Preface

In high school I read a book from the school library called Introduction to Mathematical Thought by E. R. Stabler. Early on he described a pattern in which he displayed a conditional, its antecedent, and its consequent. I stared at this display, trying to get the point. Then the penny dropped: this is a way the mind moves, from the first two to the third. That was my first experience of explicit logic, and it was eye-opening.

I took my first logic course at university from W. V. Quine. He lectured by mumbling at his stack of three-by-five cards, but the textbook was the second edition of his Methods of Logic, which I still think is the best baby logic book I have seen. As I learned gradually who Quine was, I was too awestruck to approach him again. Instead, I did a lot of work with his former student Burt Dreben. Dreben described himself as a logical positivist, but he was really a philosophical nihilist. He once said in lecture, “Rubbish is rubbish, but the history of rubbish is scholarship.” The rubbish was all of philosophy, and the scholarship was where he wanted me directed. He taught us a lot of logic and early analytic philosophy. He first taught me Gödel’s incompleteness theorem from Rudolf Carnap’s The Logical Syntax of Language, and he later taught it from Gödel’s original paper. Both now seem to me perverse pedagogy, especially the first, but it reflected his historical taste.

Here is a picture of a simple instance of truth.The example is John Austin’s. The picture has two parts: what is true, and what makes it true. As we have mentioned, the things that are true or false are articulated into smaller bits that are neither true nor false but pick out the subject matter that makes for truth. Even if one favors abstract propositions or mental judgments as bearers of the truth values, still, sentences wear their articulation into such smaller bits, words, on their inscribed faces, and that articulation seems to get read back into propositions (as concepts) or judgments (as ideas). Tarski took the sentences themselves as the truth bearers.

Traditional grammar teaches that every sentence has a subject and a predicate. While that dictum remains a basic principle of composition, the logicians reconceived it in several ways. In the first place, one of the major achievements of nineteenth-century logicians like Peirce and Schröder was to begin a systematic taxonomy of relations. It is basic here that while earlier only unary properties (like being red or being round) received sustained attention, now binary relations (like loving or being above), ternary (like a giving b to c), and so on got equal billing. It was as if where traditional grammarians had assumed each sentence has a unique subject, now logicians ignored declension ­(subject, direct object, indirect object, possessive, and so on) and treated the two, three, or however many singular terms to which a binary, ternary, or whatever polyadicity predicate is applied to make a simple sentence, all as several subjects of that single predicate. In Austin’s example, the predicate is “is on” for the binary relation of superposition, and the two subjects are “the cat” and “the mat.”

In Chapter 5 we looked at a theory about natural numbers. Since all recursive relations are n.e. in Q, and the consistency of Q is provable in elementary number theory, we concluded that Q is undecidable. But Q has only finitely many axioms specific to natural numbers, so there is a conjunction A of these axioms. So if we let P be the logic fixed by the language of Q, then a sentence B of that language is a theorem of Q if and only if A → B is a law of logic according to P. By the completeness theorem for first-order logic we mentioned in Chapter 4, being a law of logic according to P may be read equivalently either as being deducible from logical axioms of P by logical rules of inference of P or as being true in all interpretations of the language of P (which is that of Q). So if being a law of logic according to P were decidable, we could apply an algorithm for it to A → B to decide whether B is a theorem of Q, so Q would be decidable. Hence being a law of logic according to P is undecidable.

Suppose, conversely, there were an algorithm α for being a theorem of Q. The property FΣ of being a proof in Q in which only logical axioms and rules of P are used is a decidable property; just look at the proof to see which axioms and rules are used. So its gödelization FA is recursive, as is the predicate LA(x, y), which holds if and only if x is the gödel number of a proof in Q whose last line is the formula with gödel number y. Let B be any sentence in the language of Q and let b be its gödel number. Then B is a law of logic according to P if and only if (∃x)(F(x) ∧ L(x, b)) is a theorem of Q (where F and L n.e. FA and LA), so by applying α to the second we could effectively decide whether B is a law of logic according to P.

