Blind Reasoning

[DavidOdden]

Do you think concepts like "or" and "and" are also meaningless? After all, they also refer to "arbitrary" mappings of two values to one.

On the contrary, they are not meaningless, and they do refer to valid concepts. I've been very careful, you will notice, to only denigrate the connective ⊃, and the reason is exactly that while ∧ and ∨ are actual logical operators grounded in human cognition, ⊃ is not. It resembles a valid relation, but resemblance is insufficient. And my point all along has been that contemporary symbolic logic fails to distinguish between valid connectives and arbitrary connectives -- that are all arbitrary from a "logical" point of view. Thus {T,T=T; T,F=T; F,T=F; F,F=T} is a possible connective.

BTW, the number of possible mappings is 16, not 64. Two to the power of four is sixteen.

I know: I corrected that. What can I say, the ×4 was due to insufficient morning coffee.

Could you give an example?

I'll dig: most of my books are back home.

[Capitalism Forever]

That is, assuming we have warrant to believe P and P --> Q, then the warrant from these transfers to a warrant of asserting Q.

What is your standard for accepting Q, and what is your definition of "-->" ?

This is different from proving the validity of logic--i.e. proving that the system moves from logical truth to logical truths as defined in the system

What is "the system" ? If it is anything other than reality, it is not worth caring about.

For imagine this: Someone asks you what is the justification that Q if you have P, P --> Q and MPP?

The justification is that I have P and P --> Q. That should be enough for anyone who speaks English and is even remotely willing to think. (MPP is not part of the justification; it is an expression of the justification in formal logic.)

Do you actually know of any person who is generally open to rational persuasion, but has doubts about inferring Q from P and P --> Q? I have never met a person like that!

Only a person who is unfamiliar with the concept "if" would need an explanation of why this inference is justified. But accepting P --> Q presupposes a familiarity with the concept, since you cannot accept a proposition whose meaning you don't understand. And the meaning is precisely that you can infer Q from P. You have P, and you can infer Q from P--so you can infer Q. Anyone who quibbles with this is either evading something or too stupid to have a rational discussion with.

how does this then allow that logic is not just in the mind but actually objective?

Remember that "objective" is not the same as "intrinsic." You seem to have been looking for a way to make people accept ideas even if they're unwilling to do their part of the reasoning process. The mental act that MPP formalizes is an integral part of the reasoning process; you have to be willing to do it or you won't get anywhere. A person unwilling to do it is a person unwilling to think.

[Capitalism Forever]

I've been very careful, you will notice, to only denigrate the connective ⊃, and the reason is exactly that while ∧ and ∨ are actual logical operators grounded in human cognition, ⊃ is not.

How so? Don't we use the concept "if" to describe facts of reality day by day? Doesn't the connective you like to refer to as ⊃ formalize that concept just as well as ∧ formalizes "and" and ∨ formalizes "or" ?

[intellectualammo]

The task should not be termed "prove the validity of logic", though I might have slipped up and called it that anyway, so if it was my fault then I apologize. The task is to prove the warrant-transfer of any given valid inference and to define a valid inference (note, this last is clear enough in the case of MPP but until it is demonstrated that only MPP or any equivalent is a valid inference then we need to be concerned with valid inferences as such). That is, assuming we have warrant to believe P and P --> Q, then the warrant from these transfers to a warrant of asserting Q.

Do we have warrant to believe p-->q? I was looking through my logic book(Understanding Arguments: An Introduction to Informal Logic, 2005) from college, and I found this statement therein:

"Several competing theories claim to provide the correct analysis of propositional conditionals, [like, in our case, an indicative conditional or simply an "If...Then..."] and no consensus has been reached concerning which is right. It may be surprising that theorists disagree about such a simple and fundamental notion as the if-then construction, but they do."(p.152)

How are we able to "assume we have the the warrant to believe p and p-->q" in the first place? How can you transfer a warrant from p and p-->q to a warrant of asserting q, initiating from assumptions alone?

How are we able to overlook those assumptions and go right ahead and start plugging them into a modus ponens argument form? Do propositional conditionals ("if...then...") correspond exactly to material conditionals (p-->q)? You would also have to suppose that they do, according to my book, in order to be able to plug them into a truth table, if I am understanding that correctly.

With all those questions I would already owe you $15 for answers, if we weren't here!

Edited November 23, 2006 by intellectualammo

[aleph_0]

On the contrary, they are not meaningless, and they do refer to valid concepts. I've been very careful, you will notice, to only denigrate the connective ⊃, and the reason is exactly that while ∧ and ∨ are actual logical operators grounded in human cognition, ⊃ is not. It resembles a valid relation, but resemblance is insufficient. And my point all along has been that contemporary symbolic logic fails to distinguish between valid connectives and arbitrary connectives -- that are all arbitrary from a "logical" point of view. Thus {T,T=T; T,F=T; F,T=F; F,F=T} is a possible connective.I know: I corrected that. What can I say, the ×4 was due to insufficient morning coffee.I'll dig: most of my books are back home.

