Sunday, March 3, 2019

The Language of Logic

There's a classic episode of the original Star Trek series wherein Spock takes command over a stranded away team after crash landing on a distant planet [1]. As the team desperately works to repair its crippled shuttle, hostile aliens repeatedly hamper their efforts with violent attacks. Frustrated by the danger, some of the crew members demand permission to vaporize the monsters with their phasers, thereby eliminating the threat. Spock, however, refuses, and insists that a nonlethal show of force will be sufficient to scare them away without bloodshed. He immediately orders the crew to implement his plan, but it unfortunately backfires terribly. Rather than run away in terror, the aliens respond with more ferocious attacks than ever, and they even manage to kill another member of the crew. It’s a thrilling bit of drama that culminates in a heated argument between Spock and McCoy:

Spock: “Most illogical reaction. We demonstrated our superior weapons. They should have fled.”

McCoy: “You mean they should have respected us?"

Spock: “Of course."

McCoy: “Mister Spock, respect is a rational process. Did it ever occur to you they might react emotionally? With anger?”

Spock: "Doctor, I am not responsible for their unpredictability."

McCoy: "They were perfectly predictable to anyone with feeling. You might as well admit it, Mister Spock. Your precious logic brought them down on us."

Spock [some time later]: "Strange. Step by step, I have made the correct and logical decisions. And yet two men have died."

Skeptical writer Julia Galef once analyzed this episode for her presentation at Skepticon in 2011 [2], and I think she makes some very profound points about the Hollywood portrayal of logic. Remember that Spock is supposed to serve as a living embodiment of pure, logical reasoning, yet his behavior is clearly nothing of the sort. After all, how logical could your decisions really be when they consistently fail to produce an expected outcome? But rather than take responsibility for the obvious flaws in his own reasoning, the guy practically blames the world itself for not doing what he wanted.

Now in all fairness, Hollywood fiction writers are not exactly experts in logic, nor do they have much incentive to fill that role. Still, that does not excuse such a dismal portrayal of logical reasoning. Very few people have the luxury of studying this stuff in any formal capacity, which means the rest of us have little choice but to fill that void with whatever scattered fragments we can find in our popular culture. The natural result is thus a widespread confusion over what exactly logic is, how logic works, and how competent we really are at applying logic to our daily lives.

To illustrate, if we take the Vulcan philosopy of logic at face value, then a logical agent is apparently someone who just suppresses their feelings. Thus, to be logical is to simply be dispassionate, unimpulsive, and unintuitive. Any decisions based on such a mindset are good and correct, by definition, no matter the consequences. But if that’s all that logic is, then why would anyone want to adopt it? The central message seems to be that too much logic does nothing but turn us into unfeeling robots that get people killed!

It’s important to understand that logic is an essential building block to our modern lifestyle, and we only hurt ourselves by misrepresenting its inner workings to the public. Logic is not just a mere suppression of emotion, but a collection of mental tools designed to help us understand the world. Life is filled with serious problems that affect all of us on a global scale, and we cannot expect to solve them through brute intuition alone. It takes hard work to analyze this stuff and formulate solutions, yet the very tools we need for that process are being needlessly muddled and demonized by our media.

But what is logic, really? This is not an easy question to answer, and even respectable authorities are sometimes hesitant to give a truly definitive statement [3]. Some authors describe logic as the study of correct reasoning [4,5], or the study of valid inference [6]. Others describe logic as thinking about thinking [7], or maybe the science of reasoning and arguments [8,9]. These are all perfectly valid descriptions, but they also tend to lack a certain philosophical clarity. It’s a lot like trying to define what a sport is, in that plenty of vague definitions exist, but any hard answer you give will inevitably make certain groups of people very angry. Nevertheless, just because a definition can never be absolutely 100% perfect, that does not mean we should completely refrain from trying. There’s a whole world of needless confusion out there, and it only takes a few simple thought experiments to provide real insight.

To begin, suppose I were to pick up a baseball and place it in your hands. What exactly would you experience? Naturally, you can see it, touch it, taste it, and smell it. It has mass, volume, texture, and a definite position in space and time. Clearly, it’s the most tangible manifestation of a material object that there ever was.