In the last four chapters, we reviewed four major monuments of twentieth-century logic. These four results are less familiar to philosophers than Gödel’s proof of the incompleteness of arithmetic. It would be a shame if philosophy lost touch with logic. From Dedekind, Cantor, and Frege until World War II, the signal logicians were almost always both signal mathematicians and signal philosophers. It is not as if after the war such people suddenly vanished; they did not. But increasingly, mathematical logic became a recognized subdiscipline of mathematics. It is rare for our four results to be taught in a philosophy department, and mathematics courses usually move very swiftly for philosophers. Equally, many philosophers know first-rate logicians in their mathematics departments who are philosophically acute but shy about speaking up among philosophers. (A former colleague called philosophy a blood sport.) If the philosophers do not keep up as the mathematicians move on, it is inevitable that they will lose touch.

But why should a philosopher put in the significant effort needed to keep up with the mathematicians? Since the renaissance of logic in the nineteenth century, philosophers have been intrigued by its metaphysical and epistemological intimations. One thing it revealed was a whole new kind of objects, sets. Sets have more than earned their credentials in unifying and organizing disparate subjects and tracts of knowledge, so much so that by now, no education in abstract pursuits is adequate without some familiarity with sets. It can sometimes feel as if we had found a sixth and more primitive sort of vertebrate that makes sense of fish, amphibians, reptiles, birds, and mammals.

The Burali-Forti paradox was a crisis for Cantor’s theory of ordinal numbers; Cantor’s paradox was a crisis for his theory of cardinals; and Russell’s paradox was a crisis for Frege’s logicism. Had the crises been local, sets (and courses-of-values) might have gone the way of phlogiston, the stuff thought in the eighteenth century to be lost from something burning (and supplanted by oxygen taken up in burning). But sets (or their kissing cousins) were not going to go without a fight. We can make out at least four reasons for this resilience. Among philosophers, logicism retained a fascination that gave it room to evolve. Among mathematicians, Cantor’s theory of infinity retained a fascination that Hilbert, for one, would not abandon. Also among mathematicians, set theory became the framework, the lingua franca, in which – by and large – modern mathematics is conducted. Finally, there are the applications of set theory, of which those in logic are central for us.

Frege layered functions. A first-level function assigns objects to objects: doubling is a first-level function that assigns six to three; and the concept:green is a first-level function that assigns truth to all and only the green things. The derivative of the square function is the doubling function, while that of the sine is the cosine, so differentiation is a second-order function. In another example, Frege reads “There are carrots,” so its subject is the concept:carrot and its predicate is the concept:existence. Existence is thus a second-level concept whose value is truth at all and only the first-level concepts under which something falls. This allows Frege to refine Kant’s criticism of the ontological argument for the existence of God. Kant said that existence is not a predicate, which is heroic, or even quixotic, grammar. Frege could say that since existence is a second-level predicate, it is at the wrong level to be a defining feature of an object like God.

It is not true that Socrates was snub-nosed unless the flesh-and-blood nose of the flesh-and-blood man was stubby and perhaps a bit turned up. As we saw in the last chapter, whether a sentence is true depends on what the sentence is about and how it is with that subject matter. So, to take a justly famous example from Euclid, it is not true that there are infinitely many prime numbers unless there are infinitely many prime numbers, and thus unless there are numbers. This conclusion already makes some philosophers nervous, and their nerves tend to become more jangled when we add the natural reflection that numbers are neither mental nor physical but abstract objects. That reflection invites exploration and argument, but it is a view of early resort. On the other hand, we are happiest attributing knowledge to a person when we can make out how she interacted or otherwise had commerce with the subject matter of her belief so as to justify her belief. The only mode of such commerce that all parties grant is perception; sensible rationalists have an empiricist streak. H. P. Grice convinced us that perception is by its nature causal. But it is pretty close to an axiom of metaphysics that very abstract objects like numbers are utterly inert. Hence, what makes mathematical truth possible makes mathematical knowledge impossible. This antinomy is known as Benacerraf’s dilemma.

An escape from the antinomy that occurs to many people, especially mathematicians, is to say that truth in mathematics is neither correspondence to fact nor in need of reference to abstract objects but is rather just provability. Mathematicians neither perform experiments on primes in retorts nor make expeditions to examine exotic ones; instead they establish their claims by proving theorems, and that is just a mode of writing and talking. In the Middle Ages the doctrine that there are no abstracta, no Platonic forms like humanity but only predicates like “is a person,” was called nominalism. Its twentieth-century incarnation that denies mathematics abstract subject matter and assimilates mathematical knowledge to empirical knowledge of expressions arrayed in proofs is called formalism. Formalism requires that all mathematical truths be provable (and proof should not smuggle in truth conceived semantically).

