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


Litch King said...

Say we saw that baseball you were talking about. Say we thought that it was beautiful. Now, I assume we both recognize that "beauty" is not a property of the baseball in itself. Rather, beauty is something we impose on objects with our minds.

So we have this basic emotional intuition we give the linguistic label "beauty", and then communicate this idea in an English language system, "The baseball is beautiful."

I agree that the TAG form of Logical Realism from the likes of Slick, Bruggencate, and Lisle are complete and utter garbage....

BUT, I want to defend the other Logical Realist and very much Rationalist claim that objective reality, in and of itself, can be said to have "logical properties." That is to say, objective reality is, all other factors aside, at least known to be logical. And follows not just any arbitraily agreed upon rules, but the neccesary first principles of intelligibility like Identity, Non-Contradiction, and Excluded Middle.

Reality is reality, isn't not reality, and isn't both nor neither reality and not reality.

We can know this fact about objective reality a priori, as the Rationalists claim, through an exertion of pure reason.

So I would say you can define arbitrary systems of logic all day long, but there is still a Logical Realism wherein reality itself is claimed to abide by logical rules of "the one true logic", and we cannot simply jettison this logic (classical logic) and substitute it for an arbitrarily defined one because our very thinking conforms to it (they are often called laws of thought). We neccesarily abide by it in one way or another, as does reality as a whole.

Would you agree with this kind of Logical Realism?

AnticitizenX said...

Nope. I completely reject logical realism for the simple reason that what you're saying is just not meaningful. Remember, logic is a language by definition. It's a system for putting words together according to rules. So the moment people start ascribing other properties to logic, then the entire argument is immediately reduced to a fallacy of equivocation.

It's important to realize that whenever you say things like "reality is logical," you are quite literally speaking nonsense. It's like saying "the universe is grammatical." There is no law of noncontradiction "out there" in the sense that it was just waiting to be discovered. It's a human invention built on rules for connecting propositions. Whenever you violate the so-called laws of logic, you're simply putting words together incorrectly. Nothing more.

The reason why math and logic appear to describe the universe so well is because we specifically invented those systems to help us do exactly that. But that doesn't mean the universe itself is mathematical or logical. It just means that the rules of math and logic are extremely helpful at achieving the goal we intended. They provide a clarity of meaning that we require when doing science.

Litch King said...

I agree that if you define logic as strictly a property of language and language only, I am equivocating. But why is something like the Law of Identity's only viable referent a language rule? Can we not define a "logic" (if we call it that) who's signifigance is ontological, the risk of sounding too abstruse here..., that expresses this idea that it is a property of a thing that it is what it is?

This goes back to the baseball. We say the baseball is spatial. We allow that spatiality is a property the ball has objectivley. Could we not also say that it is a property of the ball,- and a neccesary one at that,- that the ball is a ball? Maybe we arn't using "logic" as you have defined it in this instance, but we are in some way applying Identity to a physical object.

So we have an inutition, a rational understanding, that "things are what they are," and we call this the "Law of Identity." And we apply it directly to reality as a whole on an a priori basis.

Is this meaningless?

AnticitizenX said...

To begin, there is no "law of identity" as you describe it in formal logic. The proposition that "a thing is itself" is basically a made-up law that Christians like to throw around, even though there is nothing in classical propositional logic that says this. The closest I could come up with is something like this:

If X is a baseball, then X is a baseball.

This is now a formal tautology in that it becomes true under all possible truth valuations. So when you come to me with something like "A thing is itself," I have to ask where on Earth this supposed law came from. It is not an accepted law of logic.

