MINDING YOUR Ps OR NOT Qs
by James Lincoln Warren
The first medieval university curriculum consisted of three subjects: Grammar, Rhetoric, and Logic, known together as the Trivium, or “three roads”. (The succeeding curriculum, necessary to complete what was then called a “classical education”, was the Quadrivium, comprising Arithmetic, Geometry, Music, and Astronomy.)
Grammar is the study of the relational structures and elements employed for meaningfully arranging words in sentences. Rhetoric is the art of using language to persuade or influence others, and by extension, those rules observed by speakers and writers to express themselves with eloquence. “Noun” and “verb” are grammatical terms. “Metaphor” and “simile” are rhetorical terms.
Which leaves Logic. Logic, which is derived from logos (the Greek word for “word”), is the branch of philosophy dealing with forms of thought. Usually people mean by it the method whereby one rationally draws conclusions from a set of observations or assumptions, but sometimes they mean something else. Republicans use it to mean thinking like a Republican, and Democrats use it to mean thinking like a Democrat. Certain men use its antonym, illogical, to describe all thought uninfluenced by testosterone; i.e., “Women are illogical.” (Certain women, on the other hand, express a sentiment which seems to me, at least, to be far easier to validate, viz., “Men are morons.”) Mr. Spock seemed to think that it meant thought uninfluenced by emotion, but that’s only because to a Vulcan, emotion means going into a hormone-induced homicidal rage prior to coitus. To me, logic means clear thinking.
Mystery readers love logic. One might say that they even demand it, especially in a good old fashioned puzzle story with a ratiocinative detective who gathers all the suspects together in the drawing room to announce who done it. Alfred Hitchcock’s Mystery Magazine even publishes a monthly logic puzzle, “Unsolved“, by Robert Kesling. (Ellery Queen offers up a crossword puzzle each month instead, but this also challenges one’s faculty of deduction.)
The most common form of logical argument, usually associated with Aristotle, is the syllogism. A syllogism consists of two premises sharing a common term that lead to a necessary conclusion: Omne animal est substantia (all animals are substantial, or have substance), omnis homo est animal (all men are animals), ergo omnis homo est substantia (therefore, all men are substantial).
Adopting the linear nature of the syllogism without its rigor can lead sloppy thinkers to a common logical fallacy, post hoc ergo propter hoc, (“after this, therefore because of this”). For example, a recent study indicated that people who drink nothing but diet sodas are much more likely to be obese than people who also imbibe other kinds of beverage. From this, some folks concluded that diet sodas are a proximate cause of obesity, but this is logically fallacious — in the first place, people might be drinking low or no calorie beverages because they were already on the chunky side, but even more likely, it’s a matter of lifestyle. Who drinks nothing but diet sodas? People who eat nothing but junk food, that’s who. That jumbo triple cheeseburger and bucket o’ fries that come with your diet soda at Burger Thing might have just a little to do with it. Guilt by association is a form of post hoc ergo propter hoc.
Logic really took off in the 19th century, when advancements in logical methodology underwent a revolution so profound that in many ways it has defined the era we live in today. A German philosopher and mathematician, Friedrich Ludwig Gottlob Frege (1848-1925) realized that he could apply mathematical-style notation to logic, thus inventing what was once called symbolic logic, but these days is more commonly known as formal logic. And formal logic is the foundation of the theory of computation, folks, and hence of computers — and civilization has grown so dependent on the computer that it’s right up there with fire, the wheel, electricity, and plumbing.
But let’s leave the starboard wingtip and climb back into the fuselage.
There were two early types of formal logic, propositional calculus (not to be remotely confused with differential or integral calculus) and predicate logic, which is really an amplification of propositional calculus, allowing for variables and declarations. Now there are more, mostly dealing with various topics in mathematical theory. But now we’ve wondered out onto the port wing, so let’s get back to the cabin again.