1) I've already shown how the paradox can be reconstructed in Normal Form so even if this were true, I'm not sure how it would contribute to the conversation.

2) How can '-->' be an invalid operator yet '&' be valid, when you can define '-->' in terms of '&' and '~'? Hell, I could define '|' to be a logical operator, (and have done so quite successfully) to represent ~P & Q (or maybe it was ~(~P & Q)), and come out with all the same results as using '&' and '~'. It's like using Roman numerals versus Arabic. Different symbols, slightly different operations, all conform to the same system.

What is your standard for accepting Q, and what is your definition of "-->" ?

What is "the system" ? If it is anything other than reality, it is not worth caring about.

My standard for accepting Q is (well... not really relevant, since MPP only concerns itself with when Q is warranted, but to give a concrete example,) warranted includes tautologies. So, "All unmarried males are bachelors," is warranted. "I am either a crouton or I am not a crouton," is warranted. It may--and in fact does--include more, but that depends upon this initial proof of warrant-transfer for MPP.

The system is the method of deriving beliefs, and depending on your status as a Platonist, it will be reality or it will not. All the same, it's worth caring about.

The justification is that I have P and P --> Q. That should be enough for anyone who speaks English and is even remotely willing to think. (MPP is not part of the justification; it is an expression of the justification in formal logic.)

MPP is not just an expression in a formal language--at least, one should hope not, least one not be able to reason syllogistically by means of it. If you ever have stated something like, "Socialists do not concern themselves with production, but destruction. They do not care about the poor, but bringing down the rich. They hate human life," you have used MPP. You have said, in effect, "If such-and-such is the attitude of socialists, then they hate life; and such-and-such is the attitude of socialists; hence they hate life." The formalization of MPP is simply an attempt at mirroring this function in the English language. The "cut rule" (a metalogical equivalent) is purely formal.

Do you actually know of any person who is generally open to rational persuasion, but has doubts about inferring Q from P and P --> Q? I have never met a person like that!

Nope. And I'm sure, before Euclid, nobody would disagree with any one of his axioms or with much of what was derived from it. Still, until you rigorize something, you're resting on an un-cashed check. If you don't have the patience or interest to rigorize logic, fine. But that's what this discussion is for.

Only a person who is unfamiliar with the concept "if" would need an explanation of why this inference is justified. But accepting P --> Q presupposes a familiarity with the concept, since you cannot accept a proposition whose meaning you don't understand. And the meaning is precisely that you can infer Q from P. You have P, and you can infer Q from P--so you can infer Q. Anyone who quibbles with this is either evading something or too stupid to have a rational discussion with.

But you get no inference in the first place without an inference rule. You can have P, and, "from P follows Q" or "it is never the case that P is true while Q is false", but still not yet have Q unless you have a meta-rule which allows you to go from these two sentences to a third, new sentence. That's where MPP comes in, and that's why confusion is not located in either P or P --> Q, but with the third part, the "glue" that tacks on the new sentence Q to the end of the whole thing.

Remember that "objective" is not the same as "intrinsic." You seem to have been looking for a way to make people accept ideas even if they're unwilling to do their part of the reasoning process. The mental act that MPP formalizes is an integral part of the reasoning process; you have to be willing to do it or you won't get anywhere. A person unwilling to do it is a person unwilling to think.

The point is that the tortoise is quite willing to do his part--he accepts everything that we accept (ostensibly). He is willing to write new lines of truth and he obeys the laws of the connectives (Note that the tortoise derived nothing we as logical thinkers would not have agreed to.) What, it turns out, he does not actually accept is MPP.

Do we have warrant to believe p-->q?

If we're dealing with an MPP induction, yes. MPP only concerns itself with the situation: P and P --> Q. If anything else holds (namely, ~P or P & ~Q), MPP is not concerned with that situation. Now certainly we have situations where we know P --> Q. They're easy to produce. If Socrates is a man then Socrates is a mammal. If a number n is represented by a numeral n, then the number n + 1 is represented (where, in the formal language, ' represents "successor" [this could be stated more rigorously, but it is fine for now]) by the numeral n'. Whenever we have such an if-then situation, and we have the antecedent (for instance, Socrates is a man and 1 is represented by 0'), then we get the antecedent.

