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Philosophy of logic deals with issues about logic that are broadly philosophical, especially metaphysical (about reality) or epistemological (about how we know). Here are examples: Are there abstract entities, and does logic presuppose them? Is logic the key to understanding the structure of reality? How do we know logical laws – are they empirical or true by convention? What is truth, and how do different views on truth affect logic? What is the scope of logic?
18.1 Abstract entities
Metaphysics studies the nature of reality. It considers broad views like materialism (only the physical is ultimately real), idealism (only the mental is ultimately real), and dualism (both the physical and the mental are ultimately real). Another issue is whether there are abstract entities – entities, roughly, that are neither physical (like apples) nor mental (like feelings); alleged examples include numbers, sets, and properties.
Logic can quickly bring up issues about abstract entities. Take this argument:
· This is green.
· This is an apple.
· ∴ Some apple is green.
In discussing this argument, we may talk about abstract entities:
· The set of green things; this set seems to be not physical or mental, but rather an abstract entity.
· The property of greenness, which can apply either to the color as experienced or to its underlying physical basis; in either case, greenness seems to be not a concrete mental or physical entity, but rather something more abstract that has physical or mental instances.
· The concept of greenness (what terms for “green” in various languages mean).
· The word “green” and the sentence “This is green,” which are abstract patterns with written and auditory instances.
· The proposition that this is green, which is the truth claim that we assert using “This is green” in English or similar things in other languages. 0369
Platonists, as logicians use the term, are those who straightforwardly accept the existence of such abstract objects. Nominalists, in contrast, are unhappy about such entities and want to restrict what exists to concrete physical or mental entities; they try to make sense of logic while rejecting abstract entities. Intermediate views are possible; maybe we should accept abstract entities, not as independently real entities that we discover, but rather as mental creations or fictions. Disputes about such matters go back to ancient and medieval debates about forms and universals, and continue to rage today.
18.2 Metaphysical structures
Does logic give us the key to understand reality’s metaphysical structure? Ludwig Wittgenstein, in his Tractatus Logico-Philosophicus (1922), argued that it does. He saw the world as the totality of facts. If we state all the facts, we completely describe reality. Facts are about simple objects. An atomic statement pictures a fact by having its elements mirror the simple objects of the world. Language, when completely analyzed, breaks down into such atomic statements. Complex statements are built from atomic ones using logical connectives like “and,” “or,” and “not.” Wittgenstein invented truth tables to show how this works. Some complex statements, like “It’s raining or not raining,” are true in all cases, regardless of which atomic statements are true; such statements are certain but lack content.
While Wittgenstein thought atomic statements were the simplest truths, he didn’t say whether these were about physical facts or experiences. In either case, complex statements are constructible out of atomic statements using the logical connectives of propositional logic (Chapter 6). Statements not so constructible are nonsensical. Wittgenstein thought that most philosophical issues (for example, about values or God) were nonsensical. Paradoxically, he thought his own theory (starting with his claim that the world is the totality of facts) is nonsensical too. He ended on a mystical note: the most important things in life (his own theory, values, God, the meaning of life) cannot be put into words.
Bertrand Russell, while impressed by Wittgenstein’s views, tried to make them more sensible and less paradoxical. Russell’s logical atomism held that an ideal language – one adequate to describe reality completely – must be based on quantificational logic (Chapters 8 and 9) and thus must include quantifiers like “all” and “some.” It must also include terms that refer to the ultimately simple elements of reality – which include objects, properties, and relations. He debated whether the basic entities of the world were physical, or mental, or perhaps something neutral between the two.
Russell thought ordinary language can lead us into bad metaphysics (§9.6). Suppose you say “There’s nothing in the box.” Some might see “nothing” as the name of a mysterious object in the box. This is wrong. Instead, the sentence just 0370 means “It’s false that there’s something in the box.” Or suppose you say “The average American has 2.4 children.” While “the average American” doesn’t refer to an actual entity, the sentence is meaningful; it asserts that the average number of children that Americans have is 2.4. “Nothing” and “the average American” are logical constructs; they’re mere ways of speaking and don’t directly refer to objects. Russell went on to ask whether sets, numbers, material objects, persons, electrons, and experiences were real entities or logical constructs. Logical analysis is the key to answering such questions. We must see, for example, whether statements about material objects can be reduced to sensations, or whether statements about minds can be analyzed as about behavior.
