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Logic applies specifically to the method, and AR discusses concepts of method at least briefly in my copy of ITOE. They don't have to refer to things in reality, they are operations or transforms that, if you plug in any "thing", will yield a predictable result.
But the question is what exactly are those methods. The notion of "non-contradictory identification" implies that some aspect of existence is being identified. A valid logical method cannot imply any contradiction, so if a method implies that X exists but in reality X does not exist -- or conversely -- then the method is invalid. A method which allows the introduction of arbitrary or actually false propositions into a logical derivation is fundamentally invalid, and such a method contradicts the nature of man's consciousness -- we don't arrive at knowledge from arbitrary or false statements, we only arrive at knowledge from true statements. So a method of consciousness which failed to distinguish "truth" from "falsity" would not be valid, since the purpose of the method is in fact to make exactly that distinction. A valid logical method must therefore prohibit the introduction of false and arbitrary propositions, even if one can blindly apply the syntax of formal deduction to false statements.
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Now, correct me if I'm wrong, but I was taught in college that a logical statement is one in which you cannot put in true premises and get a false conclusion. You can put in false premises and get a true answer, if only by accident, but if you put true premises in the answer is necessarily true. It's like a computer: garbage in, garbage out.

I'm not sure how this squares with the idea of propositions implying contradictions. I mean, a syllogism isn't a proposition. Sure, you can use a logical process to get a proposition that contradicts reality, but this doesn't necessarily mean that the logic was bad, it means that one of your initial inputs was wrong.

Wouldn't the term for a proposition that squares with reality be "true", not "logical"? Logical indicates the process by which you arrived at it, not necessarily whether it is true, i.e. it doesn't tell you whether it contradicts with reality or not.

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Wouldn't the term for a proposition that squares with reality be "true", not "logical"? Logical indicates the process by which you arrived at it, not necessarily whether it is true, i.e. it doesn't tell you whether it contradicts with reality or not.

In the metalogic of Objectivism, the only way to arrive at truth (i.e. correspondence with reality) is to remain true to reality in a logical manner (i.e. non-contradictory to reality). There is no separation of truth and logic, because logic is the means by which one obtains the truth. It is true that "logic" (as a purely syntactical process) is like a computer in the sense of garbage in / garbage out, as I illustrated with my earlier pseudo-syllogisms. But, aside from the perceptually self-evident, all truth must be arrived at via a logical mental process -- i.e. the process must be "at one" with reality every step of the way or one gets GIGO.

However, this does not mean that even using good clear reality based logic that one is always guaranteed to arrive at the truth, because man is not omniscient. There may be facts involved that he is unaware of that, the ignorance of which, will lead him to a false conclusion even when he is being logical with regard to the facts that he does know. This is a sign that one ought to check one's premises, rather than saying, "I must be right because I was being logical" even when the conclusion flies in the face of reality.

In short, if reality is not the standard, then truth is out the window.

Regarding imaginary numbers, the square root of negative numbers (i), while these do not represent any real quantity in reality, it became necessary to find a way to deal with such results because some equations, especially in electronics, lead, at least in part of their solutions, with the necessity of taking the square root of negative numbers.

Personally, I think this should have given a heads up that something was not right with the equations; that perhaps the scientists had overlooked something when they came up with certain equations dealing with (if I remember correctly) electromagnetic inductance in coils of wire.

However, they didn't do that. Instead they found a way of solving those equations by keeping the square root of -1 (i) in the equations in such a way as that when they were finished going through the calculations a real measurement could be found. Without doing this, electronics would have never advanced the way it did.

As to what they left out, well...that would take a whole revision of the equations for electromagnetism and perhaps even gravity (where one gets bent "space-time," which I think is similar to i, i.e. it works as a method to get a right answer, but doesn't represent a real quantity).

In other words, it corresponds to something, possibly a mistake, but it becomes correctable if the imaginary inputs are carried through the entire process.

[edited to clarify a paragraph]

Edited by Thomas M. Miovas Jr.
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Now, correct me if I'm wrong, but I was taught in college that a logical statement is one in which you cannot put in true premises and get a false conclusion.
No doubt that's what they said, but you may have also been instructed that a man has a right to medicine, food, transportation and housing. Don't you know not to trust college teachers? Well, generally, at any rate.

I think it's important to distinguish between the logician's formal objects "T" and "F", and actual truth and the related adjectives "true" and "false". Logic, in the Objectivist view, is non-contradictory identification (implying "of something", namely "facts of reality"), and truth is the product of a consciousness grasping a fact (falsity is then grasping that something is not a fact). Thus the subject matter of logic is "reality", and the evaluations that logic provides are about how a statement does (or does not) describe reality. This is totally at odds with the reality-detached nature of pre-contemporary formal logic.