But suppose I were to ask you to hold a game of baseball in your hands. Now what do you experience? What exactly does that even mean? Does the game occupy a particular position in space? Can you touch it? Weigh it? Measure its volume?

Of course not. But why? What’s the difference?

Obviously, the difference is that a game of baseball is not a tangible object. It’s something people do. When you observe a game of baseball, you’re not exactly watching a “game,” per se. Rather, a far more accurate description is that you are watching people as they play baseball. That is to say, they are engaging in a process defined by rules. As long as they are collectively choosing to follow the rules of baseball, then we can say they are playing baseball. And when they choose not to follow the rules, they are simply not playing baseball; they are doing something else.

By analogy, logic operates under a very similar principle. It is not a singular entity unto itself, nor does it occupy any particular location in the universe. It is, however, a process that people engage in. It’s something you do. When people choose to follow the rules of logic, then we simply say that they are being logical. And when people fail to follow the rules of logic, then they are not being logical. Unlike baseball, however, which defines the rules for an athletic activity, logic is like a set of rules built into human language. It’s a way of expressing ourselves rigorously so that ideas can be clearly communicated and then formally analyzed.

This is an important point to emphasize, because there is a huge community of hack philosophers out there who habitually fail to understand such distinctions. It’s especially common among religious apologists wherein a lack of spatial extension is immediately equated with literal transcendence beyond the limits of our material universe [10, 11]. One classic manifestation of this confusion is the famous Transcendental Argument for the Existence of God [12], which actually tries to derive God’s existence from the very laws of logic themselves. It’s all loosely based on a na├»ve viewpoint called Logical Realism, wherein logic is treated as a singular force unto itself, existing objectively and independently of any human influence---almost like an ethereal energy field that surrounds us, penetrates us, and binds the very fabric of space and time.

The reality, of course, is that logic is fundamentally a human invention. Just as English, Spanish, and Japanese are all linguistic conventions created by humans, so too is logic just another similar kind of convention. Many systems of logic are even literally described as formal languages, which is in direct contrast with informal, or natural languages.

To see why this distinction might be important, suppose you were to hear someone utter the following natural-language sentence:

I saw the man on the hill with the telescope.

This is a perfectly well-constructed English sentence, but you may have noticed that it also exhibits a peculiar quality. Namely, the meaning of this sentence is unclear. Do I have the telescope? Does the man have it? Or does no one have it, and the telescope is just sitting on the hill? There is no objectively correct answer to this question without some sort of external context to back it up.

This is a well-known property of natural language called ambiguity---the quality of allowing multiple interpretations for a given sentence. It isn’t necessarily a bad thing, mind you, and there are plenty of situations where it might even exist intentionally. For example, a pun is a specific style of joke that deliberately utilizes ambiguity for comic effect. Poets and songwriters will often deliberately exploit ambiguity to add multiple layers of meaning to their writing [13]. Hucksters and snake-oil salesmen may even use ambiguity to make extravagant claims without necessarily promising anything concrete.

That’s all fine and dandy for some, but what happens when ambiguity leads to costly misunderstandings? For example, maybe I want to borrow money from a bank, or perhaps launch rockets into space. These are situations that demand as little ambiguity as possible and so require the use of much stricter language. A formal language can therefore be thought of as any structured set of rules that attempt to mitigate ambiguity. Scientists, engineers, computer programmers, lawyers, and mathematicians are all particularly fond of formal languages because the very nature of these professions are all built on precise communication.

To demonstrate, let’s use formal language to clarify the meaning of our natural-language sentence:

I saw the man, and the man was on the hill, and the hill had a telescope.

Notice how this sentence is much clearer than its predecessor, thanks in no small part to our use of the word AND. It’s a textbook example of a little tool called the logical connective, in that it literally connects propositions together to form more complex expressions. There’s a whole bunch of them you’re probably familiar with, such as NOT, OR, XOR, IF-THEN, etc, and they all fall under the scope of classical propositional logic.

This is by no means a unique system, either, and there are all kinds of interesting sentences we could construct though alternative logical frameworks. For example, suppose I were to tell you that

For every car on the highway, there exists a driver.

