## Monday, November 26: The Scribbler

**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 true^{1}; 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 NOR^{2}. 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.

__Notes:__

- 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.

As a young girl, my mother would often refer to men as ‘animals’. After reading your article I wondered….must I mind my P.Q’s? But then, there is always Leigh’s OhMiPod……

My favorite line in this post:

“All right, I’m finished messing with your brain for now.”

JLW, thanks for linking to the women of mystery blog on your sidebar. Terrie

Belatedly scooting over to Terrie’s site, I notice that Laura at Women of Mystery published a hands-on review of the Kindle on Friday:

http://www.womenofmystery.net/2007/11/kindle-ing.html

We’re still waiting for a hands-on review of the ohMiBod.

Ps and Qs… cute!

James said ‘fun part’, which probably caused an eye-roll from most readers, but it CAN be fun in a geeky way! (Non-geeks are advised to skip immediately to the end.)

Much of computer theory and design is symbolic with little ‘real world’ visible relationship until the final product. It’s surprising how much 18th and 19th century Trivium shaped the past 50 years. Several obscure twigs of mathematics from past centuries have been discovered to have meaning within computational theory.

Most programmers never get to the point of relying upon theoretical logic, and it’s possible to write ‘applications’ (hi-level programs) without delving into ‘Boolean logic’ and ‘Boolean algebra’, the latter which roughly corresponds to propositional calculus reduced to its essential atoms, 0 and 1, FALSE and TRUE. In fact, we can assign zeros and ones to the aforementioned Ps and Qs and get meaningful answers (at least to a computer).

It’s recognized that AND is multiplicative and OR is additive, and we often use

x,+, (and–for unary NOT) when mapping binary examples. As James mentioned, systems programmers also use XOR, exclusive or, which has fascinating bit-level uses (sheesh, I sound like Spock). Hardware designers use NORs (not-or, just like English) and NANDs (not-and) in their symbolic circuit design.Clarifying the footnote, inclusive OR (rather than exclusive OR) is the far more common use, while exclusive OR (XOR) is relegated to the most elemental and intricate programming levels (plus data validation and encryption).

While another branch of the Trivium, grammar, in the last half century seems to have nose-dived in the English-speaking world, it remains ever-important to computing. Programming languages have grammars, syntax, semantics, and we even work with ‘meta-languages’ (ie, Backus–Naur form (BNF)) to describe formal languages.

As mentioned yesterday, I was a computer super-hotshot who actually wrote and worked on operating systems, telecommunications, and what we call lo-level software… the heart of computers. In college, I proposed that the traditional sentence diagrams should be abandoned in favor of modified binary trees which can be used to map sentences, but that brilliant idea (he says, modestly) never seized the imagination of linguists. They mumbled something about ‘inveterate geekiness’, I believe.

There’s a more practical use for understanding and applying old-school studies to the modern world. If you deconstruct advertisements and political talk-radio commentaries, you’ll find they don’t really say what you THINK they are saying. In fact, with just a casual word or two slipped in, talk show hosts often say the opposite of what you’re intended to believe, but if a listener doesn’t apply those old-fashioned studies of grammar, logic, and debate, he’ll never know it.

_____________________

>Alfred Hitchcock’s Mystery Magazine even publishes a monthly logic puzzle, “Unsolved“, by Robert Kesling.

NB: Dell Magazines also publishes Math Puzzles and Logic Problems:

http://www.dellmagazines.com/order/math.shtml

I intend to deal with both grammar and rhetoric in future columns.

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. ——–

So this means no beer with brie? Actually my P’s and Q’s are illogically lodged in my (un)funny bone.

The common logical fallacy, post hoc ergo propter hoc, (“after this, therefore because of this”) could also apply to politican’s “promises” (depending on the sloppy thinking voter) as well as apply to methods of writing and selling points discussed in this forum of “what sells vs. what is good” (depending on the sloppy instant gratification readers).

Then there is that omibod thingie.

Enjoyed the read. It was quite informative. Thanks.

Thanks for this cool post. As a mystery addict, a professor of logic, and a Frege scholar, I am thrilled.

It is probably a déformation professionnelle, but many logicians tend to be somewhat fussy about details. And so I can’t help but offer a comment or two.

First, even on the exclusive interpretation of OR, P AND Q is not equivalent to P OR ~Q. For let P and Q both be false. Then P AND Q is false, whereas P OR ~Q is true (~Q is true, and P false, so exactly one of P and ~Q is true).

In fact, AND cannot be reduced to just negation and exclusive OR at all. However, on the inclusive interpretation of OR (so that P OR Q means “either P, or Q, or both”), we may express P AND Q using negation and OR, thus: ~(~P OR ~Q). Your point still holds, with one modification: We can dispense with AND in favor of ~ and *inclusive* OR.

(Incidentally, exclusive OR is not the standard interpretation of OR even in mathematical logic.)

Thanks not just for an inspiring post, but also for this great blog project. I read it religiously.

And a tip o’ the hat to you, too, for putting the record straight!

Oh, dear.

Oh dear is right. Whatever happened to the Ps and Qs? This is way beyond me. I’m off to view the OhMiBod site once again. I may just be tempted to part with my hard earned cash. Then on second thought, think of how many ‘real’ books I could purchase for $69. Decisions, decisions…inclusive OR exclusive…what is a girl to do?