"Several competing theories claim to provide the correct analysis of propositional conditionals, [like, in our case, an indicative conditional or simply an "If...Then..."]and no consensus has been reached concerning which is right. It may be surprising that theorists disagree about such a simple and fundamental notion as the if-then construction, but they do."(p.152)

As I said to David, replace it by Normal Form (where you replace any statement of the form P --> Q with the form ~P v Q) and you get the same paradox. By the same token, if you find inclusive "or" to be problematic, I doubt you will find ~(P & ~Q) to be problematic. That's about as epistemically obvious as you can get. Again, replacing any one formalization for another, you get the same paradox.

Enjoooooy turkey!

[Capitalism Forever]

My standard for accepting Q is (well... not really relevant, since MPP only concerns itself with when Q is warranted, but to give a concrete example,) warranted includes tautologies. So, "All unmarried males are bachelors," is warranted. "I am either a crouton or I am not a crouton," is warranted.

I very strongly recommend reading Dr. Peikoff's essay "The Analytic-Synthetic Dichotomy." You can find it in ITOE.

It may--and in fact does--include more, but that depends upon this initial proof of warrant-transfer for MPP.

But the initial proof can only accept tautologies as warranted? It can't be done.

If you don't have the patience or interest to rigorize logic, fine. But that's what this discussion is for.

I don't have the patience or interest to attempt the impossible. You can't prove anything without accepting the basic rules of logic FIRST.

[DavidOdden]

How so? Don't we use the concept "if" to describe facts of reality day by day? Doesn't the connective you like to refer to as ⊃ formalize that concept just as well as ∧ formalizes "and" and ∨ formalizes "or" ?

You moved. Have I been asleep at the switch and not noticed? Well, congratulations.

We use the word "if" a lot; I don't think it constitutes a concept. We can separate some of the wheat from the chaff by noting that ⊃ is used where "if" is not, for example in universally quantified statements like ∀x(Man(x)⊃Mortal(x)) "All men are mortals". In addition, the best translation from "⊃" to "if" is "A⊃B"→{"if A, B", "B, if A"}. But a lot of uses of "if" don't translate into "A⊃B", for example "There's a beer in the fridge, if you want one"; "If it stops raining, could you let me know?"; "I don't know if you speak Hungarian"; "What if he gets lost?".

My opinion is that it would be more productive to look at the cases where ⊃ does correspond to one valid concept of, say, English. For example "If I support taxation, I am immoral"; "If he believes in god, he is irrational"; "if they jump into a vat of molten steel, they will die". Then we can try to identify the underlying concepts, and see whether "⊃" plays a necessary role in formalizing these sentences.

[aleph_0]

I very strongly recommend reading Dr. Peikoff's essay "The Analytic-Synthetic Dichotomy." You can find it in ITOE.

I've read it. I'm not impressed.

But the initial proof can only accept tautologies as warranted? It can't be done.

Other examples are possible, but I take tautologies to be the most easily and quickly demonstrated.

I don't have the patience or interest to attempt the impossible. You can't prove anything without accepting the basic rules of logic FIRST.

So Aristotle was wasting his time when he rigorized the rules of logic?

[y_feldblum]

You are not rigorizing logic (and neither is Boghossian). Aristotle rigorized it, and his process is very different from yours. His is integrative from observations of reality. Yours is deductive from, alternately, intuition or nothing at all.

[aleph_0]

Aristotle rigorized it, but not fully. Nor have I and nor has Boghossian, though that is the project.

But it is not true that Aristotle "integrated from observations" if, by this, you mean that he induced deduction. He did not argue for non-contradiction by so many apparent non-contradictions, but by the un-tenability of the contrary. In fact, he did not prove deduction at all but took it as primary. Indeed, I agree that that is the way to go. All that is left, then, is an explanation of how it may be justified (or put another way, how one is epistemically blameless) to use modus ponens. Put yet another way, why should we rely on the belief that modus ponens takes us from truth to truths?

I am, in lieu of furthering the discussion of the main subject, awaiting somebody to explain exactly how logic as it is presented here is a floating concept.

[DavidOdden]

I am, in lieu of furthering the discussion of the main subject, awaiting somebody to explain exactly how logic as it is presented here is a floating concept.

You have not show what aspects of reality are reflected in "⊃", nor how statements like "P⊃Q" can be validly (i.e. with reference to existence) introduced into a logical derivation. Those are examples of the floatingness problem.

[y_feldblum]

[As the foundation of logic,] we rely on the belief that modus ponens takes us from truth to truths[.]

I am, in lieu of furthering the discussion of the main subject, awaiting somebody to explain exactly how logic as it is presented here is a floating concept.

Any questions?

[aleph_0]

You have not show what aspects of reality are reflected in "⊃", nor how statements like "P⊃Q" can be validly (i.e. with reference to existence) introduced into a logical derivation. Those are examples of the floatingness problem.