The contemporary approach, as taught in probably all classes in formal logic, explicitly denies the connection with reality. For the formal logician, propositions aren't statements about reality, and truth has nothing to do with a consciousness or reality, it is purely a syntactic symbol-transformation method. Hence I insist, at the very minimum, in distinguishing between the arbitrary products of formal logic which might be "T" and "F" (and can be any number of things given ternary and higher logics). For example, the form "(P->Q)&(P)&(Q)" is a formal tautology, no matter what the content of "P" or "Q" is.

You can put in false premises and get a true answer, if only by accident, but if you put true premises in the answer is necessarily true.
That's a definitional presumption of the formalist approach, but there's no proof that it is true. The way they get around this is by stipulating certain strings of symbols as "necessarily T" (Kleene's axioms, or else a T-table). The set of logical operators is arbitrary, so you can define the operator "*" as follows:

P Q P*Q

T T T

T F T

F T F

F F T

The whole idea of "necessarily true" in a logic which is detached from reality is something totally different from what "necessarily" does in fact mean. I find many statements by so-called logicians to be very dishonest, when on the one hand they talk about real-world notions like "necessary", "must" and they use symbols like the horseshoe but don't give it the name "horseshoe" when they utter their formulae, using instead a different concept like "implies". That is dishonest because on the one hand they are denying the contingent, reality-based nature of logic but at the same time they are exploiting the real basis of logic to get people to think they are doing something non-arbitrary.

I mean, a syllogism isn't a proposition.
Formally it is: you just and the terms together and get one (long) proposition.
Wouldn't the term for a proposition that squares with reality be "true", not "logical"?
That's the "valid" vs "sound" distinction. But what constitutes a "valid" proposition depends on the rules of the logic, and a system (such as an Objectivism-based logic) which puts restrictions on what statements can be introduced in a derivation would not make such a distinction, in that "unsound" deductions are logically invalid.

I might add that the human mind does not work in terms of the formal connectives that formal logicians posit, and the symbols of formal logicians are not even very good approximations of actual cognition. This is quite clear from the fact that logicians had to create a horrifying mythical chimera, the "statement", which isn't a sentence and isn't a formula. I have never seen an actual "statement", nor have they -- the program of investigating the cognitive / logical proper properties of sentences is in its infancy. For example, the formal rules for universal quantifiers are different from the rules for "all", the symbol "horseshoe" is used to express a considerable range of cognitive relationships (the worst being the stodgey but classic old way of expressing domain restriction on quantifiers), etc. So getting back to the question of what logic is, symbolic logic is totally divorced from cognition, which is why it holds little interest

for us.

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I’d point out that the question cannot be formulated, and an answer cannot be given, without an assumption of some logic—some rules by which we go from commonly accepted, distinguishing truths, to new truths. The rules by which we move, and the truths we accept, may be controversial; but conversation degrades to meaningless chaos without some rules, something that distinguishes truth from falsehood. Look at this very response. It is appealing to something logical.

Any such meaningful system, then, defines what it is to be true in its system—in effect, reality isn’t given the option of failing to correspond. For classical sentential logic, the kernel is: Take any two sentences, call them P and Q. Consider ‘not-P’ to preclude P; and ‘P or Q’ precludes the simultaneous truth of not-P and not-Q. From this, given “Not-P and P or Q,” one can derive Q. Why is it that, using these rules, we can go from not-P and P-or-Q to not-Q? Because that’s what the rules tell us we can do! To any question, “Why can we do X in the system?” the answer is only, “It is written into the system that we can do X.”

Likewise should be the explanation of mathematics. It may be a different question to ask something like, “Why is space non-Euclidian?” or “How can space be non-Euclidean?” or “Why does gravity exert the force that it seems to?” And these questions may have some more fundamental explanation. But what one cannot hardly make heads or tails of is, “Why is it that, assuming Euclidean geometry, Euclidean geometry is true?” Never has any logical system denied the following inference: “P. Therefore, P.” Nobody cares to. Nobody even asks why we don’t challenge this. And yet, for the same reason that we accept this inference universally, we must also accept, “Provided Euclidean geometry says X, Euclidean geometry must say X.” And provided that the geometry takes some object (a point, line, closed figure, area, or such) to represent some object in reality, then what the geometry says about the object cannot fail to correlate to the object in reality. Reality isn’t given the option of defying.

