Art of Reasoning

We will employ so-called “classical logic” as we move forward. One of the key defining characteristics of classical logic is that it is bivalent, which means: Two truth values. Obviously, these are true and false.

There are other logics that employ three truth values (true, false, and null), and still others employ an infinite range of truth values between true and false, where the value is a “weighting” of “how true” or “how false” a given proposition is. And there are certainly ways that we think that motivate the development of such logics.

For example, let’s say that I claim: “Joe is bald.” You look at Joe and see a basically full head of hair. So, you respond: “False. Joe is not bald.”

I say, “Well, actually, you can’t really see it, but Joe has lost half of the hair he had when he was a teenager.”

You reply, “So what? He has full coverage, and if ‘bald’ means anything, it means visible bare areas.”

I ask you, “Okay, so how much ‘bare’ does there need to be? If we took away more hair in one area, so that it was just starting to be ‘bare,’ would you then agree that Joe is bald?”

You reply, “Well, I don’t know. There’s a sort of vagueness to the term ‘bald.’ A person might be more or less bald depending upon how the hair is distributed.”

And there is the point! In vague contexts, we often “feel” like statements can be “more or less” true. So, we are motivated to assign a “weighted” truth-value to vague statements. And such multi-valent logics attempt to codify such intuitions.

Quantum theory is another context in which multi-valent logics seem to naturally fit. It is claimed that there is only a certain likelihood that a particular particle will be in a particular place. So, assigning a weighted truth-value that correlates with the mathematically-calculated percentage chance of a given claim about a particle’s location (and/or velocity) seems to work well.

And trivalent logics are very intuitive when we are making claims about situations about which we simply don’t know (and particularly when we think that we in principle cannot know). Such logics enable us to capture inferences about in-principle unknowns with a null truth-value.

So, with all of these good reasons to employ trivalent and multi-valent logics, why should we “hamstring” ourselves with a merely bivalent logic?

There are several answers, and they all make a solid case for the adequacy of classical logic.

First, virtually all non-classical logics preserve the classical intuitions we have. These non-classical logics are not really “alternatives” to classical logic. Instead, they extend classical logic. So, for contexts in which such extensions are not necessary, it’s not like classical logic is “limited” or fundamentally “inadequate” to get the job done.

Second, the axioms upon which most of even non-classical logics are built are themselves bivalent axioms. This indicates that, at core, logic rests on a bivalent bedrock.

Finally, and more importantly, the most fundamental reason why virtually all logicians are classical logicians is that non-classical logics “build in” epistemological considerations that are both unnecessary and are even questionable. It turns out that you really can “say” anything you wish to “say” using classical logic alone.

For example, let’s return to our “Joe is bald” claim.

A classical logician will say, “Look, you might not be able to detect a particular threshold of baldness, but the problem is not one of logic! The problem is in the content, not the form. The problem is in the vagueness itself. Clearly define ‘bald,’ and the vagueness disappears! Let the vagueness disappear, and the motivation for your multi-valent logic disappears. So, there is nothing ‘limited’ or ‘inadequate’ with the classical logic itself. The problem is that we often make fluffy, vague claims out of laziness.”

In quantum theory, the classical logician will say that physicists have build a basic anti-realism into their claims. In this case, “anti-realism” means that the actual facts of the matter emerge from what we can or cannot in principle know about a particle. Because quantum theory starts by making percentage-chance claims that are based upon what we could in principle perceive or “know” about sub-atomic particles, it is not then entitled to make inferences about what is actually the case entirely beyond what we can in principle perceive!

Summarized, the idea is that it is illegitimate to move from epistemological limitations to claims that the metaphysics mirror our perceptual shortcomings.

So, the classical logician says, “Sure, you are motivated to talk in percentage-chance terms. And, yes, we totally understand that you believe that the reality you are describing ‘really is’ as you are describing it. But the theory itself builds in a basic anti-realism that is at best contentious. And, as in vague contexts, there is absolutely nothing wrong with the logic itself! Everything we really want to ‘say’ about a particle can be ‘said’ in bivalent terms. For example, if you want to say that there is a 93.0473% chance that a given particle is at space-time coordinates: w, x, y, z…. Well, make that claim, and that claim is itself either true or false! If you got the percentages off somewhat, then that claim was false. If you got it right, then that claim was true. So, bivalence remains entirely adequate, even in contexts that motivate you toward multi-valent logics.”

The same ideas hold regarding trivalent logics. The classical logician says, “There is no need to build your epistemic uncertainties into the logic itself. In those contexts where you would prefer a null truth-value, well, you can still say everything you wish in bivalent terms. That null value is a mere convenience. It does not denote an actual shortcoming in bivalence!”

So, in this course, we will be classical logicians. I personally side with the vast majority of logicians in believing that bivalence is actually how we do think, and the extensions to bivalence are not necessary to “get it right” in any context. Thus, the truth-tables and deductive methods we will rely upon in this course are indeed “classic” and sufficient for everything we will accomplish.