Why do you say that? P --> Q simply reflects any statements that have the following property: When P is true, Q is true. In all instances of P, Q as well. What is floating? I see nothing but sunkenness.

[DavidOdden]

Why do you say that? P --> Q simply reflects any statements that have the following property: When P is true, Q is true. In all instances of P, Q as well. What is floating? I see nothing but sunkenness.

What does "P" mean? What does "Q" mean" How do you know that "P⊃Q". Are you speaking of universal correlations? For example, it has been repeatedly, obsessively observed by physicists that when particles decay, the net charge of the decay products is equal to the charge of the original particle. Hence K⁺ may decay to π⁺π⁰ or π⁺π⁺π^- or π⁰μ⁺ν_μ, where the pluses and minuses all add up to +. That's an example of using ⊃ to express causation, that it is abbreviates a statement that there is a causal law that say "If you have the decay of a positive particle, that causes the creation of such-and-such pattern of positive, negative and neutral particles". Now if that is what you really mean, then I think it's possible to discuss how know that this statement is true, and might have some hope of persuading your friend to accept this use of ⊃.

[Capitalism Forever]

But a lot of uses of "if" don't translate into "A⊃B"

So what? A lot of uses of "mind" don't translate into "reasoning faculty" (for example: "Do you mind if I smoke?" "I have half a mind to slap you!") Does this invalidate the mind?

My opinion is that it would be more productive to look at the cases where ⊃ does correspond to one valid concept of, say, English. For example "If I support taxation, I am immoral"; "If he believes in god, he is irrational"; "if they jump into a vat of molten steel, they will die".

So you agree that sometimes it does correspond to a valid concept? Good, we're making progress! Now what you need to see is that the common characteristic of the sentences above is implication. Implication means that P necessarily makes Q true; in other words, that we can infer Q from P; in yet other words, that it is impossible for P to be true and Q to be false at the same time.

Now, in the examples you gave, P does always explain Q: "Why am I immoral? Well, for example because you support taxation." "Why is he irrational? He believes in God, among other things." "Why will they die? Because they're about to jump into a vat of molten steel." But there are other cases of implication where no explanation is involved, such as: "If I go at all, I'll go by car." (The explanation here would be more along the lines of "Because I hate airport security.")

And it is important not to confuse explanation with causation. Explanation is epistemological, while causation is metaphysical. "Supporter of taxation" is one way to classify people epistemologically; "immoral person" is another. The sentence you gave means that the latter category is fully contained within the former, i.e. that there are no moral supporters of taxation. (I believe this is what you've been calling domain restriction--as you can see, it is symbolized by --> because it can be expressed with an "if.") The same goes for the relationship between "believer in God" and "irrational person." The explanation of death by molten steel is the only one that has to do with causation, but what the sentence does is not express an instance of causation but rather point out a pair of events that are links in a causal chain, with the rest of the chain (the nature of molten steel as hot, maximum the heat tolerance of the human body, and so on) being familiar enough to the listener to remain unstated.

Here is an instance of causation: "The tree grew." Causation means that a certain object causes a certain action; it is not a relationship between two events but an object and something the object does. The way to express this in English is a simple indicative sentence; no connectives are necessary. I don't know of a formal logical system that can express actions (as opposed to just attributes), but if there ever is to be one, it should not use the same symbol for this as it uses for implication, as the two are two different things entirely.

Edited November 25, 2006 by Capitalism Forever

[DavidOdden]

A lot of uses of "mind" don't translate into "reasoning faculty" (for example: "Do you mind if I smoke?" "I have half a mind to slap you!") Does this invalidate the mind?

No, but ordinarily people don't make the mistake of confusing the concept of "mind" with the concept of "objecting". Unfortunately, I can't say the same about "if". That's why I demand of any person making claims about a supposed concept underlying the formal symbol "⊃" that they actually clearly identify what that concept is. In other words, as in the case of "mind", you can't base your philosophy on words, you have to base it on concepts. When a word is ambiguous, as "mind" or "ass" is (in American English -- I learned that it's not ambiguous in British English), you must make it clear what concept you are referring to.

Anyhow, I'd be pleased to see you explain what concept you think the symbol "⊃" stands for, once you figure that out. (With reference to your subsequent text, surely you will understand the ⊃ of that sentence). When you get to that point, we will have indeed made progress!