Take a physical thing which we correlate to a right triangle in some way. (Say, it is a wall.) We measure to sides which correlate to the opposite and adjacent, and find the hypotenuse as defined in the mathematical system, and compare it to the hypotenuse of the physical thing. At no step has reality been given a chance to defy the result that the mathematical object correlates to the physical object. It is no objection that space-time is curved. That just means that we measured three points assuming Euclidean geometry, and the mathematical result about the hypotenuse will correspond to something (some side, though not necessarily a side of the wall, but a side drawn through the space that correlates to the other sides of the wall in the way that the mathematical objects which we call sides correlate to the mathematical hypotenuse).

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I’d point out that the question cannot be formulated, and an answer cannot be given, without an assumption of some logic—some rules by which we go from commonly accepted, distinguishing truths, to new truths. The rules by which we move, and the truths we accept, may be controversial; but conversation degrades to meaningless chaos without some rules, something that distinguishes truth from falsehood. Look at this very response. It is appealing to something logical.

I'm not sure what you are getting at here and in the rest of your post. Logic is derived from the observation of existence using the human ability of abstraction. It is not as if we have some rules that are imposed onto our thinking or imposed upon reality that makes it comprehensive to us. There are no such "rules" without observation that leads to the fact that things are what they are and are not what they are not.

I think part of the problem is that some people tend to think of logic as a set of rules, instead of realizing that logic is based on observation and is a conceptualization of observation. That's why, in a truly logical statement it doesn't contradict reality, provided all the facts where taken into account. One actually cannot do logic without some facts behind it -- one cannot operate one's mind on nothing (though rationalists try this all of the time).

So, it is not as if the rules of Euclidean geometry exist apart from man's mind's grasp of the relationship between sides and angles of real triangles; and it works because Euclidean geometry is based upon the facts of real triangles. It is not as if Euclid just came up with some rules and they happen to apply to geometry -- no, he made observations regarding right triangles and formulated the relationship of their parts (sides and angles) to the whole.

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So, if we say math is a language, and language is used to describe concepts, then math is just another language describing processes and concepts seen in nature? For example "two apples" is the phrase applied to a pair of round red fruits; whereas as 1+1=2 could be the mathematical sentence used to describe a process in nature or a pair of apples.

And then the reason math is correct when describing nature is because it's describing existence.

And then things like calculus, irrational numbers and such could be different dialects?

Or is that too simple?

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And then things like calculus, irrational numbers and such could be different dialects?

No, I wouldn't say that mathematics is a different language, but rather that mathematics is another symbolic form of the same language. For example, 1 + 1 = 2, is just another way of stating, "If you have one thing and add another thing to it then you have two things". Mathematics is just a more condensed way of saying what can be said in English (or some other native tongue). For the more complicated mathematics, it is just even more condensed, but can be stated in English. Some work is being done as to the exact nature of the relationship between mathematics and language from the Objectivist epistemological understanding, but I think there is a ways to go before it gets to things like calculus. One of the barriers of doing this is there not being an explicit relationship between concepts in a sentence, aside from grammar. In other words, we know that abstractions can be formed into concepts via measurement omission, but what is going on at that level when we begin to modify concepts -- i.e. the brown pony is kicking. Once that barrier is broken, then mathematics written as an English phrase will fall into place, since mathematics is really just a more terse statement than can be said in English.

But, it all works -- mathematics and logic and language -- because it is all derived from observations of reality. As an abstraction, it is retained in some symbolic form, and mathematics is just another symbolic form -- however it is symbolic of mental processes grasping existence via human means.

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So, if we say math is a language
Math is not a language.
and language is used to describe concepts
Language is not used to describe concepts. (It can be so use, but is not typically or necessarily used for that purpose).

Math is a system of methods for measuring. It is a science which has progressed to the point that the specific entities being measured do not matter, and it is just pure method applicable to reality and fantasy alike.

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For the formal logician, propositions aren't statements about reality, and truth has nothing to do with a consciousness or reality, it is purely a syntactic symbol-transformation method.
I don't know which particular writers on the subject of formal logic you have in mind. I've never encountered such a view in my own (admittedly limited) readings of authoritative authors on the subject. Though, of course, there are amateurs in the subject who post endorsements of the extreme view you just mentioned. That crude view is not a philosophy of formalism I have seen endorsed by better informed writers; while, quite likely formalism is not even the most prevelent view of modern logicians who deal with formal logic, and certainly not the only one.