Cantor’s key to paradise was his proof that every set is strictly smaller than its power set, the set of all its subsets. In von Neumann’s theory of ordinals and cardinals, which is now the industry standard, the smallest infinite ordinal, ω, is the set of all finite ordinals, or natural numbers, and the smallest infinite cardinal, Cantor’s ℵ0. By Cantor’s proof, the power set of ω, P(ω), has a cardinal number bigger than ω, and iterating opens the door to a paradise of different infinite numbers, as many as there are ordinals. In von Neumann’s theory, the less-than relation between ordinals is just membership, a cardinal is an ordinal in one–one correspondence with none of its predecessors (or members), and by Cantor’s proof for every cardinal there is a bigger one. So if we let ω(0) be ω, ω(α + 1) be the least cardinal bigger than ω(α), and, when λ is a limit, ω(λ) be the union of all the ω(α) for α less than λ, we get as many infinite cardinals as there are ordinals. Our ω(α) is our version of Cantor’s ℵα; both notations are still common.

P(ω) has exactly as many members as the set of real numbers. We have seen a natural one–one correspondence between P(ω) and the reals from zero up to, but not including, one. Cartesian coordinates assume a one–one correspondence between the real numbers and the points on a line. Since lines are the basic example of continua, the cardinal of P(ω) is called the cardinal of the continuum. By our construction, ω(1) is the least cardinal bigger than ω(0) = ω, and by Cantor’s theorem, the cardinal of P(ω) is bigger than ω, so the cardinal of the continuum is bigger than or equal to ω(1). But which, bigger or equal? Cantor conjectured that the cardinal of P(ω) is ω(1), and this is known as the continuum hypothesis. The generalized continuum hypothesis says that for all α, the cardinal of P(ω(α)) is ω(α + 1), which would make the power set function like the successor function on natural numbers.

Cantor’s quarry was the infinite. The mathematics of number had always been about objects of which there are infinitely many, like natural numbers, or objects of which not only are there infinitely many but each is also itself infinite, like real numbers with endless decimal expansions. The infinities of geometry, like the infinity of points on a line or triangles in a plane, had always been there, but the applications of the calculus in geometry made its infinities more salient. The recognition of the infinity of its subject matter was always a reason not to test the conjectures of mathematics by checking the examples but rather to prefer proof. Aristotle urged that the infinite could only ever be potential, like a process with no fixed end, but that completed actual infinite wholes were ruled out. Such views look to countenance possibilities that could not be actual, which sounds contradictory, but even Gauss, the prince of mathematicians, had a horror infiniti. Cantor swam against the tide.

To work out a theory of the infinite per se, Cantor needed to figure out which things are classified as finite or infinite. That is one source of his interest in sets. For this purpose sets should be any old collections, whether unified by having something in common or not, like the Walrus’s shoes and ships and cabbages and kings. Sets should be an utterly general sort, so whether there are infinitely many such and suches can always be re-asked as whether the set of such and suches is infinite. As horses are the kind that divides into stallions and mares, so sets are the kind that divides into finite and infinite.

After Gödel proved the consistency of the continuum hypothesis with set theory, many people expected him also to prove its independence, but it was Paul Cohen who did so. There is a story, perhaps apocryphal, as such stories sometimes are, that Cohen passed by Georg Kreisel’s office at Stanford in the spring and asked for a problem to work on. To get rid of Cohen, Kreisel gave him the independence of the continuum hypothesis, and by the next fall Cohen had solved it. There is another story, certainly apocryphal, that after Gödel heard of Cohen’s success, Gödel feared that the Institute for Advanced Studies would fire him and he would starve to death as an old man. This story may be a garbled version of Gödel’s decline after the death of his wife, who had looked after him well.

We will show that if ZF is consistent, it remains so even with the addition of a sentence saying there is a nonconstructible set of natural numbers, so V = L is independent. We will show that if ZF is consistent, it remains so with the addition of a sentence saying there is a set of sets of natural numbers that is not well-ordered, so the axiom of choice is independent. We will show that if ZF is consistent, it remains so with the addition of the denial of the continuum hypothesis, so the generalized continuum hypothesis is independent. We will make constant use of the constructible sets from the last chapter, and, as there, our exposition will follow lectures given by Hilary Putnam at Harvard in the spring of 1968.