In a similar spirit, Willard Quine pursued ontology, about what kinds of entity ultimately exist. His slogan, “To be is to be the value of a bound variable,” tried to clarify ontological disputes. It means that the entities our theory commits us to are those that our quantified variables (like “for all x” and “for some x”) must range over for our statements to be true. So if we say, “There’s some feature that Shakira and Britney have in common,” then we must accept features (properties) as part of our ontology – unless we can show that we’re using an avoidable way of speaking (a “logical construct” in Russell’s sense). Quine accepted sets in his ontology, because he thought they were needed for math and science; in picking an ontology, he appealed to pragmatic considerations. He rejected properties, concepts, and propositions because he thought they were less clear.
Wittgenstein later supported an ordinary language approach and rejected his earlier basing of metaphysics on logic. His Philosophical Investigations (1953) saw his earlier work as mistakenly imposing ideas on reality instead of fairly investigating it. His slogan became “Don’t think, but look!” Don’t say that reality has to be such and such, because that’s what your preconceptions demand; instead, look and see how it is. He now contended that few concepts had strict analyses. His main example was “game,” which has no strict definition. Games typically involve a competition between sides, winning and losing, a combination of skill and luck, and so forth. But none of these family resemblances is essential; solitaire drops competition, ring-around-the-rosie drops winning or losing, throwing dice drops skill, and chess drops luck. Any strict analysis of “game” is easily refuted by giving examples of games that violate the analysis. We distort language if we think that all statements must be analyzable into simple concepts that reflect metaphysically simple elements of reality. There’s no ideal language that perfectly mirrors reality; instead, there are various language games that humans construct for various purposes. Logic is a language game, invented to help us appraise the correctness of reasoning; we distort logic if we see it as giving us a special key to understand the metaphysical structure of reality.
So we see a range of views about the connection of logic with metaphysics, with Wittgenstein holding different views at different times.1 0371
1 For more on logic and metaphysics, see §3.4 (the logical positivist critique of metaphysics), §9.2 (mind and the substitution of identicals), and §§11.2–11.4 (Aristotelian essentialism).
18.3 The basis for logical laws
Let’s consider logical laws like modus ponens and non-contradiction:
· Modus ponens: If A then B, A, therefore B.
· Non-contradiction: A and not-A cannot both be true, unless A is taken differently in both instances.
Why are such logical laws correct, and how do we know that they’re correct? Thinkers have proposed a range of answers. Here we’ll consider five: supernaturalism, psychologism, pragmatism, conventionalism, and realism. (§§17.2–17.4 discussed deviant logics that reject these two laws.)
1. Supernaturalism holds that all laws of every sort – whether about physics, morality, math, or logic – depend on God. Radical supernaturalists say that God creates the logical laws or at least makes them true. God could make a world where modus ponens and the law of non-contradiction fail; and he could violate the law of non-contradiction – for example, by making “You’re reading this sentence” and “You’re not reading this sentence” both true. So logical laws are contingent: they could have been false. Moderate supernaturalists, on the other hand, say that logical laws express God’s perfect nature. God’s perfection require that he be consistent, that his created world follow the laws of logic, and that he desire that we be consistent and logical. Since these aspects of God’s nature are necessary, the laws of logic are also necessary. Supernaturalists of both sorts hold that God builds the laws of logic into our minds, so these laws appear to us to be “self-evident” when adequately reflected upon.
Critics object that the laws of logic hold for every possible world, including ones where there’s no God; so God cannot provide the basis for these laws. Others say that, since beliefs about logic are more certain than beliefs about God, it’s wrong to base logic on God. Still others say that God accepts logical laws because they’re inherently valid; logical laws aren’t valid just because God chooses to accept them (radical supernaturalism) or because they accord with his nature (moderate supernatualism).2
2 The parallel view in ethics claims that basic moral principles depend on God’s will. See my Ethics and Religion (New York: Cambridge University Press, 2016).
2. Psychologism holds that logical laws are based on how we think. Logic is part of our biology and natural history. Humans evolved to walk on two feet, have hand-eye coordination, communicate by speech, and think logically; these promote survival and are part of our genetic and biological makeup. Radical psychologism says that logic describes how we think; logical laws are psychological laws about thinking. Moderate psychologism, in contrast, sees logic as built into us in a more subtle way; we’re so built that at reflective moments we see inconsistency and illogicality as defects – even though at other times our thinking may suffer from such defects. When we reflect on our inconsistencies, we tend to 0372 develop an uncomfortable anxiety that psychologists call “cognitive dissonance”; this is as much a part of our biology and natural history as is thirst. So the laws of logic are built into our instincts.
