17
Deviant logics reject standard assumptions. Most logicians have assumed that statements are either true or false, but not both, and that true and false are the only truth values. Deviant logics question such ideas. Maybe we need more than two truth values (many-valued logic). Or maybe “A” and “not-A” can both be true (paraconsistent logic) or both be false (intuitionist logic). Or maybe standard IF-THEN inferences are mistaken (relevance logic).
Deviant logics are controversial. Some are happy that logic is becoming, in some circles, as controversial as other areas of philosophy. Others defend standard logic and see deviant logic as promoting intellectual chaos; they fear what would happen if thinkers couldn’t assume that modus ponens and modus tollens are valid and that contradictions are to be avoided.
17.1 Many-valued logic
Most logicians assume that there are only two truth values: true and false. Our propositional logic in Chapter 6 accepts this bivalence, symbolizing true as “1” and false as “0.” This is consistent with there being truth-value gaps for sentences that are meaningless (like “Glurklies glurkle”) or vague (like “Her shirt is white,” when it’s between white and gray). Logic needn’t worry about such sentences, since arguments using them are already defective; so we can just stipulate that capital letters stand for statements that are true or false.
Many-valued logics accept more than two truth values. Three-valued logic might use “1” for true, “0” for false, and “½” for half-true. This last category might apply to statements that are unknowable, or too vague to be true-or-false, or plausible but unproved, or meaningless, or about future events not yet decided. A three-valued truth table for NOT looks like this:
|
P |
~P |
|
0 |
1 |
|
½ |
½ |
|
1 |
0 |
If P is false, then ∼P is true.
If P is half-true, then ∼P is half-true.
If P is true, then ∼P is false.
This table shows how the other connectives work: 0360
AND takes the value of the lower conjunct, and OR takes the value of the higher disjunct. IF-THEN is true if the consequent is at least as true as the antecedent and is half-true if its truth is a little less. IF-AND-ONLY-IF is true if both parts have the same truth value and is half-true if they differ a little.
Given these truth tables, some standard logical laws fail. “(P ∨ ∼P)” (the law of excluded middle) and “∼(P • ∼P)” (the law of non-contradiction) sometimes are only half true. “(P ⊃ Q)” isn’t equivalent to “∼(P • ∼Q),” since they differ in truth value if P and Q are both ½. We can avoid these results by making “(½ ∨ ½)” true and “(½ • ½)” false; but then “P” strangely isn’t logically equivalent to “(P ∨ P)” or “(P • P).”
Fuzzy logic proposes an infinity of truth values; these can be represented by real numbers between 0.00 (fully false) and 1.00 (fully true). We might define a “valid argument” as one in which, if the premises have at least a certain truth value (perhaps .9), then so does the conclusion; modus ponens then fails (since if “A” and “(A ⊃ B)” are both .9, then “B” might be less than .9) as do some other logical principles. Some propose an even fuzzier logic with vaguer truth values like “very true” or “slightly true.”
Fuzzy logic is used in devices like clothes dryers to permit precise control. A crisp-logic dryer might have a rule that if the shirts are dry then the heat is turned off; a fuzzy-logic dryer might say that if the shirts are dry to degree n then the heat is turned down to degree n. We could get the same result using standard logic and a relation “Dxn” that means “shirt x is dry to degree n” – thus using degrees-of-dryness instead of degrees-of-truth.
Opponents say many-valued logic is weird and arbitrary and has little application to real-life arguments. Even if this is so, the many-valued approach has other applications. It can be used, for example, in computer memory systems with more than two states. And it can be used to show the independence of axioms for propositional logic (§15.5); an axiom can be shown to be independent of the other axioms of a certain system if, for example, the other axioms (and theorems derived from these) always have a value of “7” on a given truth-table scheme, while this axiom sometimes has a value of “6.” 0361
17.2 Paraconsistent logic
Aristotle’s law of non-contradiction states that the same property cannot at the same time both belong and not belong to the same object in the same respect. So “S is P” and “S is not P” cannot both be true at the same time, unless we take “S” or “P” differently in the two statements. Aristotle saw this law as certain but unprovable. Deniers of the law assume it in their practice; wouldn’t they complain if we bombarded them with contradictions?