The form "(P->Q)&(P)&(Q)" is a formal tautology, no matter what the content of "P" or "Q" is.
No, "(P->Q)&(P)&(Q)" is not regarded as a tautology. Maybe you have in mind "((P->Q)&P)->Q)"?

That's a definitional presumption of the formalist approach, but there's no proof that it is true.
That the defintion of 'tautology' is stipulative in that regard is correct. But given that definition, it is proven that "((P->Q)&P)->Q)" is a tautology.

The way they get around this is by stipulating certain strings of symbols as "necessarily T" (Kleene's axioms, or else a T-table).
That "((P->Q)&P)->Q)" is a tautology as confirmed by a truth table doesn't require any stipulation about necessity.

The set of logical operators is arbitrary, so you can define the operator "*" as follows:

P Q P*Q

T T T

T F T

F T F

F F T

There are 16 binary Boolean functions. That logic books usually focus on only a few of them is arbitrary in a certain sense but reflects mainly that certain of these functions more often enter into everyday argumentation, particular mathematical arguments. Meanwhile, it is fully recognized by logicians that your * operation is indeed a full fledged member of the set of 16 binary Boolean functions.

I find many statements by so-called logicians to be very dishonest, when on the one hand they talk about real-world notions like "necessary", "must" and they use symbols like the horseshoe but don't give it the name "horseshoe" when they utter their formulae, using instead a different concept like "implies". That is dishonest because on the one hand they are denying the contingent, reality-based nature of logic but at the same time they are exploiting the real basis of logic to get people to think they are doing something non-arbitrary.
Dishonesty, I would think, is attempt to decieve or mislead. I've not seen grounds to infer such a motivation. Of course, for many words, logicians have special technical definitions that in different ways differ from ordinary, everyday, non-technical definitions of those words. But I've not seen any intent by logicians to fool anyone about that, especially as so many books on logic indeed discuss differences between the technical sense and everyday senses, especially about difference between the technical sense of material implication (that associated with the horseshoe symbol) and the sense of 'implies' in which causality or relevence is at play. Indeed, any informed logician will quite quite readily grant that the notion of material implication does not capture the notion of relevance between antecedent and consequent; that sense is investigated in the study of relevance logic.

I might add that the human mind does not work in terms of the formal connectives that formal logicians posit, and the symbols of formal logicians are not even very good approximations of actual cognition.
I don't know of anyone who claims that sentential connectives capture the scope of human cognition. However, it seems that the classical first order system of connectives and quantifiers does permit a formalization of a basic mathematical reasoning; or, at least, that portion of mathematical basic reasoning that we find in the proof of mathematical theorems in the literature. (Of course, that is not to deny the importance of alternatives to classical first order logic.)

This is quite clear from the fact that logicians had to create a horrifying mythical chimera, the "statement", which isn't a sentence and isn't a formula.
Such an invention is not REQUIRED for studying, understanding, or using formal logic as formal logic is conveyed in ordinary modern study.

For example, the formal rules for universal quantifiers are different from the rules for "all", the symbol "horseshoe" is used to express a considerable range of cognitive relationships (the worst being the stodgey but classic old way of expressing domain restriction on quantifiers), etc.
I wonder what specifically you have in mind there.