Critics object that radical psychologism, which claims that logical laws describe our thinking, makes it impossible for us to be illogical or inconsistent. But people often reason invalidly or express inconsistent ideas; so logical laws don’t necessarily reflect how we think. Moderate psychologism recognizes this; it sees logical laws as reflecting norms about thinking that are built into us and that we recognize at reflective moments. This approach gives a plausible evolutionary and biological explanation of how logic can be instinctive in us; but it fails if it’s taken to explain what makes logical laws true or solidly based. Suppose evolution gave us an instinctive belief in the flatness of the earth; it wouldn’t follow that the earth actually was flat – or that this belief was so solidly based that we couldn’t criticize it. Similarly, the instinctiveness of the laws of logic wouldn’t make these logical laws correct or solidly based; maybe our instincts on these matters are right or maybe they’re wrong – we’d have to investigate further.
There’s also a problem with basing our knowledge of logical laws on evolutionary theory. We need logic to appraise the correctness of scientific theories like evolution; so our knowledge of logic cannot without circularity rest on our knowledge of evolutionary theory. In addition, our knowledge of logic is more solidly based than our knowledge of scientific theories.
3. Pragmatism holds that logical laws are based on experience. The broad consensus of humanity is that logic works; when we think things out in a logical and consistent way, we’re more apt to find the truth and satisfy our needs. This pragmatic test gives the only firm basis for logic or any other way of thinking.
Critics agree that, yes, logical thinking does work. But logic works because its laws hold of inherent necessity; so logical laws cannot be based on experience. Our experience can show us that something is true (for example, that this flower is red); but it cannot show us that something must be true (that its opposite is impossible). Compare logic to mouse traps. We can test various mouse traps to see how well they work; a given trap might catch a mouse or might not – both are possible. But it’s not possible for a logical law to fail – for example, for “If A then B” and “A” to both be true while “B” was false. The necessity of logical laws shows that they cannot be based on experience.
Besides, we cannot know that logic works unless we appeal to observation and reasoning – where the reasoning presupposes logical laws. So the pragmatist defense of logical laws is ultimately circular.
4. Conventionalism holds that logical laws are based on verbal conventions. We use logical words like “and,” “or,” “if-then,” and “not” according to rules that can be expressed in basic truth tables (§§6.2–6.6). Given these basic truth tables, we can show modus ponens to be valid (since its truth table never gives true premises and a false conclusion); we can similarly show the law of non- contradiction to be true (since its truth table comes out as true in all cases). So we can justify logical laws using conventions about what the logical words mean. 0373 Conventionalism explains why logical laws are necessary; if we deny them, we contradict ourselves, since we violate the meaning of words like “and,” “or,” “if-then,” and “not.” It also explains how we can know logical laws in an a priori manner, independently of sense experience; logical laws are true by virtue of the meaning of words (§§3.6–3.7), and so we can grasp their truth by becoming clear on what they mean. Conventionalism explains the necessity of logical laws without appealing to controversial beliefs about God, evolution, abstract entities, or our ability to grasp abstract truths. Logic’s conventionality also allows for alternative logics that are equally correct but follow different conventions.
Critics raise objections to conventionalism. First, the attempt to prove modus ponens using truth tables is circular:
· If the truth table for modus ponens never gives true premises and a false conclusion, then modus ponens is valid.
· The truth table for modus ponens never gives true premises and a false conclusion.
· ∴ Modus ponens is valid.
· If A then B
· A
· ∴ B
This argument itself uses modus ponens; so it’s circular, since it assumes from the start that modus ponens is valid. Second, conventionalism confuses the logical laws (which are necessary truths) with how we express them (using language conventions). If we changed our language, the logical laws would still be true, but we’d have to express them using different words. Third, conventionalism makes logical laws too arbitrary, since they could fail if we changed our conventions; for example, both modus ponens and the law of non-contradiction fail on some many-valued conventions (§17.1). But logical laws seem to have an inherent correctness that doesn’t depend on which language conventions we adopt.
5. Realism holds that logical laws are objective, independent, abstract truths. We discover logical laws; we don’t construct or create them. Logical laws aren’t reducible to the mental, the physical, usefulness, or conventions. Logical laws govern our world, and every possible world, because violating them is impossible; it cannot be, for example, that A and not-A are both true. Logical laws become self-evident to us when adequately reflected upon. This doesn’t mean that logical intuitions are infallible; beginning logic students tend to have poor intuitions about whether an argument is valid. But logical intuitions can be trained; we can test proposed inference forms through concrete examples where the validity or invalidity is more obvious. The best evidence for a logical principle is that a well-trained mind finds it evident and can’t find counterexamples.