This charming little sentence was brought to you by First-Order Logic, which tells us how to use tools like the universal quantifier (for every) and the existential quantifier (there exists).

Another fun system you may have heard of is called Modal Logic, and it basically gives meaning to words like possible, necessary, and actual. These words are called modal operators, and they allow us to construct such happy sentences as

It is possible to paint a car red, but it is necessary to put wheels on it.

That’s all well and good so far, but there’s much more to logic than a bunch of wordy tools for constructing fancy sentences. Often times, we need to analyze the interplay between ideas, which makes it awfully nice to define some formal way of expressing those relationships. That’s why no system of logic is ever truly complete without some corresponding deductive system to go with it. In its simplest form, a deductive system is very similar to the idea of grammar that we typically associate with natural languages. Only rather than govern the flow of words in a single sentence, a deductive system governs the flow of sentences within an argument.

To see how this works in practice, let’s borrow a page from the classic Mel Brooks' film, Robin Hood: Men in Tights, by considering the following English sentences [14]:
  1. The king’s illegal forest to pig wild kill in it a is.
  2. It is illegal to kill a wild pig in the king’s forest.
Notice that both of these sentences contain the exact same collection of words, but in different arrangements. The first arrangement is generally considered “bad” in the sense that the words failed to follow the proper rules of English grammar. As a result, your brain was most likely unable to derive any coherent meaning from it. In contrast, the second arrangement is generally considered “good” because it correctly followed the rules of grammar. That’s why your brain was able to make sense out of it in accordance with established conventions.

By analogy, an argument behaves very much the same way. Simply begin with a collection of sentences, called premises, and then apply some rule of inference to see whether or not a conclusion supposedly follows. To demonstrate, consider the basic structure of a classic syllogism:
  1. All men are mortal.
  2. Socrates is mortal.
  3. Therefore, Socrates is a man.
Now compare that against the following:
  1. All men are mortal.
  2. Socrates is a man. 
  3. Therefore, Socrates is mortal.
Once again, we have the exact same scenario as before in that both arguments contain identical words, but with different arrangements. Just as grammar dictated the proper flow of words in a sentence, we can clearly see that logic dictates the proper flow of sentences in an argument. The first argument is thus said to be invalid for the simple reason that it failed to follow the rules of a formal syllogism. Likewise, the second argument is said to be valid because it does follow the rules.

Bear in mind now that there is no universally correct way to stick words together in a natural language sentence. Languages around the world happily mix and match the flow of nouns, verbs, adjectives, conjugations, and the like, and no one complains about which arrangements are objectively “real.” The only thing that matters is for us to agree on a given convention so that meaningful communication can take place. Anyone who refuses to follow the agreed-upon rules for English grammar will thus find themselves unable to talk effectively with other English-speaking humans.

By extension, the exact same principle applies to logic. There is no universally correct way to stick sentences together in an argument, but there are rules we have agreed upon for the sake of communication. The very sentence All men are mortal, is really just a declaration of a simple rule: You show me an example of a thing that is a man, and I shall henceforth agree to label that thing as a mortal. Why exactly should anyone feel compelled to do that? Because that’s just what it means to say that all men are mortal! So when you finally do come to me with the proposition that Socrates is a man, then all we have to do is follow the rule by declaring Socrates to be a mortal as well. It has nothing to do with some objective state of external affairs, but a convention of language and understanding. In principle, I could even violate that convention outright by refusing to accept the mortality of Socrates, but all that would result is a bunch of needless confusion and frustration. It would like saying “hey guys, let’s play some hockey” before throwing a football at the goalie and then shouting “checkmate” at the referee. It’s not playing by the rules.

This idea is important, because it directly conflicts with a classical philosophical principle known as rationalism---the idea that pure, deductive logic is the ultimate source of all human knowledge. According to many popular schools of thought, such a doctrine would actually have you believe that the deepest mysteries of life, the universe, and everything, can all be perfectly well-understood by sitting in an armchair and thinking really hard about them. Descartes, Spinoza, and Leibniz were all particularly famous for holding this sort of view, and we can easily spot their influence in more modern philosophy as well. For example, the Ontological Argument for the Existence of God is a classic manifestation wherein the very idea of God Himself can supposedly be used to deduce His own existence. It's a textbook case of blatant philosophical question begging because pure logic, in and of itself, will never tell you anything about objective reality. At best, it can only tell you whether or not your attempts to describe reality have been formulated correctly.

