Logic for a Fuzzy World

For the purposes of formal logic in this class, we make 2 assumptions about truth value (truth-or-falsity), namely, (i) that the truth value of a sentence is not relative to persons, places, times or circumstances and (ii) that there are just two truth values, true and false and that they do not admit of degree, i.e. there’s no such thing as a little bit true or a lot true, more or less true or false or anything in between true and false.

Now on the face of it this seems unrealistic, if not downright naïve. We live in a world where things aren’t black and white so (ii) seems crude and dogmatic. Moreover, the truth value of lots of sentences does seem relative to persons, places, times and circumstances so (i) seems to be false. So what should we do? I argue that these claims aren’t as unrealistic, naïve or dogmatic as they may seem by considering and responding to putative counterexamples.
A counterexample is a case or example that shows a general claim to be false. So, suppose you say “All dogs have brown eyes.” I can show that to be false by producing a counterexample: “Nope. Most Siberian Huskies have blue eyes.” Sometimes though what seem to be counterexamples really aren’t. Consider: “All monkeys have tails.” “Nope. Chimpanzees don’t have tails.” “Sorry, chimps aren’t monkeys—they’re apes—so that’s not a counterexample.”

To test claims (i) and (ii) let’s consider what seem to be counterexamples and see if we can respond to them.

Apparent counterexamples to (i):

(1) I like chocolate

(1) is true when Geraldine says it but not when David says it.
Response: We exclude sentences that are context-dependent. When we want to deal with sentences involving indexicals we consider the statements they make given their contexts of utterance. When Geraldine says, “I like chocolate” she’s making the statement that Geraldine likes chocolate; when David says, “I like chocolate” he’s making the statement that David likes chocolate. “Geraldine likes chocolate” is true whoever says it. “David likes chocolate” is false whoever says it. The truth value of these sentences isn’t relative to persons.

(2) There’s not a cloud in the sky.

(2) is true in some places at some times but not in other places at other times—and it doesn’t include any indexicals.

Response: (2) doesn’t express a complete thought and in logic we restrict ourselves to sentences that express complete thoughts. When we fill out (2) to make it express the complete thought the speaker likely has in mind we see that the claim is context dependent: the statement the speaker probably intends to make is that at the time of his utterance, the sky in his immediate vicinity is cloudless, e.g. “At 11:35 am Monday, February 20, 2006 the region of the sky visible to observers on the 600 block of 2nd Avenue, Chula Vista is cloudless.” The truth value of that sentence is not relative to persons, places, times or circumstances—it is, as it happens, false regardless of the context of utterance.
Apparent counterexamples to (ii)

(3) For Sale: 1996 silver, 4-door Nissan Sentra with 5 speed manual transmission, new clutch, recent brake job, low mileage.

I’m trying to sell my car and this is almost true—everything except the bit about low mileage (this car has a little over 187,000 miles on it). Well, in any case, it’s not entirely false—it’s somewhere between true and false. It’s not, um, a complete lie.

Response: (3) is a conjunction (and-statement) of several different claims, of which some but not all are true. We resolve to assign true to a conjunction only if all of its parts (“conjuncts”) are true; otherwise we call it false. This seems a sensible policy because it means you know what you’re getting. If the conjunction is true you know all of its parts are. If it’s false then you can go on to ask why and consider those parts individually.

(4) God exists.

Well no one knows for sure, do they? So how can you say (4) is either true or false?

Response: See “Truth, Belief and Justification.” Knowing that something is so is quite a different thing from it’s being so. Even if no one knows, or ever will know, or for all practical purposes can know, whether a proposition is true or false it doesn’t follow that it’s neither true nor false. Consider “Lucy,” an early hominid who lived about 3.2 million years ago, whose bones were discovered in Ethiopia in 1974. Did she have exactly 4 children or not? No one will ever know. But there is a fact of the matter—it’s either true or false that she had exactly 4 children. And it’s either true or false that God exists—we just don’t know and, at least in this life, never will know.

(5) Stealing is wrong.

Well maybe not all the time. What about Jean Valjean who stole that loaf of bread when he was starving? Or what if you’re captured by the Bad Guys and locked up. Wouldn’t it be ok to steal the key to get out so that you can run back to your outfit and warn them that the Bad Guys are planning a raid?

Response: (5) is not a complete thought. Is it supposed to mean that stealing is always wrong? Then it’s false. Is it supposed to mean that stealing is sometimes wrong? Then it’s true.

(6) Teal is a shade of blue.

Well, some people think it’s blue but others think it’s green so (6) isn’t really entirely true or false.

Response: Those who call teal blue and those who call teal green don’t mean the same thing by “blue”—they classify colors differently and are really saying different things when they say (6). If we get precise about what we mean by “blue”—whether the range of colors between x and y wavelengths or between w and z wavelengths we’ll recognize that the disagreement is merely verbal, that we’re not really saying the same thing. Teal is in the x to y range – false; teal is in the w – z range – true.

(7) David is bald.

David has grown the remaining hair from one side of his head to a length of 6 inches and combs it over the top of his head. It looks fairly plausible until the wind blows. Well he does have some hair on top. So (7) isn’t either entirely true or entirely false.

Response: This is a tough one but arguably not because there’s a problem about truth and falsity. Rather the problem is with “bald”—it’s a vague predicate like “tall,” “fat,” and many others. We resolve to avoid vagueness and to make the terms we use precise, even if only by stipulation. Once we do that, stipulating perhaps that “bald” is to mean having fewer than x hairs per square inch on a given region of the head then we can either judge that (7) is true or false. The problem is that adopting a precise criterion for baldness along these lines, whatever the value of x, whatever cranial region we specify, leads to some very counterintuitive results. In particular, we’ll have to say that there is a baldness threshold such that losing one hair can make a guy bald. This is a hard problem, for reasons I hope to discuss—before we put it aside and get on with business.