Is there any value in studying "Symbolic Logic"?


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No. I think it is enlightening, but unnecessary for most production. It is fun for many of us, and I think it begins to be fun and enlightening right off the bat with learning to translate natural language statements into the logical formulas. That gives one a new insight into thought and language.

There is a delightful T-shirt available lately, which says "Another day without using calculus." Everyday we use logic, but for most of us most days, we'll not use symbolic logic.

Should you be studying philosophy at the graduate level, mastery of symbolic logic up through first order predicate calculus (and with identity) is necessary, and I'd say that one ought to get a grip even on the limitative theorems discovered in modern logic so as not to speak the layperson junk that goes around concerning their implications for epistemology.

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No, "layperson junk" in this context alludes to the appeals made to the Gödel Incompleteness Theorems to fake one's way to a confirmation of one's plain old belief in the impotence of reason. Such theorems imply no such thing, and competence in the theorems and the scaffolding education in modern logic required to understand them is needed to seriously expose such misuses of the theorems.

Here is a somewhat earlier theorem, discovered in 1920, the Löwenheim-Skolem Theorem:

If all of a class of logical quantificational schemata come out true together under an interpretation in a nonempty universe, they come out true together under some interpretation in the universe of positive integers.

This theorem is reached in chapter 33 of Quine's text Method's of Logic (4th edition), and to understand what it is saying one first needs to learn the preceding chapters, which will introduce all the terminology, including the specialty meanings for terms also in ordinary usage, such as universe, schemata, and interpretation. The good news is that the positive integers will just be what we already know from grade school.

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I remember Bill Dwyer and others breaking paragraphs down into symbolic logic to prove that it was correct or not in a logical sense. I liked that. Perhaps there is an Artificial Intelligence program that can do it automatically. I can see an immediate application in translation and diplomatic circles. The leaker Edward Snowden is claiming some knowledge that aliens are continually communicating with us. He sounds like he is looking for publicity. But, a symbolic logic translator might be a step towards a *universal translator,* to use if aliens ever do ask, "Hello? Are you there?"

Peter Taylor

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I found it helpful. I had to take semester of symbolic logic for my associate's degree in criminal justice. The instructor was a doctoral candidate at the University of Michigan; and she taught it as a class in symbolic logic. The other instructor (the department chair), taught it as a class in critical thinking, and only introduced symbolic logic at the end of the class.

What does "necessary and sufficient" mean? After Hurricane Katrina, President Bush insisted that "our response was necessary and sufficient." "Necessary and sufficient" means "equal to". Suppose there had been no hurricane at all. Nothing. But the federal government sent the response anyway. Would the consequences have been as destructive as a hurricane? I do not mean that we must always analyze every plain statement for logical flaws. ("How are you?" Fine. "Are you fine, as in not coarse, fine as measure of purity as sterling silver is .925 fine, fine as a penalty to be paid?") But I do insist that if you have some experience with logic, you can see through more fallacies in the so-called "ideas" of other people. (And, of course, seeing your own would be "the giftee some Pow'r gie us".)

English has only one "or" though in computing we have the "Xor" which also exists in other languages such as Latin. Working freelance, I sign a lot of contracts. It matters whether "or" could mean both, rather than just one xor the other. If Mom says that Sue or Nancy will bring you a chocolate cake and they both show up on your doorstep with the treat, you are unlikely to raise a fuss. However - "You can have two weeks' sick leave or two weeks of vacation." That, you need to get clear before you sign.

The negation of (A or B) is (not-A and not-B). I figured it out on my own and wanted to wave my homework at the instructor, but the girl behind me figured it out on her own and waved hers first. It is well known as "De Morgan's Theorem."

Maybe the shortcoming was only mine, but when someone said "If A, then B" I took that to mean that the creation of A would cause B. "If you go over the speed limit (A), you will get a ticket (B)." That is not what it means. It means that if you got a ticket (A), then you went over the speed limit (B). The predicate identifies the cause. That is why symbolic logic is also called predicate calculus. (If A then B) does not imply (if B then A). You can get away with speeding.