So getting back to the question of what logic is, symbolic logic is totally divorced from cognition, which is why it holds little interest for us.
Of course, one may find certain things unsuited or interesting, but "totally divorced" strikes as overstatement here. Also, who do you mean by 'us'? Objectivists? You don't think there are Objectivists who are interested in symbolic logic? As to reality, symbolic sentential logic is applied to such subjects as electronic switching circuits, et. al. You are yourself typing at a computer whose advent was enabled by such developments. Edited by Hodge'sPodges
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I've never encountered such a view in my own (admittedly limited) readings of authoritative authors on the subject.
De gustibus. I encounter this perspective virtually daily in a professional context. If you find the purely syntactic reality-detached perspective ludicrous, so much the better.
No, "(P->Q)&(P)&(Q)" is not regarded as a tautology. Maybe you have in mind "((P->Q)&P)->Q)"?
I doubt that was exactly the tautology I had in mind, but thanks for pointing out the typo -- this many months later I can't remember what the exact intended form was.
That "((P->Q)&P)->Q)" is a tautology as confirmed by a truth table doesn't require any stipulation about necessity.
I'm sorry you didn't understand what I said. This pertains to the claim that formal deductions are "necessarily truth-preserving". Of course you claim you've never heard of this position, so it's not surprising that you didn't understand. Although note that you yourself refer to a so-called "truth" table, and as you surely must know, truth is a semantic notion not a syntactic one.
There are 16 binary Boolean functions.
I would not be perturbed if there were 99 Boolean functions. They simply have no relationship to logic. My * function is dysfunctional from a logical perspective; are you attempting to bolster my position by pointing out that there are a number of other dysfunctional functions?
Dishonesty, I would think, is attempt to decieve or mislead. I've not seen grounds to infer such a motivation.
I suppose that's because you deny that logicians are all really either closet Objectivists / Aristotelians or ultra-Platonists?
I don't know of anyone who claims that sentential connectives capture the scope of human cognition.
Then you agree that symbolic logic is not useful for anyone interested in actual logic, I presume.
However, it seems that the classical first order system of connectives and quantifiers does permit a formalization of a basic mathematical reasoning; or, at least, that portion of mathematical basic reasoning that we find in the proof of mathematical theorems in the literature.
That's a pretty weak justification for a lousy system. Why don't we just agree that FOP logic is wrong-headed and try again, this time focusing on the proper function of logic?
I wonder what specifically you have in mind there.
I'd be happy to explain if you would first say how you think these symbols have actually been used in the symbolization of natural language. Let's see if there is a shred of experiential common ground. How about giving me a formalization of the classical deduction "All men are mortals, Socrates is a man; therefore Socrates is mortal."
You don't think there are Objectivists who are interested in symbolic logic?
In fact I myself am interested in a reality-based formalization of reasoning and cognition.
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You wrote, "For the formal logician, propositions aren't statements about reality, and truth has nothing to do with a consciousness or reality, it is purely a syntactic symbol-transformation method."

I encounter this perspective virtually daily in a professional context.
I am curious whether you have found this in the literature of the subject; and if so, where. In any case, if by 'formal logician' you mean one who works in the field of formal logic, then it is not a uniform position of formal logicians that "propositions aren't statements about reality, and truth has nothing to do with a consciousness or reality, it is purely a syntactic symbol-transformation method" and probably not even a very prevalent view in the field, moreover that I don't know what author on the subject would describe truth as a [certain kind of] METHOD.

I'm sorry you didn't understand what I said. This pertains to the claim that formal deductions are "necessarily truth-preserving". Of course you claim you've never heard of this position
No, I made no such claim. As to the tautologies, whatever one holds about deductions being necessarily truth preserving (or even more simply that certain deductive systems are truth preserving), it is not required to subscribe to anything about necessity simply to see that "((P->Q)&P)->Q" is a tautology upon a certain definition of 'tautology'.

truth is a semantic notion not a syntactic one.
Yes, and the notion of 'tautology' (in a particular technical sense), in ordinary classical mathematical logic, refers both to semantical and syntactical considerations, but can be reduced to purely syntactical (That is, the set of tautologies can be specified purely syntactically. Note: That is a quite limited technical matter and does NOT in itself to make such a broad claim as "truth has nothing to do with a consciousness or reality, it is purely a syntactic symbol-transformation method"). Anyway, your comment does not refute any remark I've made.

are you attempting to bolster my position by pointing out that there are a number of other dysfunctional functions?
I'll leave to you to say what a "dysfunctional function" is in mathematics. In any case, your * operation is just a garden variety Boolean function. To regard it as "dysfunctional" seems to be an odd way of anthroprormorphizing it.

I suppose that's because you deny that logicians are all really either closet Objectivists / Aristotelians or ultra-Platonists?
Whatever I think logicians are in their closets, I don't see an attempt to deceive merely by using special terminology, especially as I described what many an introductory textbook says on such subjects as the material conditional.

Then you agree that symbolic logic is not useful for anyone interested in actual logic, I presume.
Please don't presume. I say what I mean as clearly as I can. And what I have said is quite limited. I've not made the claimed any sweeping views such as what may be useful to ALL people and as to what "actual" logic is.

That's a pretty weak justification for a lousy system.
It was not my intent in that remark to justify a system.

I'd be happy to explain if you would first say how you think these symbols have actually been used in the symbolization of natural language. Let's see if there is a shred of experiential common ground. How about giving me a formalization of the classical deduction "All men are mortals, Socrates is a man; therefore Socrates is mortal."
I am not interested in performing excercises for you. I just asked what you have in mind with a certain statement of yours. More specifically, I wonder what actual text from book or article in the field of formal logic you are referring to. It's fine if you'd rather not say; I am interested though. Edited by Hodge'sPodges
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P.S., you wrote, "symbolic logic is totally divorced from cognition, which is why it holds little interest for us."

So I asked who 'us' refers to there. Objectivists? And I asked, "You don't think there are Objectivists who are interested in symbolic logic?"