Critics object that realism makes logical laws too mysterious. Suppose you’re a materialist: you hold that all facts are expressible in the language of physics and chemistry. How do objective, irreducible logical facts fit into such a universe? Are logical facts composed of chemicals, or what sort of weird thing are they? And how could we ever know such mysterious logical facts? In addition, objective, abstract logical laws seem to presuppose abstract entities (§18.1), which 0374 have no place in a materialistic world. A dualist view that accepts only mind and matter would have similar doubts about realism.
Logicians for the most part (except for deviant logicians – see Chapter 17) agree on the logical laws. But logicians differ widely on what these laws are based on and how we can know them to be correct.
18.4 Truth and paradoxes
Truth is important to logic. A valid argument is often defined as one in which it’s impossible to have the premises all true and conclusion false. Truth comes up further in propositional logic (with truth tables and the truth-assignment test) and in refutations of invalid arguments (which are possible situations making the premises all true and conclusion false).
There are many issues about truth. For example, is classical logic right in assuming that statements are true or false, but not both, and that true and false are the only truth values? Some deviant logics deny these assumptions (Chapter 17).
What do “true” and “false” apply to? Suppose you point to a green apple and say “This is green.” Is what is true the sentence “This is green,” or perhaps the sentence as used on this occasion (where you point to a certain object)? If so, then is this sentence concrete physical marks or sounds, or is it a more abstract pattern that has written or auditory instances? Or perhaps what is true-or-false is not sentences, but rather propositions, which are assertions that we use language to make. But then are propositions something mental, or are they abstract entities, like the meaning of “This is green”?
What does “true” mean? On different views, being “true” is:
· corresponding to the facts (correspondence theory),
· cohering with our other beliefs (coherence theory),
· being useful to believe (pragmatist theory),
· being verified (verification theory), or
· being what we’d agree to under cognitively ideal conditions (ideal consensus theory); or perhaps
· “It’s true that A” is just a wordy way to assert A (redundancy theory).
The pragmatist and verification analyses reject the law of excluded middle, since it can happen that neither a statement nor its negation is useful or verified. These two analyses could also support many-valued logic (§17.1), since a statement can be useful or verified to various degrees. Thus different answers to “What is truth?” can support different logics.
Alfred Tarski proposed an adequacy condition, called “convention T,” that any definition of truth must satisfy; here’s an example: 0375
The sentence “Snow is white” is true, if and only if snow is white.
This equivalence raises problems for definitions that water down truth’s objectivity. For example, the view that “true” just means “accepted in our culture” leads to an absurdity. Imagine a tropical island where snow is white (in high-mountain cracks that are never visited or seen) but yet people don’t believe that it’s white; on the proposed view, snow could be white while “Snow is white” wasn’t true – which is absurd. A similar objection works against the pragmatist and verification views. Imagine that “Snow is white” was neither useful to believe nor verified; then, on pragmatism or verificationism, snow could be white while “Snow is white” wasn’t true – which is absurd.
Further issues are raised by the liar paradox, a statement that asserts its own falsity (and so appears to be both true and false). Consider claim P:
(P) P is false.
Is P true? Then things must be as P says they are, and thus P has to be false. Is P false? Then things are as P says they are, and thus P has to be true. So if P is either true or false, then it has to be both true and false.
Graham Priest and others claim that P is both true and false, which requires rejecting Aristotle’s law of non-contradiction (§17.2). The more common view is that P is neither true nor false, which requires rejecting or qualifying Aristotle’s law of excluded middle. Bertrand Russell proposed a theory of types that outlaws certain forms of self-reference. Very roughly, there are ordinary objects (type 0), properties of these (type 1), properties of these properties (type 2), and so on. Any meaningful statement can talk only about objects of a lower type; so no speech can talk meaningfully about itself. P violates this condition, and so is meaningless – and thus neither true nor false.
But Russell’s view seems to refute itself. “Any meaningful statement can talk only about objects of a lower type,” to be useful, has to restrict all statements, of every type; but then it violates its own rule and declares itself meaningless.
Tarski, to deal with the paradox, proposed that no language can contain its own truth predicate; to ascribe truth or falsity to a statement in a given language, we must ascend to a higher-level language, called the metalanguage. P violates this condition and so is meaningless – and thus neither true nor false.
Opponents say Tarski’s view is too restrictive. English and other languages do contain their own truth predicates, and they need to do this for many purposes. So it would be better to have a less sweeping restriction to take care of the liar paradox. But there’s little agreement about what this restriction should be.