You might get something else entirely different out of studying logic.

I see that Wolf jumped in ahead of me here. Let me say that so-called "Boolean Algebra" is more than what we learn in computers. He is right about Boolean algebra being the symbolic logic of computing. But the or xor and nand if then of computering is a very restricted application. (See here.)

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No, "layperson junk" in this context alludes to the appeals made to the Gödel Incompleteness Theorems to fake one's way to a confirmation of one's plain old belief in the impotence of reason. Such theorems imply no such thing ...

Indeed. The analogy I like is that the incompleteness theorem invalidates reason no more than the presence of a singularity invalidates physics.

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I guess I should mention that the 19 valid syllogisms are often stated in symbolic form, for example: All A is B, some B is C, therefore some A is C. Thomas Aquinas' Restatement is an excellent reference and coincidentally an important book that illustrates how Aristotle blundered with "essential" and "accidental" qualities..

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Here is an interesting letter. How much has changed since this was written? Has there been much work into adding *time* to logical syllogisms?

From: Ram Tobolski To: OWL Subject: OWL: Can rationality be formalized? Date: Sun, 06 Oct 2002 00:04:29 +0200

As a response to Robert Campbell's interesting post (10/4) on John Pollock and the philosophy of AI: Can rationality be "formalized"? Can it turn out to be no more (and no less) than performing deductions from axioms, within a formal logic theory?

I may have something to contribute here, since I've recently written a paper that is related to this subject (I am studying for M.A. in philosophy).

First, let me comment on one argument that Robert used. He wrote:
First, there *has* to be more to rationality than following the rules in some system of formal logic (or applied probability theory). I would argue that logic captures in symbolic formulas and context-independent rules *some* of the constraints that we apply when we think rationally. No system of formal logic captures all of them. Indeed, if every constraint that human beings ought to apply in their thinking (no matter how specialized or expert) somehow got codified into a system of formal logic, the thinking *about* those constraints that enabled all of them to be codified, and the constraints by which *it* was guided, would have escaped being codified--and so on.

How strong is this argument, depends on the question, whether one is supposed to be consciously or _unconsciously_ guided by formal logic. If we allow it to be unconscious (and this is perhaps the serious variant, although it may not be Pollock's variant), then the above argument is not conclusive. If our underlying logic is unconscious, then it can guide us in gradual conscious codification, until every bit of thinking becomes, perhaps, codified... In this case, "codification" is not a creation of something new, but an uncovering of something that already existed, below consciousness. It is a "recalling", as in Plato's doctrine that one is "recalling" the Ideas, because they were instilled into one before one was born...

However, there are other arguments against rationality being formalized at its origin. What we have to consider here is, first, semantics, and, second, epistemology.

Semantics: If rationality is formal logic, then it is merely a manipulation of symbols. But then the following question arises: What about the _meanings_ of the symbols, which we manipulate by logic?!

Epistemology: And how does one _know_ what are the meanings of the symbols one is using?!

Formal logic, when taken in the radical sense that we are talking about now (sometimes called formalism), goes hand in hand with _nominalism_ about universals. In formal logic systems, it is almost universally assumed, that the meaning of a predicate-word, a universal-word (e.g. 'red') is a fixed _set_ of individuals (the set of red objects); or a function, which assigns a fixed set of individuals to any possible world (the set of red object is a possible world).

However, if this is so, a question arises, how does each predicate-word gets to be associated with a certain set?

One possibility, is that every rational agent simply _decides_ which set to associate with any word. But what would the decision be based on? This alternative turns out to be subjectivistic.

Another thesis of formalists (from e.g David Hilbert to Donald Davidson), is that the language itself, the sum total of symbols, somehow "fixes" its own meaning. One may believe, or hope, that even though a single word can be associated with any set, when we consider a complete inter-connected theory (which is nothing but a system of words), all the meanings of the words will be simultaneously fixed.