You replied, "In fact I myself am interested in a reality-based formalization of reasoning and cognition.'

But that's not what I asked. Rather I am interested to know whether by "it holds little interest for us" you mean Objectivists generally, mostly, or uniformly, and whether you rule out that certain Objectivists might be interested in the symbolic logic you find uninteresting.

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I'm not sure what you are getting at here and in the rest of your post. Logic is derived from the observation of existence using the human ability of abstraction.

I would disagree. From whence do we derive the ability to derive--what is the nature of this derivation? Do these derivations obey rules, or are they chaotic? If they obey rules, then from whence do we get these rules? And we proceed so forth until, fundamentally, either human knowledge is fundamentally chaotic or fundamentally rule-following. If rule-following, these rules are logic and they are not derivative of anything.

So, if we say math is a language, and language is used to describe concepts, then math is just another language describing processes and concepts seen in nature? For example "two apples" is the phrase applied to a pair of round red fruits; whereas as 1+1=2 could be the mathematical sentence used to describe a process in nature or a pair of apples.

And then the reason math is correct when describing nature is because it's describing existence.

And then things like calculus, irrational numbers and such could be different dialects?

Or is that too simple?

This seems largely right to me.

Math is not a language.Language is not used to describe concepts. (It can be so use, but is not typically or necessarily used for that purpose).

Math is a system of methods for measuring. It is a science which has progressed to the point that the specific entities being measured do not matter, and it is just pure method applicable to reality and fantasy alike.

How do you define language and measure (and system)? It seems to me I can communicate by means of mathematics, unless you take mathematics to be different from what's communicated when it's discussed. Is this insufficient for language? It certainly is insufficient for natural language, but for language in general?

I don't know which particular writers on the subject of formal logic you have in mind. I've never encountered such a view in my own (admittedly limited) readings of authoritative authors on the subject.

Wouldn't Frege endorse just this thesis?

This pertains to the claim that formal deductions are "necessarily truth-preserving". Of course you claim you've never heard of this position, so it's not surprising that you didn't understand. Although note that you yourself refer to a so-called "truth" table, and as you surely must know, truth is a semantic notion not a syntactic one.

It seems that they are indeed truth-preserving, though. Forgetting truth-tables, just consider the operators on true and false sentences. We must agree that there are such sentences, and let us now represent any member of the set by P and Q. Take 'P' as the identity operator over the truth-value of P, so that /'P'/ = true iff P is true and /'P'/ = false otherwise, and so forth with disjunction. Modus ponens must certainly preserve truth.

I would not be perturbed if there were 99 Boolean functions. They simply have no relationship to logic.

It would be absurd to think that they characterize all of logic, but from the above it seems right that they have some relationship to logic, and in fact serve quite well in our mathematical and computational uses of them.

You wrote, "For the formal logician, propositions aren't statements about reality, and truth has nothing to do with a consciousness or reality, it is purely a syntactic symbol-transformation method."

Yeah, this I'm not so sure would be Frege, though maybe. Maybe modern computer scientists and such would think so, but just about every modern analytic philosopher would distance himself from this due to the Incompleteness proof.

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I would disagree. From whence do we derive the ability to derive--what is the nature of this derivation? Do these derivations obey rules, or are they chaotic? If they obey rules, then from whence do we get these rules? And we proceed so forth until, fundamentally, either human knowledge is fundamentally chaotic or fundamentally rule-following. If rule-following, these rules are logic and they are not derivative of anything.

Logic is the method of non-contradictory identification of the facts of reality as given by perception. There are no pre-existing rules in the human mind apart from observation. The most fundamental observation is that a thing is what it is and cannot be what it is not; we observe that a cat is a cat and not a dog or a car or an rock. From this we arrive at the law of identity. When one says P is not Q, we are making those kinds of observations and abstracting from the observation, where P and be anything, and Q is anything else. All logical statements must be reducible to something one can point to -- i.e. one must reduce one's logical statements to the facts of reality, otherwise they are not logic, but rather rationalism. Rationalism consists of thinking in terms of floating abstractions that are not tied to reality; and if one is making arguments about logic apart from observation and non-contradictory identification of the facts of reality as given by perception, then one has the term "logic" in one's mind as floating abstraction -- or method without anything to use that method on or having been derived from.

You can put P's and Q's into all sorts of equations or statements, but if those statements are not tied to reality they are completely meaningless.

Edited to add: We do not start off with the rules of reason and logic planted firmly in our heads, and then apply logic to make observations; rather it is the other way around. One makes observations about the consistency of existence and then formalizes those observations into the method of logic.