Aristotle mentions Heraclitus as denying the law of non-contradiction. The 19th-century thinkers Georg Hegel and Karl Marx also seemed to deny it and are often seen as proposing an alternative dialectical logic in which contradictions are real. Critics object that such a logic would confuse conflicting properties in the world (like hot/cold or capitalist/proletariat) with logical self-contradictions (like the same object being both white and, in the same sense and time and respect, also non-white).
In standard propositional logic, the law of non-contradiction is “∼(P • ∼P)” and is a truth-table tautology – a formula true in all possible cases:
|
P |
~(P · ~P) |
|
0 |
1 |
|
1 |
1 |
“This is false: I went to Paris and I didn’t go to Paris.”
“P and not-P” is always false in standard logic, which presupposes that “P” stands for the same statement throughout. English is looser and lets us shift the meaning of a phrase in the middle of a sentence. “I went to Paris and I didn’t go to Paris” may express a truth if it means “I went to Paris (in that I landed once at the Paris airport) – but I didn’t really go there (in that I saw almost nothing of the city).” Because of the shift in meaning, this would better translate as “(P • ∼Q),” which wouldn’t violate the law of non-contradiction.
Some recent logicians, like Graham Priest, claim that sometimes a statement and its contradictory are both true. Such dialethist logicians don’t say that all statements and their denials are true – but just that some are. Here are examples where “A and not-A” might be claimed to be true:
· “We do and don’t step into the same river.” (Heraclitus)
· “God is spirit and isn’t spirit.” (the mystic Pseudo-Dionysius)
· “The moving ball is here and not here.” (Hegel and Marx)
· “The round square is both round and not-round.” (Meinong)
· “The one hand claps and doesn’t clap.” (Eastern paradox)
· “Sara is a child and not a child.” (paradoxical speech)
· “What I am telling you now is false.” (liar paradox)
· “The electron did and didn’t go in the hole.” (quantum physics)
Most logicians contend that these aren’t genuine cases of “A and not-A,” at least 0362 if they’re taken in a sensible way, since we must take the two instances of “A” to represent different ideas. For example, “Sara is a child and not a child” can be sensible only if it really means something like “Sara is a child-in-age but not a child-in-sophistication.” Paradoxical speech, although sometimes nicely provocative, doesn’t make sense if taken literally. Dialethists try to show that some of their allegedly true self-contradictions resist such analyses.
In standard propositional logic we can from a single self-contradiction deduce the truth of every statement and its denial. But then, if we believed a self-contradiction and also all its logical consequences, we’d contract the dreaded disease of contradictitis – whereby we’d believe every statement and also its contradictory – bringing chaos to human speech and thought. Here’s an intuitive derivation showing how, given the contradictory premises “A is true” and “A is not true,” we can deduce any arbitrary statement “B” (this “A, ∼A ∴ B” inference is called the explosion principle):
1. A is true. {premise}
2. A is not true. {premise}
3. ∴ At least one of these two is true: A or B. {from 1: if A is true then at least one of the two, A or B, is true}
4. ∴ B is true. {from 2 and 3: if at least one of the two, A or B, is true and it’s not A, then it’s B}
Dialethists respond by rejecting standard logic. They defend a paraconsistent logic that rejects the explosion principle; this lets them contain an occasional self-contradiction without leading to an “anything goes” logical nihilism. In the above argument, they reject line 4 and thus the “(A ∨ B), ∼A ∴ B” inference (disjunctive syllogism). Suppose, they say, B is false and A is both-true-and-false (!); then, they say, “(A ∨ B)” is true (since “A” is true), “∼A” is true (since “A” is also false), but “B” is false – and so disjunctive syllogism is invalid.