Implication means that P necessarily makes Q true; in other words,

Uh, let's not have any other words right now. So first, I'm wondering what the difference is between P necessarily making Q true, and P unnecessarily making Q true. Maybe you meant simply that P makes Q true, that is, if P is a fact, or, P exist, then that fact causes Q, or causes the metaphysical state described by "Q" to exist. It is possible for P to "imply" Q but not cause Q? Since we're dealing in floating abstractions like "P" and "Q", I'm not sure if this make sense. I just think that you should see that you basically said "Yes, 'P⊃Q' is a statement of causal relationship". Let me point you to some evidence that I've found that does support my contention that the only valid use of '⊃' is to express causal relations. To wit, using bold to clarify: "Why am I immoral? Well, for example because you support taxation." "Why will they die? Because they're about to jump into a vat of molten steel."

But there are other cases of implication where no explanation is involved, such as: "If I go at all, I'll go by car."

Meh. That's an excellent example of a totally different use of the word "if", which isn't subsumed under what I see as the only valid use of "⊃". I can't for the life of me see how this is an example of P necessarily making Q be true.

(The explanation here would be more along the lines of "Because I hate airport security.")

Excellent!! Back on track. You've ultimately reduced it to causation.

Here is an instance of causation: "The tree grew." Causation means that a certain object causes a certain action; it is not a relationship between two events but an object and something the object does.

So are you really saying that if P lawfully causes Q, then there is no way to talk about that relationship?

I don't know of a formal logical system that can express actions (as opposed to just attributes), but if there ever is to be one, it should not use the same symbol for this as it uses for implication, as the two are two different things entirely.

Since there doesn't seem to be any such thing as "implication" which is separate from causation on the one hand or domain restriction / subset identification on the other, I would agree to an even stronger statement, namely that no formal system should even symbolize "implication". Are you saying that you don't know of a way to symbolically say "John kissed Mary", a sentence which describes an action?

[Capitalism Forever]

Since we're dealing in floating abstractions like "P" and "Q"

That's an excellent example of a totally different use of the word "if", which isn't subsumed under what I see as the only valid use of "⊃". I can't for the life of me see how this is an example of P necessarily making Q be true.

You've ultimately reduced it to causation.

I'm sorry, David, but you are waaaaaaay to confused on these things and I don't find it to be in my interest to pursue the matter further with you. Thank you for the conversation.

[aleph_0]

What does "P" mean? What does "Q" mean" How do you know that "P⊃Q". Are you speaking of universal correlations? For example, it has been repeatedly, obsessively observed by physicists that when particles decay, the net charge of the decay products is equal to the charge of the original particle. Hence K⁺ may decay to π⁺π⁰ or π⁺π⁺π^- or π⁰μ⁺ν_μ, where the pluses and minuses all add up to +. That's an example of using ⊃ to express causation, that it is abbreviates a statement that there is a causal law that say "If you have the decay of a positive particle, that causes the creation of such-and-such pattern of positive, negative and neutral particles". Now if that is what you really mean, then I think it's possible to discuss how know that this statement is true, and might have some hope of persuading your friend to accept this use of ⊃.

'P' and 'Q' are not yet assigned values. 'Tis the nature of variables. They are able to vary. By definition, however, modus ponens is only concerned with those Ps which are true sentences. Any true sentence. At all. And we do not know that 'P --> Q'. It is only that, by definition, modus ponens is only concerned with such instances where P --> Q is true.

Yes, it is possible to discuss causal correlation. Yet there are other correlations. For instance, a map correlates to geography, and P --> Q correlates to those sentences such that one may only be true when the other is true though the other may be true when the one is false. Again, if you have some kind of strange epistemic objection to '-->', replace everything with Normal Form. MPP will become P, ~(P & ~Q)/ Q.

I don't see what's so difficult to grasp about that. I have already explained the meanings involved, none of which are any more abstract than algebra; and I already gave plenty examples, the validity of none of which has been shown objectionable.

Oh yes, but I would like to draw attention to one aspect of the response, namely, the belief from induction. Why do you believe that, when a particle decays its net charge will be equal to the charge of the original particle? Because you've reasoned: "If such-and-such conditions are observed then we derive such-and-some-other conclusion. And such-and-such conditions are observed. Hence, such-and-some-other conclusion is justified."

[DavidOdden]

It is only that, by definition, modus ponens is only concerned with such instances where P --> Q is true.

But recall also that the problem has something to do with the recalcitrant person who does not freely introduce lines in a logical derivation for no reason, other than to be in accord to some stipulation. As I have pointed out a number of times, "P⊃Q" cannot be introduced ad libitum, rather it has to be built up inductively, and the process of doing so actually introduces Q anyhow. Had you pointed this out to the recalcitrant gentleman, you would have been able to see if he was simply being uncooperative to the program of reality-independent logic, versus actually embraces a contradiction.

I don't see what's so difficult to grasp about that. I have already explained the meanings involved, none of which are any more abstract than algebra; and I already gave plenty examples, the validity of none of which has been shown objectionable.