Edited by Thomas M. Miovas Jr.
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Logic is the method of non-contradictory identification of the facts of reality as given by perception.

Assuming that's true, it seems to presuppose non-contradiction. That's not a rule given by perception, unless it's to be circular (we use perception to prove non-contradiction, and we use non-contradiction to come to conclusions about perception). Thus, it seems non-contradiction is a law of logic antecedent to perception.

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Assuming that's true, it seems to presuppose non-contradiction. That's not a rule given by perception, unless it's to be circular (we use perception to prove non-contradiction, and we use non-contradiction to come to conclusions about perception). Thus, it seems non-contradiction is a law of logic antecedent to perception.

I think in Thomas' quote that "facts" are the objects that "are given by perception".

Anyway, you don't prove non-contradiction - it's the Law of Identity. It seems to me that the phrase "non-contradictory identification of facts" could be considered doubly redundant. Identification is the determination of a thing's identity (which is definite and non-contradictory), and all facts are subject to the Law of Identity. If your 'indentification' gives you a result such as "A = not A", then it was actually mis-identification.

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Assuming that's true, it seems to presuppose non-contradiction.

As Jake mentioned, one does not prove non-contradiction because on the perceptual level it is obvious that a cat is not a bicycle, and therefore there is no reason to come up with a concept "cat-bicycle" or to try to reason how a cat can be sat upon and peddled to one location or another. There are no contradictions in reality on the perceptual level means that you never see a thing being what it is not; you only see it being what it is. So the point of the non-contradiction aspect of identification is basically saying do not identify it being other than what it is. I don't know that this is directly implied in the term "identification", but at least on the perceptual level, one can say that it is. But non-contradictory identification holds for all discoveries of existence and not just the perceptual level of knowledge. The perceptual level gives us a clue that a thing is what it is by observing it, and the same principle applies for those things we cannot directly perceive. So, as he said, if in your process of reasoning you wind up with a thing is not what it is, then you've made a mental mistake somewhere.

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As Jake mentioned, one does not prove non-contradiction because on the perceptual level it is obvious that a cat is not a bicycle, and therefore there is no reason to come up with a concept "cat-bicycle" or to try to reason how a cat can be sat upon and peddled to one location or another.

Well then there's your law of logic which precedes observation.

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From whence do we derive the ability to derive--what is the nature of this derivation?
That is the faculty of reason that you're speaking of; its ultimate nature is a research question for cognitive science. One example is the capacity for concept formation. Before answering the question "what physical fact gives rise to our ability to form concepts", we have to be able to answer the question "what is the physical process behind forming concepts", which we cannot do at present.
How do you define language and measure (and system)?
Why do I need to define either? I seriously don't understand how that's useful, since the terms "language" and "measure" should be obvious at least to anyone who speaks English.
It seems to me I can communicate by means of mathematics, unless you take mathematics to be different from what's communicated when it's discussed.
That is because language and mathematics are species of the genus "symbolic system". You (and the earlier poster) simply made the error of equating "language" and "symbolic system". They are different. It would be like me saying that geometry is a kind of number theory.
It seems that they are indeed truth-preserving, though.
You're not addressing the point I made -- the word "necessarily" was crucial. A proper logical derivation does of course make the transition from true premises to true conclusions, because a proper logical derivation only allows true premises and independently validated laws of logic. Such laws aren't rendered "necessarily so" because of the fact of arbitrary symbolic stipulation, but because they do state actual relations between proposition and reality.

Then the question is, what are the actually valid laws relating man's consciousness to reality; can they be understood well enough that they can be given unique forms such as ∃ ∀∧⊃? The symbol "⊃" does not; I believe that ∃ does not. ∀ is related to something valid, but that thing is best understood in terms of a theory of concepts and the notion "essential characteristic" of a concept.

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Well then there's your law of logic which precedes observation.
I think I see the problem. Observation precedes the construction of a putative law. The law may be factually mistaken and thus withdrawn from consideration, or as in the law of identity may be actually correct, and thus we do accept this law as a law of logic.
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Well then there's your law of logic which precedes observation.

You seem to be laboring under the idea that one must have logic or a logical structure to see the difference between a cat and a bicycle, and that isn't so. It is not logic imposed on perception that makes it possible for you to see the difference between THIS and this, but rather the difference is given in perception without the necessity of logic operating on the senses. It is a Kantian notion that the rules of logic are in our heads and are imposed upon our sensory manifold giving rise to the observation of what we observe. All that is necessary is for you to LOOK, you don't have to think, to see the difference between a cat and a bicycle or the difference between this and THIS -- you just look and you see it. Perception is not controlled by logic or any logical structures of the mind -- it's just a causal interaction between what is observed and the observer requiring no logical interface in-between them.