Paraconsistent logicians have developed their own truth tables. One option lets “A” and “not-A” have any combination of true or false, independently of each other; so we have four possibilities:
P ∼P
0 0
0 1
1 0
1 1
P and not-P are both false.
P is false and not-P is true.
P is true and not-P is false.
P and not-P are both true.
This approach rejects the usual understanding of “not,” whereby “not-A” has the opposite truth value as “A.” In paraconsistent logic, disjunctive syllogism is invalid, since it can have true premises and a false conclusion:
0363 Similarly, the explosion principle, which permits us to deduce any arbitrary statement from a self-contradiction, is invalid:
Paraconsistent logic lets logic go on normally for the most part – so most of the arguments in this book that are valid/invalid on standard logic would come out the same as before; but it also permits an occasional self-contradiction to be true. Thus it denies that a strict adherence to the law of non-contradiction is necessary for coherent thought.
Critics object that it makes no sense to permit “A” and “not-A” to both be true, at least if we take “not” in anything close to its normal sense. If we reject the usual truth table for “not,” which makes “not-A” always have the opposite truth value of “A,” then what is left of the meaning of “not”?
Critics also object that permitting “A” and “not-A” to both be true lets irrational people off too easily. Imagine politicians or students who regularly contradict themselves, asserting “A” and then a few breaths later asserting “not-A,” and yet defend themselves using the “new logic,” which lets both be true at once. Surely this is lame and sophistical.
Some who accept the law of non-contradiction see value in paraconsistent logic, since people or computers may have to derive conclusions from inconsistent data. Suppose that our best data about a crime is flawed and inconsistent; we still might want to derive the best conclusions we can from this data. The “anything and its opposite follows from inconsistent data” approach of classical logic is unhelpful. Paraconsistent logic claims to do better.
Critics question whether paraconsistent logic can do better. If our data is inconsistent, then it has errors and can’t provide reliable conclusions. So we need to clear up the inconsistency first, perhaps by rejecting the least solidly based statements. We need to see what follows (using standard logic) from the most probable consistent subset of the original data.
Critics also claim that rejecting disjunctive reasoning lessens the real-world usefulness of paraconsistent logic. Suppose we know that either A or B committed the murder, and then we find out that A didn’t do it. We need to conclude that B then did it; but paraconsistent logic says that this is invalid!
Logicians defend the law of non-contradiction in different ways. Some see it as a useful language convention. We could imagine a tribe where vague statements (like “This shirt is white”) in borderline cases are said to be both true and false (instead of neither true nor false). Some might speak this way; and we could easily translate between this and normal speech. If so, then perhaps a strict adherence to the law of non-contradiction is at least partly conventional. But, even so, it’s a convention that’s less confusing than the paraconsistent alternative.
Others see the law of non-contradiction as a deep metaphysical truth about reality. They see paraconsistent logicians as offering, not an alternative way of 0364 speaking, but rather an incoherent metaphysics. Regardless of our verdict here, dialethism and paraconsistent logic do offer interesting challenges that make us think more deeply about logic.
17.3 Intuitionist logic
Aristotle held the law of excluded middle, that either “S is P” or “S is not P” is true. Standard propositional logic expresses this as “(A ∨ ∼A)” (“A or not-A”), which has an all-1 truth table and is true in all possible cases. Intuitionist logicians, like the mathematicians Luitzen Brouwer and Arend Heyting, reject this law when applied to some areas of math. They similarly reject the law of double negation “(∼∼A ⊃ A)” (“If not-not-A, then A”). They think “A” and “∼A” are sometimes both false in cases involving infinite sets. To emphasize these differences, intuitionists use “¬” for negation instead of “∼.”
