Phosphorus” expressed predication,then the sentence would mean that Hesperus has the property of Phosphorus, which verges on the meaningless and is certainly not true—“Phosphorus” is not a predicate but a singular term. So “is” must be ambiguous between the two cases, sometimes meaning identity and sometimes meaning predication. That would be a serious failing in natural language, requiring linguistic reform: our language systematically confuses two very different concepts.

But this conclusion is far too hasty; there is no need to adopt the ambiguity thesis in order to account for the meaning of “is”. For, first, it is not difficult to construe the “is” in identity statements as simply the predicative “is”, by expanding such statements in the obvious way, viz. “Hesperus is identical to Phosphorus”. Here we have a predicate expression, “identical to Phosphorus”, coupled with the “is” of predication, so that the sentence is saying “Hesperus has the property of being identical to Phosphorus”. We don’t need a separate meaning for “is” to account for its use in identity statements; we just need to fill out the predicate in the obvious way. Clearly “is” cannot express identity in the expanded version, or else the sentence would be saying that Hesperus is identical to identity with Hesperus, which is nonsense. The point is even clearer if we add a sortal term to statements of identity, as in “Hesperus is the same planet as Phosphorus”: here “same planet” carries the attribution of identity, with “is” just acting as the predicative copula. When we use “is” alone in an identity statement this is just a shorter version of the explicit expansion thatemploys the identity concept directly. There is no “is” of identity.

Can we enforce uniformity of meaning from the other direction? That is, can we claim that “is” always expresses identity? It would certainly be difficult to do that if we read the sentences in question naively, as saying (for example) that the cup is identical to redness; but a simple paraphrase can resolve this problem. What if we rephrase “the cup is red” as “the color of the cup is(identical to) red”? That is a straightforward identity statement, and it is straightforwardly true. The same trick can be applied to all predicative uses of “is”, as in “the species of Felix is cat” or “the job of John Smith is philosopher”. Put in stilted philosopher’s language, we are paraphrasing “a is F” as “among the attributes of a is F-ness”, where “is” expresses simple numerical identity. We can take this as a quantified statement along the following lines:

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“there are attributes F that a instantiates and one of these F’s is identical to G- ness”. Thus: “there is a (unique) color C such that the cup has C and C is identical to redness”. This sounds rather ponderous, no doubt, but it corresponds quite well with the intuitive meaning of the original statement, more colloquially expressed as, “the color of the cup is red”. (As to “the statue is bronze”, this comes out as, “the material composing the statue is (identical to) bronze”. Equally, we could paraphrase the sentence as, “the statue is composed of bronze” where the “is” here is just the usual “is” of predication, not a special “is” of composition.)

So there is nothing compulsory about finding ambiguity in “is”; in fact, it is quite easy to provide paraphrases that employ “is” in one meaning for all sentences that contain “is”. And surely that is the preferable position, since it is hard to believe that natural language could harbor such a disreputable ambiguity—why not simply have two words for such very different concepts? There is the question which of the two theories we should prefer, given that both appear adequate. I incline to a mixed position, combining both types of paraphrase. The second type offers a convincing expansionary analysis, spelling out the underlying meaning of the sentence; but the first type makes it clear that the so-called “is” of identity is really short for “is identical to” or “is the same as”, which contains the “is” of predication. Thus “the cup is red” has the same meaning as, “the color of the cup is identical to red”. We turn the original sentence into a statement of identity, but that statement itself contains in its expansion a predicative use of “is”, with identity conveyed by the attached predicate “identical to red”. Predicative sentences turn out to be identity sentences, but identity sentences turn out to contain the “is” of predication. So in the final analysis “is” is always predicative, but ordinary predicative sentences are equivalent to identity sentences.