Notice what the presumption is here: Formal logic provides us with rules of transformation, transformation of truth from one sentence to another: If p1 is true, and p2 is true, or p3 is true... then p7 is true. A formal theory apparently fixes all the truth values for all the sentences that can be legally formed within the theory. [Even this turns out to be unachievable, btw, as is implied by Godel's incompleteness theorems.]

All this concerns the _sentences_ in the theory, not the _words_ that are combined to form these sentences. And now, the formalist thesis is that fixing the truth values of the _sentences_ fixes also the meanings of the _words_ in those sentences. [btw, this shows that the formalist thesis is closely associated with, if not equivalent to, the so-called truth-conditional theory of linguistic meaning.]

However, this turns out not to work, also. A methodical work on this problem was initiated by W.V.O. Quine, around the so-called problem of the indeterminacy of translation. Subsequently, Hilary Putnam showed [in his book "Reason, Truth and History" (1981)] that for any formal theory, we can always exchange individuals between the sets in its semantics, such that the truth-values of all the sentences in the theory remain the same! And since it is assumed that the meaning of the predicate-words are (fixed) sets, this conclusively proves exactly what we need: We cannot fix the meanings of (predicate) words by fixing the truth-values of sentences.

There are also more exotic, although philosophically less significant (or so it seems to me) results from mathematical logic, which testify to the limits of formalization: We have Godel's incompleteness theorems (In every theory which includes number theory, there is a proposition which is true but cannot be proved!) and the Skolem-Lowenheim theorem (Every first-order formal theory has different semantic models with different, infinite magnitudes). I am not elaborating on those here.

So the idea that rationality is (already) formalized, falls right here, on the semantic-epistemological considerations: If the meanings of words are not simply _decided_, and if they are not fixed by language itself, then the rational agent has to _know_ how to assign a meaning to each word. Then what does this knowledge consist of? How does the agent get to have this knowledge, about the meanings of words? (And especially about the meanings of predicate-words, universal-words. Whether it be words used by others, or words that one is using by oneself.)

A better approach, and a proper solution is already contained in the objectivist epistemology: the meanings of predicate-words is in that they are representing _concepts_. And concepts are not formal, they are not symbolic structures. Concepts are _identification_ algorithms. They are means to _identify_ objects, to identify properties of objects, to identify relations between objects. [The identification of empirical objects is not deductive, but inductive, and fallible.]

And so, it seems to me that rationality cannot be formalized. And, quite surprisingly (don't you think?), this turns out to be a variation on the problem of universals, and a demonstration of the strength of the objectivist epistemology.

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Xray once wrote on Objectivist Living: Rand was quite committed to logic, but her claim that "It is the use of logic that enables man to determine what is and what is not a fact" disregards that "logical" conclusions can be also drawn from false premises not corresponding to fact. For example, a child drawing conclusions based on the premise that monsters exist, it is a logical act to look under the bed to check if one is lurking there.
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Ayn Rand’s character John Galt speaking on page 989 of “Atlas Shrugged”: The man at the top of the intellectual pyramid contributes the most to all those below him, but gets nothing except his material payment, receiving no intellectual bonus from others to add to the value of his time. The man at the bottom who, left to himself, would starve in his hopeless ineptitude, contributes nothing to those above him, but receives the bonus of all of their brains. Such is the nature of the “competition” between the strong and weak of the intellect. Such is the pattern of “exploitation” for which you have damned the strong.
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Jennifer Burns in ”Goddess of the Market,” responded to the above passage page 173-174: In these passages Rand entirely drops the populism, and egalitarianism that characterized her earlier work, reverting to the language used by earlier defenders of capitalism. Although she did not use explicit biological metaphors, her arguments were like a parody of social Darwinism. “Atlas Shrugged” was an angry departure from the previous emphasis on the competence, natural intelligence, and ability of the common man that marked the Fountainhead.” Why such a dramatic shift in thirteen years? Partly Rand was simply tending back to the natural dynamics of pro-capitalist thought, which emphasized (even celebrated) innate differences in talent. These tendencies were exaggerated in Rand’s work by her absolutist, black-and-white thinking. Her views on the “incompetent” were particularly harsh because she was so quick to divide humanity into world-shaking creators and helpless idiots unable to fend for themselves. This binarism, coupled with her penchant for judgment, gave the book much of its negative tone. Because she meant to demonstrate on both a personal and a social level, the result of faulty ideals, Rand was often merciless, with her characters, depicting their sufferings, and failings with relish. In one scene she describes in careful detail the characteristics of passengers, doomed to perish in a violent railroad crash, making it clear that their deaths are warranted, by their ideological errors. (566-568). Such spleen partially explains the many negative reviews that Rand received. After all, by renouncing charity as a moral obligation she had voluntarily opted out of any tradition of politeness or courtesy. “Atlas Shrugged” demanded to be taken on its own merits, and most book reviewers found little to like.
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Back to me. I have listened to liberal pundits my whole life and concluded that their reasoning starts from their ideal premise then logically rejects all contrary evidence and they do this their entire lives. You can never get them to “Check Their Premises.” Here is an everyday example of the use of logic. Some snipped clips from Walter Williams.