Intuitionist mathematicians see the natural numbers (0, 1, 2, …) as grounded in our experience of counting. Mathematical formulas are human constructs; they shouldn’t be considered true unless the mind can prove their truth. Goldbach’s conjecture says “Every even number is the sum of two primes.” This seems to hold for every even number we pick: 2 (1 + 1), 4 (3 + 1), 6 (5 + 1), 8 (7 + 1), and so on. But no one has proved or disproved that it holds for all even numbers. Most think Goldbach’s conjecture must be true or false objectively, even if a proof either way may be impossible. Intuitionists disagree. They say truth in mathematics is provability; if neither Goldbach’s conjecture nor its negation is provable, then neither is true. So intuitionists think that, in some cases involving infinite sets (like the set of even numbers), neither “A” nor “∼A” is true, and so both are false. The law of excluded middle does apply if we use finite sets; so “Every even number under 1,000,000,000 is the sum of two primes” is true or false, and we could write a computer program that could in principle eventually tell us which it is.
Some non-realists reject the law of excluded middle in other areas. Suppose you think the only basic objective truths are ones about your individual experience, like “I feel warmth” or “I sense redness.” You might accept objective truths about material objects (like “I’m holding a red pen”), but only if these can be verified by your experience. But often your experience can verify neither “A” nor “not-A”; then neither would be true, and both would be false. So you might reject the law of excluded middle on the basis of a non-realist metaphysics.
Realists think this is bad metaphysics. Goldbach’s conjecture about mathematics is objectively true or false; and our experience supports (but doesn’t conclusive prove) that it’s true. It’s wrong to identity “true” with “verified,” since we may imagine unverifiable truths; there may be a whole world of truths and falsehoods that aren’t accessible to our finite minds. 0365
17.4 Relevance logic
Classical propositional logic analyzes “If P then Q” as simply denying that we have P-true-and-Q-false:
· (P ⊃ Q) = ∼(P • ∼Q)
· If P is true, then Q is true = We don’t have P true and Q false
An IF-THEN understood this way is a material implication and is automatically true if the antecedent is false or the consequent is true. This leads to the so-called paradoxes of material implication:
· From “not-A” we can infer “If A then B.” So from “Pigs don’t fly” we can infer “If pigs fly, then I’m rich.”
· From “B” we can infer “If A then B.” So from “Pigs don’t fly” we can infer “If I’m rich, then pigs don’t fly.”
While many logicians see such results as odd but harmless, relevance logicians see them as wrong and want to reconstruct logic to avoid them.
Relevance logicians oppose evaluating the truth of “If A then B” just by the truth values of the parts; they say an IF-THEN can be true only if the parts are relevant to each other. While they’re vague on what this means, they insist that logic shouldn’t prove theorems like “If A-and-not-A, then B,” where antecedent and consequent share no letters. Since paraconsistent logic (§17.2) rejects the related explosion principle that a self-contradiction entails every statement, there’s a natural affinity between the approaches; many relevance logics are also paraconsistent. Relevance logics often symbolize relevant implication as “→,” to contrast with the “⊃” of material implication.
Defenders of material implication appeal to conversational implication to diffuse objections based on the paradoxes of material implication. Paul Grice claims that what is true may not be sensible to assert in ordinary speech. When we speak, we shouldn’t make a weaker claim rather than a stronger one unless we have a special reason. Suppose you tell your five children, “At least three of you will get Christmas presents” – while you know that all five will. The weaker statement suggests or insinuates that not all five will get presents. This is due to speech conventions, not logical entailments. “At least three will get presents” doesn’t logically entail “Not all five will get presents”; but saying the first insinuates the second. Similarly, there’s little point in saying “If P then Q” on the basis of knowing not-P or knowing Q – since it’s better to say straight off that not-P or that Q. There’s generally a point to saying “If P then Q” only if there’s a special connection between the two, some way of going from one to the other. But, again, this has to do with speech conventions, not with truth conditions for “If P then Q.” 0366
Some defenders of material implication claim that the so-called paradoxes of material implication are perfectly correct and can be defended by intuitive arguments. We can derive “If not-A then B” from “A”:
1. A is true. (Premise)
2. ∴ Either A is true or B is true. {from 1}
3. ∴ If A isn’t true, then B is true. {from 2}
Relevance logic must reject this plausible derivation; it must deny that 2 follows from 1, that 3 follows from 2, or that deducibility is transitive (if 3 follows from 2, and 2 from 1, then 3 follows from 1). Doing any of these violates our logical intuitions at least as much as do the material-implication paradoxes. So relevance logics, although they try to avoid unintuitive results about conditionals, cannot achieve this goal; they all result in oddities at least as bad as the ones they’re trying to avoid. Another problem is that a wide range of conflicting relevance logics have been proposed; these disagree much on which arguments involving conditionals are valid.