We are all post-Kantians, not perhaps because we believe the space in which we live and move and have our being is created by the action of our senses, but because Kant set an agenda for philosophy that we are still working through. Kant says (at 260 in his Prolegomenon to Any Future Metaphysics) that it was Hume who first interrupted the dogmatic slumbers into which Leibniz and Wolff had lulled him. Hume had argued that nothing in the idea of causation can guarantee that for each event there exists an earlier one that caused it; ideas do not guarantee that anything answers to them. But if it is not a relation of ideas that every event have a cause, then, if it is true, it can only be a matter of fact, and so known, if known at all, by experience. This conclusion was an affront to the dogma that the principle of sufficient reason is known and justified independently of any appeal to experience.

Kant composed his agenda for overcoming skepticism in terms that endured. Assume the anachronistically labeled traditional analysis according to which a person A knows that p (where “p” marks a blank to be filled by an indicative sentence) just in case A believes that p, it is true that p, and A is justified in believing that p. The point of the justification clause is that for A to know that p, his belief’s being true should not be just a lucky guess. The focus of epistemology, or theory of knowledge, has usually been more on the nature of justification than on details in the analysis of knowledge. Kant divides knowledge into two sorts that differ according to how the belief is justified. He calls knowledge a posteriori when it is justified, even in part, by appeal to sense experience. Knowledge is a priori when it is indeed knowledge but not a posteriori, that is, when it is justified but not justified even in part by sense experience. Note two points here. First, the account of a priori knowledge is wholly negative; it says only how a prioriknowledge is not justified. This raises the question how a priori knowledge is justified, and a refinement of that question will become the first agendum of Kant’s critical philosophy. Second, while nearly all parties agree that some knowledge is justified by sense experience, it may not be equally evident that there is a priori knowledge. This issue was on the philosophical agenda before Kant, and is still there two centuries after his death (in 1804). Kant himself thought we know a priori that for every event there is a prior event that caused it, and he worked hard to elaborate this thought. That principle was, he thought, necessary for natural sciences like physics, and while much of the physicist’s knowledge of nature is a posteriori, he could not have that a posteriori knowledge unless he knew a priori that every event has a cause. But Kant also believed that there are whole, systematic bodies of knowledge that are entirely known a priori. The two leading examples were logic and mathematics. We will return in a moment to what Kant thought logic was. From the Greeks to Kant, mathematics was first and foremost geometry. Geometry was not just Euclid’s system of figures in the plane, but also the solid geometry of the space in which we live and move and have our being. Kant is quite explicit that there is only one space; the idea of lots of spaces is later and thoroughly un-Kantian. In his Elements, Euclid’s Book IX is about number theory, like the infinity of primes. But the Greeks thought of number geometrically, and the mathematics of number achieves independence from geometry only in the arithmetization of analysis during the nineteenth century. It was central for Kant that our geometrical knowledge is a priori.

In this book I present an unconventional perspective on some of the most interesting problems of logic and philosophical analysis. The philosophy of logic concerns itself with every aspect of the logic of thought and language, and the logic of thought and language, properly understood and applied, in turn provides a key that can help unlock philosophical puzzles and involve us in deeper, more interesting ones.

That there are forms of thought capable of being symbolized, and that these formalizations can be used to establish the logical, semantic, and other structural properties of sentences and deductive inferences, is itself a phenomenon worthy of reflection. Philosophical problems and logical conundrums are always expressed in a relatively sophisticated language. Sometimes philosophical difficulties are themselves self-created products of the languages and conceptual frameworks in which they arise. A crisis in logic or philosophy often points toward expressive limitations or unclarities in established patterns of thought and language by which its problems are imagined and articulated.

Where difficulties of expressive and problem-solving ability are encountered, we may try to expand or restrict our resources so that we can deal adequately with the challenges immediately at hand. We may introduce new or compress established distinctions, add new concepts, or eliminate troublesome categories, or confused or inapplicable terminologies. Logical analysis of the terms and sentences in which philosophical problems are formulated often provides a testing ground for innovations in the syntax and formal mechanisms of symbolic logic and in less rigorously developed refinements of ordinary thought and colloquial language.