Liberal Reasoning: Idiotic or Dishonest? Walter E. Williams | Sep 23, 2015
Many people argue that liberals, socialists and progressives do not understand basic economics. I am not totally convinced about that. Take the law of demand, for example, one of the fundamental principles of economics. It holds that the lower the cost of something the more people will take or do of it. Conversely, the higher the cost the less people will take or do something. By their actions, liberals fully understand the law of demand. Let's look at some proof.

The Seattle City Council voted unanimously to establish a tax on gun and ammunition sales. Hillary Clinton has called for a 25 percent tax on gun sales. In Chicago, Cook County Board President Toni Preckwinkle proposed "violence taxes" on bullets to discourage criminals from buying guns. Let's ignore the merit of these measures. They do show that gun grabbers acknowledge the law of demand. They want fewer gun sales and thus propose raising the cost of guns.

. . . . In the Ohio Legislature, Rep. Bill Patmon, a Democrat from Cleveland, introduced a bill to make it illegal to manufacture, sell or display toy guns. The ban would apply to any toy gun that a "reasonable person" could confuse with a real one. A $1,000 fine and up to 180 days in jail would be imposed for failure to obey the law. That's more evidence that liberals understand the law of demand. You want less of something? Just raise its cost.

Even San Francisco liberals and environmentalists understand the law of demand. They've proposed a ban that over the next four years would phase out the sale of plastic water bottles that hold 21 ounces or less in public places. Violators could face fines of up to $1,000.

Former U.S. Secretary of Energy Steven Chu once said, "We have to figure out how to boost the price of gasoline to the levels in Europe" in order to make Americans give up their "love affair with the automobile." If gas prices rise high enough, Chu knows that Americans will drive less.

There you have it -- abundant evidence that liberals, socialists and progressives understand the law of demand. But wait a minute. What about raising the cost of hiring workers through increases in the minimum wage?
Aaron Pacitti, Siena College professor of economics, wrote that raising the minimum wage "would reduce income inequality and poverty while boosting growth, without increasing unemployment." The leftist Center for Economic and Policy Research has written a paper whose title tells it all: "Why Does the Minimum Wage Have No Discernible Effect on Employment?" The U.S. Department of Labor has a page on its website titled "Minimum Wage Mythbusters" (, which relays a message from liberal economists: "Increases in the minimum wage have had little or no negative effect on the employment of minimum-wage workers."

What the liberals believe -- and want us to believe -- is that though an increase in the cost of anything will cause people to use less of it, labor is exempt from the law of demand. That's like accepting the idea that the law of gravity influences the falling behavior of everything except nice people. One would have to be a lunatic to believe either proposition.
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