Relevance logicians have found other conditional arguments that, while valid on the traditional view, seem to them to be invalid. Some even question the validity of modus ponens (“If A then B, A ∴ B”). One allegedly questionable modus ponens inference involves measles:
· If you have red spots, then you have measles.
· You have red spots.
· ∴ You have measles.
· (R ⊃ M)
· R
· ∴ M
This is claimed to be invalid because you might have red spots for some other reason. Another objection, from Vann McGee, is more complex. In 1980, three main candidates ran for US president: two Republicans (Ronald Reagan, who won with over 50 percent of the vote, and John Anderson, who got about 7 percent of the vote and was thought to have no chance to win) and a Democrat (Jimmy Carter, who got just over 40 percent). Consider this argument, given just before the election:
· If a Republican will win, then if Reagan does not win then Anderson will win.
· A Republican will win.
· ∴ If Reagan does not win, then Anderson will win.
· (W ⊃ (∼R ⊃ A))
· W
· ∴ (∼R ⊃ A)
Here it seems right to believe the premises but not the conclusion (since clearly if Reagan doesn’t win, then Carter will win, not Anderson). Again, this instance of modus ponens is claimed to be invalid.
Defenders of modus ponens think such examples confuse a genuine IF-THEN with other things. Compare these three ways of taking “If you have red spots, then you have measles”: 0367
1. Genuine IF-THEN: “If you have red spots, then you have measles.”
2. Conditional Probability: “The probability is high that you have measles, given that you have red spots.”
3. Qualified IF-THEN: “If you have red spots and other causes can be excluded, then you have measles.”
The premise about measles, if a genuine IF-THEN, has to mean 1, and not 2 or 3; but then its truth excludes your having red spots but no measles. The truth of this IF-THEN doesn’t entail that we’re certain that there are no other causes; but if in fact there are other causes (so you have red spots but no measles), then the IF-THEN is false. A similar analysis takes care of the Reagan argument.
Even if we reject relevance logic, still we have to admit that some conditionals, or their near relatives, cannot plausibly be interpreted as material implications. We already mentioned conversational implication (where saying A suggests or insinuates a further statement B) and conditional probability (where fact A would make fact B probable to a given degree). There are also logical entailments (“B logically follows from A” – which Chapter 10 symbolizes as “☐(A ⊃ B)”) and counterfactuals (“If A had happened then B would have happened” – often symbolized as “(A ☐→ B)”). So conditionals and their near relatives form a diverse family, going from very strong logical entailments, through standard IF-THENs, down to probability or to mere suggestion or insinuation. Even apart from relevance logic, conditionals raise many logical issues.
It shouldn’t surprise us that central logical principles raise controversies. Even “I see a chair” raises controversies if we push it far enough. But not all alternative views are equally reasonable. I’d contend that, despite controversies, I really do see a chair. And I’d contend that most assumptions about logic that have been held since Aristotle’s time are solid.1
1 I do think, however, that in quantified modal logic there’s much to be said for free logic (§11.4), which is somewhat deviant. For more on deviant logics, see Graham Priest’s An Introduction to Non-Classical Logic, 2nd ed. (Cambridge: Cambridge University Press, 2008) and J. C. Beall and Bas van Fraassen’s Possibilities and Paradox (Oxford: Oxford University Press, 2003).