Propositional calculus is essentially syllogistic logic expressed in symbols, and that’s where we’re going to have us some fun. Using the traditional letters to represent propositions (although Greek letters are common, too), let us call the premise omnis animal est substantia Proposition P, and the premise omnis homo est animal Proposition Q. What we want to prove is the conclusion, ergo omnis homo est animal, which we will label Proposition A.
Now comes the fun part, operators. These are analogous to operating signs in good ol’ arithmetic, like the plus (+) and minus (-) signs.
In formal logic, there are a number of these, but the most common are AND, usually notated with a dot or period (.), which means that both arguments must be true for the conclusion to be true; OR, usually noted with “V”, meaning one of the arguments must be true and the other false for the conclusion to be true1; and NOT, usually written as a tilde (~), a “unary” operator that changes a preposition’s state from true to false or vice versa — e.g., “~P” means “omnia animal non est substantia”, “not all animals are substantial” or “all animals are not substantial”. Anyway, to prove Proposition A, we must apply a “Truth Test” to Propositions P and Q. This can be written as:
P . Q
Pronounced “P AND Q”.
This statement evaluates as true only if both P and Q are true. (Bear with me here. It may seem like egghead B.S., but I really am going somewhere with this (eventually).)
But this can also be written as
P V ~Q
Pronounced “P OR NOT Q.” In this case, if P is true, then NOT Q must be false for the statement to be evaluated as true, because the OR operator allows only one of the prepositions to be true. And if NOT Q is false, then Q must be true — in other words, both P and Q must be true.
This is an example of what logicians call reduction to primitives, i.e., ideas that can be broken down further into composite parts. The smallest parts are the primitives. So, since “OR NOT” is semantically equivalent to “AND”, we can dispense with “AND” altogether, and instead of having three operators (. V ~), we only need two (V ~).
As it turns out, all logical operators can be reduced to a single primitive, called joint denial or NEITHER NOR2. This discovery was made by mathematician Henry M. Sheffer in 1913. The symbol for it, usually called a “Quine dagger” after logician Willard Van Orman Quine who introduced it, is a down-pointing arrow 3, but I’m going to use the exclamation mark as a typographically available substitute. Using joint denial, here’s what “P . Q” looks like:
(P ! P) ! (Q ! Q)
Pronounced “Neither neither P nor P nor neither Q nor Q.” I leave it as an exercise to the Gentle Reader to parse it out and convince himself it’s semantically identical with “P AND Q”. Anyway, as you can plainly see, in simplifying our language by reducing terms to their simplest state, we’ve actually made our expression far more complex and less intuitive.
All of which brings me to a couple points: (1) The clearest expression of something is not necessarily the one using the simplest language, and (2) Thinking clearly ain’t easy.
Which brings me to Warren’s Syllogism: Clear thinking is difficult. Clear writing requires clear thinking. Therefore, clear writing is difficult. Q.E.D.
All right, I’m finished messing with your brain for now. But the next time you get into a high dudgeon because some author writes a compound sentence instead of a series of simple declarations, or uses an unusual word rich in innuendo and association instead of a plain word stripped to its bare denotation, ask yourself if you’re being logical.
- Computer programmers might be offended by this, because in most programming languages, “OR” stands for “inclusive OR”, meaning that if either or both of the prepositions be true, the entire statement is true. The operator they use in the sense of only one of the prepositions being true is “XOR”, or “exclusive OR”. But in standard mathematical logic, OR means the exclusive OR.
- In computer science, this is usually abbreviated to simply NOR.
- Lest ye be disturbed that Quine had something named after him that was really another man’s idea, Sheffer had another operator named after him, the “Sheffer stroke”, an up-pointing arrow or vertical line, representing “NOT AND” a.k.a. “NAND”. Quine points out in his classic textbook Mathematical Logic that Charles Sanders Peirce discovered reduction to joint denial in 1880, but Peirce’s notes on the subject weren’t published until almost twenty years after his death, in 1933.