Well, I've shown that all valid cases of ⊃ reduce to entirely different concepts, namely "causation" and "context", and I also don't see why that point should be so hard to grasp.

Why do you believe that, when a particle decays its net charge will be equal to the charge of the original particle? Because you've reasoned: "If such-and-such conditions are observed then we derive such-and-some-other conclusion. And such-and-such conditions are observed. Hence, such-and-some-other conclusion is justified."

In fact I believe it because I trust the experimental physicists, insofar as I have never personally induced kaon decay, or at least knowingly. This has been established empirically, observing a wide variety of specific instances of positive particles decaying and the results of that decay, so that there may be an instance of K⁺ → π⁺π⁰, K⁺ → π⁺π⁺π^- and so on. Based on the perceptual evidence that there is such a law, the law has been tested to the point that all of the evidence points in the direction of that conclusion (the law), and no evidence, perceptual or conceptual, suggests that an alternative conclusion is possible. That means that the conclusion is logically validated, i.e. is certain (see OPAR ch. 5). I now have as a cognitive tool a causal principle that informs me of the nature of things that I have not yet observed.

[aleph_0]

But recall also that the problem has something to do with the recalcitrant person who does not freely introduce lines in a logical derivation for no reason, other than to be in accord to some stipulation. As I have pointed out a number of times, "P⊃Q" cannot be introduced ad libitum, rather it has to be built up inductively, and the process of doing so actually introduces Q anyhow.

How do you start from induction? How do you induce something without a previous rule for inductions, if your inductions are not to be random? Given a rule for induction, you have an abstract rule of deduction which you did not build up inductively.

Another way to ask the question: How do you induce induction? For you seem to believe that induction gets you from truth to justified derivations. Upon what do you base this belief? And upon what do you base the belief that the belief is justified, ad infinitum? You must start from a basic point which was not discovered from induction. This basic point can be abstractly applied to all propositions P and Q. Again, it is no more disconnected from reality than arithmetic expressed in algebra.

Had you pointed this out to the recalcitrant gentleman, you would have been able to see if he was simply being uncooperative to the program of reality-independent logic, versus actually embraces a contradiction.

No need. The recalcitrant gentleman has been cornered and dealt with. We (the royal "We") showed that he was treating a derivation rule as if it were an axiom schema. Problem solved.

Well, I've shown that all valid cases of ⊃ reduce to entirely different concepts, namely "causation" and "context", and I also don't see why that point should be so hard to grasp.

Because you haven't shown how "Socrates is a man; if Socrates is a man then Socrates is a mammal; hence Socrates is a mammal," is reducible to causation or context. There are infinite instances of valid '-->' situations which need not introduce causation or context.

In fact I believe it because I trust the experimental physicists, insofar as I have never personally induced kaon decay, or at least knowingly. This has been established empirically, observing a wide variety of specific instances of positive particles decaying and the results of that decay, so that there may be an instance of K⁺ → π⁺π⁰, K⁺ → π⁺π⁺π^- and so on.

Fine, then you have the inference: If scientists document such-and-such conditions, then such-and-some-other conclusion is warranted (etc.). In fact, you may reduce all derived beliefs to some form of, "If such-and-such conditions hold, then such-and-some-other conclusion is warranted; and such-and-such conditions hold; hence, such-and-some-other conclusion is warranted." If you do not use this form, then any alternative form of derivation that I can imagine (such as "It is true because I believe it," or more simply, "It is just true! I need no argument!") I will simply reject out of hand.

Moreover, how did scientists establish this empirical fact? Did they squint really hard at a particle? Or did they hold the proposition, "If our readings tell us such-and-such, then we may conclude such-and-some-other; and our readings tell us such-and-such,"?

and no evidence, perceptual or conceptual, suggests that an alternative conclusion is possible.

Perhaps so, though that does not indicate that an alternative is impossible (i.e. no positive claim has been made here). Also, I suppose it depends on your definition of "suggestion", since one might consider the contradictions in the sciences today to suggest that something in our scientific conception is wrong and that it might well be located in our theory of particles. I wouldn't argue so, but that's a whole other discussion.

[DavidOdden]

How do you start from induction? How do you induce something without a previous rule for inductions, if your inductions are not to be random?

Are you asking about formal symbolic induction, or cognitive induction? Either way, it ultimately emerges from realising "existence is identity". If you could imagine that not being true in some sci-fi kind of way, then induction (or deduction) could not have been discovered. In accepting the axiomatic principle "existence is identity", I can discover more specific facts about reality such as "causality". As for the transduction of these kinds of basic epistomological / metaphysical principles into a system of formalized symbolic logic, the "how" is partly a historical question (blame Frege, for example); but from the perspective of modern formal logic, it is "just another system", and not all logics accept inductive generalization. From that POV, it's arbitrary whether you include the law of the excluded middle or not; whether you include inductive generalization or not; idem ◊p→□◊p.