Secondly, many of your questions are akin to asking "Why should all bachelors be unmarried?" I really don't know how to answer this any way other than "because that's what the word means." If you were to walk into a college-level classroom and study logic, then you will pretty much learn all the things I just talked about above in this essay. There is no metaphysical wanking that goes on, but there will be a bunch of playing around with sentences and truth valuations. Heck, most courses/textbooks will even state outright that logic is a system of "formal language" designed to help us communicate. So the moment you go on about something other than language rules, I don't know how to respond to that. It's like trying to ascribe some kind of deep, metaphysical significance to the rules of commas and capitalization. English has rules, and we use those rules to communicate effectively. Isn't that enough?

Dref Plinth said...

"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!"

Should this be a "brute" fact as defined approximately at 00:41 in the video by ?

Or should this be a consequence of the definition of the idea of an equivalence relation given that reflexivity, symmetry, and transitivity are the properties of an equivalence relation?

I like your online efforts to spread philosophical / logical awareness - if that is a fair description of your endeavor. "On the Failure to Eliminate Hypotheses in a Conceptual Task" by Wason is an interesting paper that I had not seen before. Thank you for bringing that to my attention.

Dref Plinth said...

"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 = 4, 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."

Do you mean to imply the use of addition mod 12 in all cases?
7 + 8 = 15
15 = 1*12+3

AnticitizenX said...

Dref: When I speak of a "rote fact," I'm simply saying that the Peano axioms define what it means for n to be a natural number and that human beings made it so. It's like saying that all bachelors are unmarried men. That's just a fact of the English language because people had to pick some combination of letters and sounds when building the definition. I don't think "brute fact" is the right word, however, because it is something that could potentially change overnight, and because we human beings are most definitely the reason it exists.

AnticitizenX said...

Dref: Yes, I'm using a base-12 clock for the modular arithmetic. I guess that means 7+8 = 3, yes? I can fix that.

Andrew Wells said...

So it’s interesting that you (correctly) define the existential quantifier as expressing ontological commitment “there exists...”. Are you aware that mathematicians, and physicists quantify over mathematical objects like sets? So:

(1) the existential quantify expresses ontological commitment
(2) we quantify over mathematical objects
(C) mathematical objects exist (Platonism/mathematical realism).

(1) is the standard semantics of logic, which you apparently already agree with. (2) is an undeniable fact to anyone who knows a bit of math. So either you accept platonism, or you’re in full fledged conflict with physics and math (hence why the majority of people who study either are platonist).

AnticitizenX said...

"the existential quantify expresses ontological commitment"

The existential quantifier does nothing of the sort. In mathematics, to say "there exists an X" is really a roundabout way of saying "the set of all X is nonempty." Mathematical sets can easily refer to many things that are not objectively real, like "the set of all reindeer driving Santa's sleigh." So just because I can say things like "there exists a Rudolf in the set of all reindeer," that does not literally mean "Rudolf" is an objectively real entity that was discovered rather than invented. Your argument is a fallacy of equivocation.

Andrew Wells said...

Yes, in standard semantics it does.

Propositions that have the same semantic structure should have very similar truth conditions, so consider "there are two towns smaller than Harrow". This statement is true because (a) there exists a town called "Harrow" and (b) there are, minimally two towns smaller.

So when I say "there are at least two integers smaller than 63" ... I'm committing myself to the existence of integers (assuming I stick to standard semantics).

"Mathematical sets can easily refer to many things that are not objectively real"

Yes, it's not whether you can construct a set, but if you quantity over that set of say "reindeer's driving Santa's sleigh" the statement will come out false, because "Rudolf" or "Santa" have no reference.

P.S. sorry, I deleted an earlier version of this comment due to unreadable spelling/grammar mistakes

AnticitizenX said...

That kind of thinking can very quickly lead to bizarre outcomes. For example, suppose I were to tell you that

"Harry Potter is a student at Hogwarts."

In principle, you could say that this proposition is false because there is no actual Harry Potter or Hogwarts. But if you do that, then the negation of the proposition becomes true.

"Harry Potter is NOT a student at Hogwarts."

So now you have a statement about purely fictional things that is suddenly true again.