To demonstrate, suppose I were to fill an entire argument with complete, nonsensical gibberish like so:
  1. All flurbles are snuffins.
  2. Zarky is a flurble.
  3. Therefore, Zarky is a snuffin.
Notice that we again have a perfectly valid argument in the simple sense that it merely follows the correct rules of logic. Never mind the fact that flurble, snuffin, and Zarky have no accepted meaning within the English language. The conclusion follows logically from the premises in accordance with the syllogistic convention. You show me an example of a thing that is a flurble, and I will agree to categorize such a thing as a snuffin.

Clearly, something very important still seems to be missing from our logical framework. After all, why should I, or anyone else for that matter, accept the proposition that all flurbles are snuffins? By who’s authority should anyone feel bound by this declaration? Does the dictionary contain some entry that categorizes them accordingly? Is there a children’s show where the characters follow this rule? Maybe there’s an obscure corner of Madagascar where scientists have experimentally uncovered this phenomenon? Or what if some old lady next door to me just happened to utter that little fact the other day, and she’s never been wrong before?

This is why you generally can’t do logic without some formal system of semantics to go with it. After all, we can agree all day on the basic structure of a given deductive system, but it won’t do much good without some authority by which to establish premises in the first place. To that end, it generally helps to associate our propositions with some kind of indicator that officially denotes their authoritative “correctness.” In logic, this is known as a truth-value, and is typically expressed through a binary set containing the elements True and False {T,F}. Thus, to say that a proposition is “true” is to basically say that you accept it as a premise, and so you agree to abide by the formal conventions of some deductive system. Propositions that are “false” would then naturally fail in that regard.

That’s pretty intuitive so far, but it’s important to always keep in mind that we don't have to adopt a binary set of truth values. For example, some systems of logic actually use three truth values instead of two, and are thus referred to as tri-state logics {T,F,U}. Another well-known system is called Fuzzy logic, and it utilizes an entire spectrum of truth-values by mapping them across all real numbers between 0 and 1. In principle, you could even walk to a chalkboard right now and invent your own completely original logic that uses 17 truth values on alternate Thursdays. Again, there is no objectively correct system to use, other than whatever collection of rules we happen to agree upon for the sake of communication. And since the binary system just so happens to be simple, familiar, and functional, it almost always ends up being the de facto presumption under most situations.

Once we’ve finally agreed on an official system of truth values, the next step is to formally establish which propositions are true and which ones are false. In logic, this is known as a truth assignment function, though many references will also call this an interpretation. Thus, to interpret a logical proposition is to assign it a truth value accordingly.

To demonstrate, consider again the simple proposition that all men are mortal. Is that true or is that false? One interpretation could be that every human being we’ve ever encountered has been mortal so far, and so we might as well just take it for granted that all future humans will be mortal as well. Alternatively, you could say that the English dictionary specifically defines “mortality” as an inherent property of human beings, thus making it true by definition. For that matter, maybe you think the old lady next door is the ultimate authority on all things mortal, and so if she says it, then it must be true.

These are all perfectly valid interpretations in the simple sense that they tell us how to assign truth values to a given proposition. So if you happen to abide by one of these interpretations, then great. We can finally build arguments on premises that are officially true. If, however, you reject these interpretations entirely, then that’s great too. All it means is that you have arbitrarily chosen to assign truth to propositions in accordance with some other set of rules.

Remember that in the formal context of propositional logic, truth is just a label that we assign to propositions. That means propositions can either be true or they can be false, but there is no such thing as raw “essence of truth” unto itself. So whenever you come to me with a simple proposition like all men are mortal, then sooner or later that proposition must be interpreted if we ever expect to do any logic on it.