Epimenides of Crete in the sixth century BC proposed the liar paradox, and St Paul mentioned it in his letter to Titus (1:12). It has been widely discussed ever since. While most logicians think that a theory of truth must deal with the paradox, how best to do this is still unclear. 0376
18.5 Logic’s scope
“Logic” is often defined in ways like “the analysis and appraisal of arguments” or “the study of valid reasoning.” The term “logic” can be used in a narrow and a broad sense. Logic in the narrow sense is the study of deductive reasoning, which is about what logically follows from what. Logic in the broad sense includes also various other studies that relate to the analysis and appraisal of arguments, like informal logic, inductive logic, metalogic, and philosophy of logic (Chapters 3–5, 15, and 18).
Even taking “logic” in this narrow deductive sense, there’s still some unclarity on what it includes. Suppose you say, “I have $30; therefore I have more than $20.” Is this part of logic, part of math, or both?
Willard Quine suggested that we limit “logic” to classical propositional and quantificational logic (Chapters 6 to 9), which he saw as fairly uncontroversial and as focusing on topic-neutral terms like “and” and “not” that arise in every area of study. Modal and deontic logic (Chapters 10 to 12) focus on terms like “necessary” and “ought” that are too colorful and topic-specific to be part of logic; these areas, if legitimate at all (and he had doubts) are part of philosophy in general, not part of logic. Mathematical extensions, like set theory and axiomatizations of arithmetic, belong to math. And deviant logics (Chapter 17) are illegitimate.
Most logicians today tend to use “(deductive) logic” in a broader way that’s hard to pin down. Deductive logic is commonly taken to include, besides syllogisms and classical symbolic logic, extensions like modal and deontic logic, deviant logics, and sometimes even mathematical extensions like set theory. Logic is part of at least three disciplines – philosophy, math, and computer science – which approach it from different angles. Any attempt to give sharp and final boundaries to the term “logic” would be artificial.1
1 For more on philosophy of logic, see Willard Quine’s Philosophy of Logic, 2nd ed. (Cambridge, Mass.: Harvard University Press, 1986), which is a good introduction from an influential and controversial thinker, and Colin McGinn’s Logical Properties: Identity, Existence, Predication, Necessity, Truth (Oxford: Clarendon, 2000), which gives an opposing view.
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For Further Reading
If you’ve mastered this book and want more, consult my Historical Dictionary of Logic (Lanham, Md.: Scarecrow Press, 2006). This brief encyclopedia of logic has nontechnical, alphabetized articles on branches of logic, figures and historical periods, specialized vocabulary, controversies, and relationships to other disciplines – and a 13-page chronology of major events in the history of logic. It also has a 52-page bibliography of readings in logic, a list of 63 recommended works in various categories, and this smaller list of very helpful works:
· P. H. Nidditch’s The Development of Mathematical Logic (London: Routledge & Kegan Paul, 1962): the history of logic from Aristotle onward.
· Willard Quine’s Philosophy of Logic, 2nd ed. (Cambridge, Mass.: Harvard University Press, 1986): a contentious introduction from a major thinker.
· Colin McGinn’s Logical Properties: Identity, Existence, Predication, Necessity, Truth (Oxford: Clarendon, 2000): an opposing view from Quine’s.
· Graham Priest’s An Introduction to Non-Classical Logic, 2nd ed. (Cambridge: Cambridge University Press, 2008) a defense of deviant logic by its most eloquent defender (technical parts may be skipped).
· Ian Hacking’s An Introduction to Probability and Inductive Logic (Cambridge: Cambridge University Press, 2001): a solid introduction.
· George Boolos and Richard Jeffrey’s Computability and Logic, 3rd ed. (Cambridge: Cambridge University Press, 1989): topics like Turing machines, uncomputable functions, the Skolem-Löwenheim theorem, and Gödel’s theorem – technical but clear and doesn’t assume much math.
If you’re just starting, you might pick one or two of these that interest you. For further suggestions, consult my Historical Dictionary of Logic.
As advanced students go through various chapters, they might want to pursue further readings in this book.Chapter 6 (basic propositional logic) goes well with metalogic §§15.1–2, deviant logic Chapter 17, and philosophy of logic §18.4. Chapter 7 (propositional proofs) goes well with metalogic §§15.3–5 and perhaps informal and inductive Chapters 3 to 5. Chapters 8 and 9 (quantificational logic) go well with metalogic §15.6, history of logic Chapter 16, philosophy of logic §§18.1–3, and syllogisms Chapter 2. And Chapters 10 to 14 (modal/deontic/belief logic and a formalized ethical theory) go well with history of logic §16.5 and philosophy of logic §18.5.