What I think is missing from all of this discussion is any notion of "logical ethics", namely, what should you do with a logic; why; what is your standard of evaluation? "Correspondence with reasoning" is my standard in logic, hence it's not hard to justify inductive generalization.

For you seem to believe that induction gets you from truth to justified derivations. Upon what do you base this belief?

Correspondence with reality. Well, actually I don't deal in "justified" or "derivations", but if I correctly interpolate what you're referring to, formal logical principles are validated by reference to whether they state "what's happening" in reality. If you're asking for a self-contain, self-justifying complete formal system, I haven't had time to construct it this afternoon.

Because you haven't shown how "Socrates is a man; if Socrates is a man then Socrates is a mammal; hence Socrates is a mammal," is reducible to causation or context.

Oh, okay, I thought we were only dealing in universally quantified propositions. In this case, you can't get the second proposition without first entering the conclusion. Nothing allows you to introduce arbitrary, unproven statements. To simplify the matter, we could shorthand the proof of the proposition "Socrates is a man" to the fact that you can see that he is a man. I don't know of any way to directly validate the claim "if Socrates is a man then Socrates is a mammal", without first proving "All men are mammals". But you're asking how to induce a singular proposition independent of a domain-restriction rule of the "All men are mammals kind". It can't be done, so "if Socrates is a man then Socrates is a mammal" can't be validly entered without reference to a rule regarding mammals.

Moreover, how did scientists establish this empirical fact? Did they squint really hard at a particle?

I don't know, but that's a scientific question and not a philosophical one. Surely there must be an expert in the ares of charge-detector construction in this forum, so perhaps they can explain the details of how it works. My knowledge is rather rudimentary, reducing to the operation of magnets.

[aleph_0]

I think, like Capitalism, I should simply adjourn this conversation. I have fully (even more fully than should be necessary for the most untrained of students) explicated the nature, function, and necessity of logic as formalized by propositional logic and still you do not seem [to want?] to understand. So I must simply welcome other people to a discussion of the main topic or the other topic title Blind Reasoning, if they have any further insight.

[y_feldblum]

In closing, then - DavidOdden and I understand the nature, function, and necessity of formalized logic fully well; we understand it from the viewpoint of Objectivist epistemology. And we understand, fully well, that any attempt to ground formalized logic in itself or in whim or in nothing at all must fail. Formalized logic must be grounded, but not in "rational intuition" (msyticism/intrinsicism). In particular, this is why Boghossian in "Blind Reasoning" failed.

We understand that the principles of logic are not self-evident and are not axiomatic, and that they must be grounded in reality, by means of observation and integration. That formalized logic, as the art of deduction, is a subset of logic in general, the art of identification. That the principles of formalized logic can be grounded intermediately in the principles of logic in general, and in particular in the methods of integration and conceptualization.

The syllogism is the most basic principle of formalized deductive logic, as the father of logic Aristotle discovered, and the closest principle to the referential and hierarchical relationships of concepts and concretes - and therefore the closest principle to the concrete entities, attributes, and actions in reality -, and it is the principle upon which others such as modus ponens rest.

[aleph_0]

I actually take it back, I’m not done arguing over the primacy of logic. I want to confine our conversation to the most basic ingredients and steps of deduction. Here you have introduced “existence is identity” (or more properly worded, existence has identity). Yet this is not a derivation rule, but a key axiom schema. Note, I cannot go from this truth applied to any particular observation within my visual field [or otherwise] to any new, nontrivial truth. From this I can only take in an observation and note that it has identity, and vice versa, suppose an identity and note that by provisional assumption it must exist within the context of the provisional assumption. I cannot, from this alone, move from an observed patch of red in my visual field to the conclusion that it is a shirt with x, y, and z physical properties which will be useful in keeping warm and maintaining modesty. For this we would need some argument of the form, “If I observe some specific patch of red in some defined set of circumstances, then I may conclude that it is a shirt; and I observe some specific patch of red in an appropriate set of circumstances; hence it is a shirt.” Or more fundamentally, such as when learning about shirts in general, we might say, “If the patch of red that I saw exhibiting the x, y, and z properties of a shirt, today exhibits the same properties, then all things being equal, tomorrow I may successfully use it as a shirt.” Of course, an infant learning about shirts wouldn’t be able to verbalize this so clearly, but the concepts would be present or else how does the infant learn about shirts?

The point being, however, observation alone does not get us anywhere—nor does any axiom or even any axiom schemata. For the content of observations are in fact axiomatic, and so not methods of derivation.