At least one common method of interpretation is to simply assert a small handful of propositions outright and then see what happens. Propositions like this are called axioms, and they serve as very powerful building blocks for many formal languages. For example, according to the language of natural numbers, it is simply a rote fact of life that, for any natural number n, n=n. Why exactly should that be the case? Because we say so, that’s why! It’s just one of the things we demand to be true whenever we talk about natural numbers. It's no different from demanding that all bachelors be unmarried men. All languages are built on rules, and there is nothing wrong with declaring those rules as a foundational property of the language itself.

Once the axioms of a language have been officially established, the next step is to begin deriving new propositions in accordance with some deductive system. Any new true propositions generated in this fashion are called theorems, and they represent the heart and soul of virtually all mathematical inquiry. That's why we say mathematics is an invention and not a discovery. Everything you were ever taught about the nature numbers, sets, functions, etc, all began as little more than a collection of arbitrary axioms being operated upon by logical rules of inference.

If that all sounds a bit circular to you, then just remember that axioms technically have nothing to say about objective reality. Rather, they simply define the foundational rules for a particular language. If that language happens to work well at describing practical scenarios with functional precision, then all the better. But it’s important to always keep in mind that there is nothing physically forcing us to adopt a particular axiomatic system. In principle, we could easily mix and match the rules all day, and there are plenty of situations where it might even be beneficial to do so.

For example, take the classic Peano axiom that there is no natural number n such that n+1=0. Where exactly is it etched in stone that we must absolutely adhere to this axiom from now until eternity? Why not just discard this rule entirely and then replace it with something whacky like 12+1=1? Now we suddenly have a perfectly self-consistent system where all sorts of funny things can happen, like 7+8 = 3, and 10+11 = 9. It’s nothing mysterious or unfathomable. It’s just modular arithmetic, and people around the world use it every day as a practical system for telling time.

This has all been a tiny, oversimplified sampling of the rich tapestry that exists within the world of modern logic, but hopefully we can begin to recognize the essential properties that make logic distinct. When all is said and done, a logic is little more than a formal language coupled with a deductive system and semantic interpretations. Granted, this is not necessarily a perfect definition, and there are probably plenty of experts who would take issue with the finer details. That’s fine, but we can clearly see that logic is far more than a mere absence of emotional impulsivity. It’s not an ethereal force that governs the universe, either, but a linguistic convention built on rules---rules that are designed help express ideas rigorously and then analyze the interplay between them.

So the next time you find yourself stranded on a distant planet surrounded by hostile alien monsters, just remember that it’s okay to feel a little bit emotional. But no matter what feelings may be aroused in any given moment, you’re eventually going to have to start making decisions, and presumably you’d like those decisions to improve the situation. Often times, we may not even have the luxury of careful deliberation, but must instead act upon brute, intuitive impulse. However, on the rare occasions when we do have time to think about a problem in detail, then it usually helps to have some official system in place for evaluating information, formulating a plan of action, and then coordinating that plan among your peers. So step back, take a breath, and use your logic!  

  1. "The Galileo Seven," Star Trek, Season 1, Episode 16
  2. J. Galef, "The Straw Vulcan," Oral Presentation at Skepticon 4 (2011) [link
  3. C. DeLancey, A Concise Introduction to Logic, Open SUNY Textbooks (2017)
  4. I. M. Copi, C. Cohen, and K. McMahon, Introduction to Logic, 14th ed, Pearson Education Limited (2014)
  5.  P. Suppes, Introduction to Logic, Van Nostrand Reinhold (1957)
  6. Columbia University, The Columbia Encyclopedia,  8th ed, Columbia University Press (2018)
  7. S. Guttenplan, The Languages of Logic: An Introduction to Formal Logic, 2 ed, Wiley-Blackwell (1997)
  8. P. Hurley and L. Watson, A Concise Introduction to Logic, 13th ed, Cengage Learning (2017) 
  9. L. T. F. Gamut, Logic, Language, and Meaning, Volume 1: Introduction to Logic, University of Chicago Press (1991) 
  10. W. L. Craig, "Do the laws of logic provide evidence for God?" (2016) [link
  11. J. Warner, "Is God real? Evidence from the laws of logic," (2019) [link]
  12. M. Slick, "The Transcendental Argument for the Existence of God" (2008) [link
  13. "The machinations of ambiguity are among the very roots of poetry." - William Empson
  14. Video clip link