How then should we analyze “Hesperus is Phosphorus”—what is its underlying logical form? It turns out to mean the same as, “Among the attributes of Hesperus one of them is that of being identical to Phosphorus”. We quantify over attributes and declare one of them to be identical to identity with Phosphorus—where “is” occurs in its predicative meaning. Thus: “There are attributes that Hesperus has and one of them is identical to identity with Phosphorus”. This sentence expresses an identity proposition concerning the attribute of identity with a given object, but in order to state that identity we need to use “is” predicatively. Given that the “is” in an identity statement so clearly means, “is identical with”, this seems to be just what we would expect on the assumption that identity is at the root of all predication. All propositions

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are really identity propositions, on this view, formed by quantifying over attributes or properties. The recipe for constructing the underlying identity proposition is simply to refer to a property and declare it one of the properties an object has, as in “the color of the cup is (identical to) (the color) red”. If the cup has many colors, it would be better to say, “a color of the cup is red”, so as to avoid falsely imputing uniqueness, which can then be expanded into, “there are color properties that the cup has and one of them is identical to the property of being red”. Second-order quantification plus identity therefore enter even into ordinary subject–predicate sentences—which is not what we have been taught to expect. But, as we know from Russell’s theory of descriptions, language can be more complex than it seems on the surface when it is properly analyzed. First-order logic really embeds second-order logic (with identity) in underlying logical form. Still, “is” remains uniformly a device of predication, even as it occurs in second-order identity sentences. The impression that “is” is ambiguous disappears once we carry out the requisite analysis.

(5) Mereological Arithmetic

Résumé. Il est très commun de parler des nombres comme s’ils possédaient des parties composées d’autres nombres. Mais il s’avère qu’une fois nous aurons mis au clair la nature de la relation partie-tout, il s’avérera qu’il s’agit bien là d’une erreur qui résulte de la confusion entre les nombres et les marques que nous utilisons pour nous référer à eux.

ملخّص. إ ّنه من العاد ّي ج ّدا القول بأ ّن الأعداد تعتبر جزءا مك ّونا لاعداد أخرى ، غير أ ّن ذلك يكون خاطئا متّى و ّضحنا طبيعة الجزء و الك ّل. يظهر الخطأ من ج ّراء الخلط بين الأعداد و العلامات الدالة على تلك الأعداد.

Abstract. It is common to speak as if numbers have parts consisting of other numbers. But it turns out that this is a mistake, once we are clear about the nature of part-hood. The error appears to arise from a confusion of numbers with the marks we use to refer to them.

The way we talk about numbers resembles the way we talk about physical objects in one respect: we talk as if numbers have parts. This is written into the language of arithmetic. Just as we say that a cake can be divided into parts, so we say that a number can be so divided. We also speak of adding and subtracting in relation to numbers, as we speak of adding and subtracting in

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relation to a physical thing or collection of things. You can subtract a slice from a cake or some marbles from a pile of marbles, or add to the cake or the marbles, as you can add or subtract numbers. Thus we think mereologically about numbers.

Pursuing this mereological way of thinking, we can (and some do) push it further. Take the number 15: this number can be divided by 3 to give 5, so we can say that 15 divides into three equal parts of 5. These parts are traditionally called “aliquot parts”, meaning that 5 divides into 15 exactly three times, where 3 is a whole number (an aliquot part of a number is defined as an exact divisor of that number). This notion is also used in chemistry, where it means dividing a chemical sample into parts of equal quantity. Mathematicians also speak of “aliquant parts”, which do not divide to produce whole numbers—8 divides into the aliquant parts of 5 and 3. The numbers into which a given number divides can themselves be divided, ending in the whole number 1. Thus every whole number can be divided by 1, which is thus a part of every number. But these are not the smallest parts, because whole numbers can be divided—hence fractions. In the limit each number has infinitely many parts, as we keep dividing. The picture here is that a number is like a physical object in that it can be divided successively into smaller and smaller parts, the sum of which add up to the number in question. Thus there are part-whole relations between numbers, as there are part-whole relations between chemical quantities.