This is why all propositions require an interpretation (i.e., truth assignment function) in order to make sense in logic. If we interpret the proposition to mean something like,

"In the fictional story about Harry Potter, it is said that there is a place called Hogwarts that Harry attends."

Now we can say that the proposition is true without invoking any existential quantifiers. However, this gets really cumbersome, and the whole point of the existential quantifier is to basically communicate that stuff in the first place without necessarily being so verbose and technical.

AnticitizenX said...

Again, the existential quantifier as absolutely nothing to do with existence in any literal, ontological sense of the word. It just refers to sets, and there is nothing wrong with sets comprised of fictitious entities. When you say "there exists an integer," you are not saying that integers are objectively real things. You're just saying that the set of all integers is nonempty. That's a very, very big distinction.

Let me give you another example:

"For every English sentence, there exists a subject and a verb."

I think we can all agree that English does not exist in any objectively real sense. Subjects and verbs are also purely fictitious entities that only exist because we say they do. Language is an instrument of pure, human invention. It was not discovered. It was invented, and it can change tomorrow for no other reason than because we collectively decide it should. Yet if we follow your reasoning, this sentence has to be false, because subjects and verbs are not real things.

So by your reasoning, this sentence has to be false.

Andrew Wells said...

It's been two years since I sat my last exam on logic, but from memory, the negation of (a) (Ey)(Py ^ Dy) "there exists a Harry Potter which is a student and which attends Hogwarts" is NOT (b) (Ey)~(Py ^ Dy), "there exists a Harry Potter who is not a student at Hogwarts".

For (a) the negation goes before the quantifer, for (b) the proper negation is using De Morgan's law, i.e using the universal quantifer. The negation of (b) is "It is not the case that for all H, if H is a student then he does not attend Hogwarts ..." I'm clearly not committed to the existence of Harry Potter, because the universal quantifier is not ontologically committing.

"Now we can say that the proposition is true without invoking any existential quantifiers."

The statement is literally false, it may be true within the narrative of J.K Rowing's fictional universe, but it not literally true, mathematical propositions are literally true. Unless, you're say a fictionalist who denies the truth of all mathematical propositions (but somethings you've said indicate you're not, so this is very odd as an example).

We use Quine's criteria for ontological commitment in standard logic, and of course that means those things used as bound variables under (existential) quantifiers are something we're ontological committed to.

Andrew Wells said...

Again, that's not the argument. No one denies that sets can be composed of non-existent objects.

What you've said about quantification is flat out false, look up "Quine's criteria of ontological commitment", it is standard practice in logic. It is what almost every logicain (bar a tiny handful) understand by the "existential quantifier".

"I think we can all agree that English does not exist in any objectively real sense."

I agree.

"So by your reasoning, this sentence has to be false"

No it doesn't, because the truth conditions of that statement are not the same as my example, i.e. "two Fs smaller than G". I can express the truth of the statement without committing myself to "verbs".

AnticitizenX said...

I like how you want you try and express things very technically. That's neat to see someone who actually knows this stuff on a formal level. Let's poke at it an see what happens.

For starters, my first proposition wasn't exactly a strict, first-order sentence. It was really more like a statement of an abstract set. For example, we could let H indicate the set of all students at Hogwarts. We could then let h denote Harry Potter. Thus, the sentence "Harry Potter is a student at Hogwarts" could be expressed simply as:

(1) h ∈ H.

I think it is safe to interpret this sentence as TRUE, provided that it is simply true within the fictional world of Harry Potter. We can also interpret its negation to mean "Harry Potter is NOT a student at Hogwarts."

So in this sense, we have yet to even touch first-order logic because we have not invoked at existential quantifiers. However, it seems to me that if we let (1) be true, then it seems natural to me that

(h ∈ H) → ∃h (h ∈ H)

This is now getting super technical, and it might be fun to explore some of the axioms of and rules of inference in this context.

Andrew Wells said...