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I don't follow.

You don't see a thing not being what it is not without seeing it being what it is.

I'd say its a precondition to seeing tings as things but not as proceeding seeing at all.

It precedes it logically in the sense that one need not know anything about content that is put into the law to know that the law holds of everything. That is why we symbolize it with variables like A = A--you can fill in the 'A' with "cat" or "A cat", or "Jeff", or "Pegaus", or "round sqaure". In this way, you can have your law before any applications of it.

That is the faculty of reason that you're speaking of; its ultimate nature is a research question for cognitive science. One example is the capacity for concept formation. Before answering the question "what physical fact gives rise to our ability to form concepts", we have to be able to answer the question "what is the physical process behind forming concepts", which we cannot do at present.

I'm not certain how cognitive science could explain the subject matter at hand, since one would need to use the tools of study in order to study them. Any analogy to how we can see our eyes in a mirror or talk about language would break down at the relevant, crucial point. That is, what we are after is to identify those basic rules which make thought possible. Without the law of non-contradiction we would not be able to think of anything. Cognitive science admitted has a large role to play in explaining our experiences and reasoning patterns. But this is largely confined to the task of seeing what kind of material mechanisms determine brain functions, or what kind of reasoning fallacies we tend to make, etc. These studies do not, and cannot, investigate such things as the law of non-contradiction any more than the computer sciences can (merely because they study a given piece of hardware that may perform operations which we interpret as application of non-contradiction).

Why do I need to define either? I seriously don't understand how that's useful, since the terms "language" and "measure" should be obvious at least to anyone who speaks English… That is because language and mathematics are species of the genus "symbolic system"… You (and the earlier poster) simply made the error of equating "language" and "symbolic system". They are different. It would be like me saying that geometry is a kind of number theory.

If you feel so strongly about it, really, you needn't define anything. I was just trying to understand your statements. For I’m not sure how a language differs from a symbolic system. Please forgive me if I am not proficient enough in English.

You're not addressing the point I made -- the word "necessarily" was crucial. A proper logical derivation does of course make the transition from true premises to true conclusions, because a proper logical derivation only allows true premises and independently validated laws of logic. Such laws aren't rendered "necessarily so" because of the fact of arbitrary symbolic stipulation, but because they do state actual relations between proposition and reality.

I wouldn't state that they are necessary because of the symbolism. I doubt anybody holds that thesis. But the symbolism is supposed to reveal their necessity. Indeed, it is unfathomable to me how they might possibly be false. So I contend that it's impossible for them to fail to preserve truth, and that naturally leads me to hold that they're necessary. I can't find a weakness in my own position.

Then the question is, what are the actually valid laws relating man's consciousness to reality; can they be understood well enough that they can be given unique forms such as ∃ ∀∧⊃?...

I think I see the problem. Observation precedes the construction of a putative law. The law may be factually mistaken and thus withdrawn from consideration, or as in the law of identity may be actually correct, and thus we do accept this law as a law of logic.

I know that nobody believes those symbols are sufficeint--you need negation! (tee-hee) Anyway, I don't think anybody holds that we have a collection of symbols representing the whole of valid reasoning. What is claimed is that these symbolic representations show some part of valid reasoning, or at the very least, of rule-following.

But it seems we are simply operating on two different idiolects. Very good, then. What modern philosophers refer to as logic is merely the laws upon which all reasoning is conducted (including the reasoning we use to go from observations to conclusions about the observations). The laws which are based on observations--what we call physical laws--indeed have many of the properties you ascribe to them and it would be exasperating to try to capture them with anything like ⊃.

You seem to be laboring under the idea that one must have logic or a logical structure to see the difference between a cat and a bicycle, and that isn't so.

Not to see the difference, but for it to be intelligible that there is a difference. Suppose you have no laws of logic, and so you have no law of non-contradiction, and so you see a cat and at the same time it is a bicycle. Nothing is to stop the catcycle from being the dogskates that they are. Only non-contradiction, or something very like it, can keep away such absurdity.

It is a Kantian notion that the rules of logic are in our heads and are imposed upon our sensory manifold giving rise to the observation of what we observe.

That's not necessarily the claim being made here, though. This logic, outside of which thought is impossible, may be (in fact, I suspect, is) a feature of reality, like mathematics.

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