But there is a crucial disanalogy here, which undermines this whole way of thinking. Suppose I divide a cake or chemical sample into thirds: none of these third parts are identical to the others—we have three separate physical entities, which together compose the original object. But if I divide 15 into thirds I get the number 5, which is identical to the other (alleged) parts. The parts areidentical to each other, being just the number 5. If we call the parts “P1”, “P2” and “P3”, we have “P1 = P2 = P3”. It actually makes no sense to speak of combining 5 with itself–all that could ever give is 5. If it did make sense, we would have to conclude that 15 = 5. Similarly, if each number divides into 1 when divided by itself, we would have to say that every number is identical to 1, since every number would resolve into a collection of 1’s, i.e. the number 1. It is of course true that adding 5 to itself three times gives 15, but adding is not mereological combining—combining 5 with itself can only give 5. What are called aliquot parts are not parts at all: 5 is indeed a divisor of 15, but it is not literally a part of 15. By contrast, aliquot parts of a chemical sample are genuine parts of that sample, being numerically distinct from each other, and combinable into the whole sample. If we think about it in terms of set theory, the set {5, 5, 5} is really just the set {5}, since 5 = 5; but the set {five molecules

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of C, five molecules of C, five molecules of C} is not identical to {five molecules of C}, since the former set specifies fifteen molecules of C. The set {5, 5, 5} is just like the set {Aristotle, Aristotle, Aristotle}—a peculiar way to represent the set whose only member is 5 or Aristotle. Combining Aristotle with Aristotle gives you only Aristotle, and similarly for the number 5.Numbers really don’t have other numbers as parts, not literally.

How could this error have arisen? I suspect confusion between use and mention lurks behind it. I can certainly write down fifteen tokens of the numeral “1”, and the set or aggregate of these tokens is not identical to a set or aggregate consisting of a single token of “1”. We can rightly view these inscriptions (physical marks) as wholes with parts: the aggregate of the tokens is composed of each token or sub-aggregates of tokens. A collection of written tokens of a given numeral does indeed have aliquot parts, just like a chemical sample. The set {“5”, “5”, “5”} is not the set {“5”}, since each token is distinct from the others. But we must not confuse tokens of numerals with numbers, which do not form aggregates in this way. Of course, to claim that a number like 15 is an aggregate of numeral tokens is both highly implausible and also not what those who speak mereologically of numbers intend. But if we fall prey to a use-mention confusion we could easily slip into the error I have identified. Then we will find ourselves saying things like, “Consider three occurrences of the number 5”, which is really quite meaningless. In short: fifteen tokens of “1”are not fifteen parts of 15. And the number 1 is not a part of 15 at all, but simply one of its divisors.

Another source of potential confusion is that numbers can be attached to collections and the collections can be distinct from each other. Thus a collection of fifteen dogs can be divided into three collections of five dogs, where these collections are not identical with each other. We could reasonably assert that the fifteen-dog collection is made up of three five-dog collections. But again, it doesn’t follow that the number 15 is itself made up of three fives; and there is really no such thing as “three fives” (unless that means “three times five”), since 5 is just itself and no other thing (there is only one five). If we confuse numbers with collections that numbers number, then we will be prone to misplaced mereological thinking about numbers themselves. To repeat: numbers cannot be composed of their divisors because the divisors don’t aggregate in the right way. The number 5 is a divisor of 15, but there are not three of these numbers that aggregate to give 15. Aggregates of physical objects have part-whole relations, but numbers are not like that.

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Finally, we must not confuse arithmetic with geometry. Abstract geometrical figures can be conceived in mereological terms, as when we divide a triangle into two parts by drawing a line from one angle to the midpoint of the opposite side. Here the two parts really are parts that aggregate to form a whole, neither being identical to the other. But we cannot likewise say that 10 can be divided into two non-identical equal parts, since 5 is simply identical to 5. We cannot think of 10 as literally composed of two halves both consisting of 5, because those halves would simply be 5. It is quite true that 5 is a half of 10, but it is not true that 10 is composed of 5 twice (whatever that may mean). It is really a category mistake to describe numbers in part-whole terms. But it is a very tempting category mistake, being embedded in the very language we employ to describe arithmetic relations, and abetted by perennially tempting confusions, particularly the use-mention confusion. At best talk of part and whole in relation to numbers is a metaphor, and a highly misleading one.

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