Yes, thanks I enjoyed this too but I think we've reached an impasse because there's not much of an analogy left there with what the platonist is saying.

"... provided that it is simply true within the fictional world of Harry Potter."

You're not saying something similar about mathematics though, are you? If mathematical propositions are "fictionally true", they are literally false. Something like that might work for pure mathematics but it makes applied math seem at best, mysterious (see for example, the Quine-Putnam indispensability argument).

Your position is also really confusing, it sounded as though you were a deflationary nominalist (someone who just denies that we have commitments to the reference of singular terms and so forth), and when you spoke about language I thought you might go down the route of paraphrase nominalism (reducing math to pure logic, definitions, syntactical arrangements and so on - maybe I read too much into that). This comment sounds a lot more like Mary Leng's sort of soft fictionalism. Not all of these can be true, they're at odds with each other just as much as they're at odds with platonism.

AnticitizenX said...

I do not think it is appropriate to say that mathematics is "fictionally true." I was only using fiction as an example of how it is perfectly appropriate to speak of true propositions, even if the entities referred to by the propositions are not real.

In mathematics, it would be more appropriate to think of it as "conventionally" true. That is to say, everything we know to be true about mathematics is just as meaningful as the things we know to be true about the English language. For example, the Peano axioms of arithmetic are only true because they define the properties of natural numbers. There is nothing forcing us to adopt them one way or another, except for the properties that we feel are meaningful to assert. Thus, for all practical purposes, it is really no different than declaring that all bachelors must be unmarried men. It's a perfectly true sentence, but its truth value is assigned by the virtue of language and meaning. So it's not really a "fiction," but it's not exactly "real" in any objective sense of that word.

Andrew Wells said...

Then the analogy breaks down because the proposition “Harry goes to Hogwarts” is false. It’s “true” in a non-existent fantasy world (which is another way of saying that it is false in the real world).

*”There is nothing forcing us to adopt them one way or another, except for the properties that we feel are meaningful to assert. Thus, for all practical purposes, it is really no different than declaring that all bachelors must be unmarried men”*

There’s a lot in here I don’t agree with. It’s clear from the above that you’re not just denying an object realism about mathematical entities, but even a semantic realism about mathematical propositions.

The English language and the designation of certain combinations of letters to meaning, may be a human construction but the proposition it asserts in “all bachelors are unmarried” is objectively true, not itself a mere construction.

Similarly with mathematical statements if we say, for example that “2 + 2 = 4” is the consequence of axioms that are just a construction, then (contrary to the almost universally held belief that such propositions are necessarily true), we could have just as easily “decided” that it should equal 7 instead. That’s a heavy price, just to avoid platonism.

AnticitizenX said...

It's not really proper to speak of truth in a non-existent fantasy world and then falsehood in the real world. All we can really say is that there is some proposition and that there are different interpretations we assign to it. Since there is no such thing as an "objectively correct" interpretation, then this distinction of yours doesn't really make sense. All we can say is that the proposition is true under some interpretations and false under others.

---the proposition it asserts in “all bachelors are unmarried” is objectively true, not itself a mere construction---

I have no idea what you're trying to say here. Do you deny the fact that the definition of the word "bachelor" is an arbitrary human construction and that it could hypothetically change tomorrow? Or are you just saying that it is "objectively" true that all bachelors are unmarried, simply because it is objectively true that English speaking humans have adopted this convention? (a point I would agree with).

---we could have just as easily “decided” that it should equal 7 instead. That’s a heavy price, just to avoid platonism.---

Except we can very easily "decide" that fact right now, and I have already given you a textbook example of how that works in the real world. Every time you look at a clock, you are exercising a numeric system where 8+6=2. If you ask me, the "heavy price" here would be someone ignoring the plain reality that they've already accepted as part of their daily life. It would be no different than everyone in the world simply deciding to change the definition of "bachelor."

Unknown said...

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Francesca Paolucci said...

Happy Holidays from Italy!