As for formalizing the logic of statements of the form, “If P then Q; and P; thus Q,” yes, some syntactic systems do not have inductive generalization—in fact, most do not. In fact, propositional logic does not. It merely has propositions and, depending on how you want to construct it, a few logical connectives and a single derivation rule. And while this does not reach very far into the form of specific concepts, such as the content of any given P, it is complete and sound. Anything true within the system is provable by the system. Now we do not yet have any P, though a P could be given as has been done previously. And we do not want to yet give a particular P because we want to prove a general proposition about all P’s and so remain ambiguous about the specific nature of any particular P—yet note that, as of yet, we have not done anything which is not already done in a perfectly intelligible way in basic English grammar: Identified a statement or sentence-clause. P is any, possibly true and possibly false, statement that you could produce in English, a concept equally mirrored by algebra, where x is any number, possibly equal to zero and possibly not equal to zero. Furthermore, there is a more perfect analogue which every Objectivist accepts: A = A. We do not specify the A, but before any observation, induction, or ratiocination we have the reflexive relationship of identity. And this, moreover, also does not exceed basic English grammar. We give a noun-clause and conjoin it to itself with a representation of the verb “is” of identity. So here we have it, P is just some true or false sentence or sentence-clause. This is a perfectly intelligible formalization.

Moreover, we can take two statements and represent them by P and Q. And we can deny them. So where you might say, not-P or not-Q, we can instead write ~P or ~Q. These will have a truth-value depending on the truth-value of the statements P and Q. And moreover, we can conjoin them. We can write, instead of P-and-Q, P & Q. This, too, will have truth-value based on the truth-value of each. Well how do you find the truth value of each? We’re not concerned with that just yet. Maybe we’ll talk about it later, but for right now, I’m perfectly content with the truth value being undecided because I haven’t made a claim about one or the other being true or false. So the question is entirely irrelevant. But now give me a situation where P is true. Well we know, then, that P & P must be true. How do we know P & P is true? Well I just gave you that P is true and you know that a conjunction is true when both sentence letters are true. If you have some sci-fi way of imagining this false, I just cease to be able to communicate with you. Now we didn’t get at this by a derivation within the formalization, but we can see that it must be true. It’s an axiom. All things are true or false, but not both. So we could either have written the statement as I did above or as ~(P & ~P). This is always going to be true. How do we know? I already told you, “Filter out all situations where P is false. Now we have only that P is true. Now I give you the new sentence, P & P. This is always going to be true.”

So also take a situation where we know that P is true, and we know that ~(P & ~Q) is true. Well we can see that Q must be true, and so now we produce a purely formal system: Now that we have these collections of very rudimentary symbols, whenever you face a sentence phi that is given by an axiom or derivation from axioms, and a sentence psi which we know never to be false when phi is true, then we know that psi is true.

I may write more later, but if you can possibly deny anything in this, I can only imagine insanity is the cause, in the same way that denying excluded middle and reflexivity of identity is simply insane.

[DavidOdden]

In closing, then - y_feldblum and I understand the nature, function, and necessity of formalized logic fully well; we understand it from the viewpoint of Objectivist epistemology. And we understand, fully well, that any attempt to ground formalized logic in itself or in whim or in nothing at all must fail. Formalized logic must be grounded, but not in "rational intuition" (msyticism/intrinsicism). In particular, this is why Boghossian in "Blind Reasoning" failed.

We understand that the principles of logic are not self-evident and are not axiomatic, and that they must be grounded in reality, by means of observation and integration. That formalized logic, as the art of deduction, is a subset of logic in general, the art of identification. That the principles of formalized logic can be grounded intermediately in the principles of logic in general, and in particular in the methods of integration and conceptualization.

I mean, how many times do we have to say this? I understand that you would like us to accept primacy of consciousness, and especially to reduce logic to being an arbitrary, self-contained and self-justifying system of choices. I just can't do that and maintain a shred of intellectual honesty and self-respect. I've explained to you in great detail everything that's relevant regarding the connective ⊃. I've pointed out that the ability to write strings of symbols according to a mechanical rule it no proof that such an act is of any value. I've pointed out the obvious fact that you cannot compel a person to act according to principles, and that in fact a person should not act accoding to arbitrary dictates that do not correspond to graspable and justified principles. I've also pointed out that all of this why and how jibber-jabber is meaningless without a standard of evaluation, and you know what the correct standard of evaluation for amy logical system is.

However, I notice that I have been remiss in not pointing out that rules of formal logic are not truth-preserving, they are T preserving. T is one of two values in formal systems, along with F, and the combination of certain rules of inference (if you use inference) or axioms (if you're Kleene) has an abstract algebraic property of T-preservation. Truth, on the other hand, is different from T. Truth, in contrast, is the product of the identification of the facts of reality. Only an insane person would deny that.