Truth (Was:Re: PROOF INFINITY DOES NOT EXIST!...)
in [Logic]
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From: Curt Welch on 23 Jul 2010 19:11 herbzet(a)cox.net wrote: > Curt Welch wrote: > > herbzet wrote: > > > herbzet wrote: > > > > Wolf K wrote: > > > > > > > > The rules of inference in e.g. PQ (Boolean) logic are designed > > > > > to prevent transformation of a "true" sentence into a "false" > > > > > one. > > > > > > > > Yes. (The question, for me, is how this magic is performed!) > > > > > > The broader question of this thread, of how truth or falsity is > > > assigned to sentences in the first place, is not my concern here. > > > > > > Given that a truth-value has somehow been assigned to a sentence, the > > > question then becomes, by what magic do these particular > > > transformation rules, the "logical" transforms, manage to pull off > > > the trick of preserving "truth" throughout all the allowed > > > transformations of that sentence, regardless of what truth-value was > > > originally assigned? > > > > > > Let us recall that one and the same set of transforms allow: > > > > > > 1) the transformation of a false statement into another false > > > statement 2) the transformation of a false statement into a true > > > statement 3) the transformation of a true statement into another true > > > statement > > > > > > but do *not* allow > > > > > > 4) the transformation of a true statement into a false statement. > > > > I don't get your point here. It's trivial to transform a true > > statement into a false statement. Just add not to it front of it. > > > > A -> not not A > > ??? > > 'A --> not A' is *not* standardly allowed as a logical transform. > 'A --> not not A' *is* standardly allowed. I just failed to grasp what you were saying. I see it now. You were talking about transforms that maintain the truth. I missed that. > > > How is this done? What is the common feature of these "logical > > > inference" rules by which they can magically distinguish and filter > > > out "truth" from "falsehood"? > > > > > > Let there be no mistake, this is a very non-trivial question -- > > > the nature of "logical implication" has been a matter of debate > > > for thousands of years, and is very much a live question still. > > > > > > My answer, arrived at after a great deal of consideration (trust me), > > > is as I posted the other day: > > > > > > "The only thing further I would want to say here is to broaden > > > somewhat the proposed definition of logical truth -- not only > > > to have the logical equivalence of statements X = Y, but also > > > to have the logical inclusion of one statement in another. This > > > is typically indicated with an arrow '->' and read as "implies"; > > > thus we have X -> Y (X implies Y) which IMO is the assertion > > > that what is meant by statement X includes what is meant by > > > statement Y. > > > > > > [We might say that Y is analytic with regard to X.] > > > > > > "For example, "John is a bachelor" includes as part of its > > > meaning "John is a man", so we can infer the latter from > > > the former. > > > > > > [I point out: this inference is valid /regardless/ of whether > > > 'John is a bachelor' is assigned the value "true" or "false".] > > > > > > "Again, it naturally falls out that if X is true, so too will > > > Y be true (though not necessarily the reverse). We can then > > > define the concept X = Y as meaning that X -> Y and Y -> X. > > > > > > "The "logical" transforms of a sentence X to a sentence Y then > > > are those transforms which preserve some (or all) of the meaning > > > of X in the meaning of Y, without adding anything extra." > > > > The word "meaning" doesn't fit there in my view. > > You are not alone in that -- in modern logic we are used to > abstracting away the meaning of propositions and dealing > with their abstract forms -- that is, only the meanings > of the "logical" elements of the propositions are retained > for the purpose of analysis. Right. > Nevertheless, the net effect is as I have described above: some > (or all) of the meaning of X is preserved in the meaning of Y, > with nothing being added -- regardless of what interpretation > (meaning) we may subsequently assign to the abstract logical > forms of X and of Y. Ah, ok, you aren't just being sloppy, you are actually trying to say something about the meaning once you have abstracted a truth value from the statements and concluded that X -> Y. Ok, yeah, I think I agree with your point though I might have to think a bit more. That conclusion however includes a big assumption. That is, the assumption that the truth value was correctly abstracted from the meaning (which it probably never can be). > In fact, the word "effect" is misleading -- it is the preservation > of meaning that determines what are the logical transforms, and > a fortiori what are considered the logical elements of the language. > > > The total meaning in > > "John is a bachelor" is far more complex than any questions about the > > binary truth of the statement. > > Sure. > > > Maybe, "logical truth value" is better than "meaning" in your > > sentence? > > The logical transforms do not necessarily preserve truth-value -- in > some instances they allow the derivation of true statements from false > statements, e.g. from the false compound statement "Lassie is a man, > and all men are mortal" we conclude correctly that "Lassie is mortal". Yeah, I'm with you now. > > > Or to put it differently, the relation of logical implication > > > holds between two statements X and Y when what X asserts > > > includes all of what Y asserts. > > > > In natural language, what a sentence asserts is it's meaning > > Right. > > > and again, > > the meaning is far more complex than anything to do with trivially > > simple binary logic. > > Ok. > > > Simple logic is not a good tool for understanding meaning. > > But it is the reverse that is the case. > > Though we may have some doubt about what precisely it means to assert > "John has a big ego", it is indubitable that this proposition includes, > and hence implies, the proposition that "John has an ego" -- regardless > of whether or not John has a big ego, or indeed whether he has an ego at > all. Well, yes and no. I agree with the idea you are suggesting, but meaning is actually always more complex than that. Meaning to me is defined in two ways. It's defined first by the physical chain of events which caused a specific language stament to be produced (the senders meaning), and it is defined by the physical effect it has on the universe (the receivers meaning). There is no (absolute) "generic" meaning to a statement such as "John has an ego". Each time the words are generated by someone, it has a different meaning - the meaning which is associated with why that person spoke those words. And each time someone reads the words, it creates meaning in the receiver. That meaning will never be the same twice even for the same person (even though it will likely be very similar for consecutive times). We can talk about the average meaning in a society - that is, what the most likely average expected meaning of someone speaking those words in that language to be. But in truth, meaning is so highly complex, and so different from person to person, that there's little truth in talking about expected meaning. If I hear a teenage girl say "John has an ego" in a conversation she is having with another teenage girl that seems to be talk about boyfriends, my brain will make all sorts of educated guesses about what the meaning of the sender was. I might guess the real meaning might be closer to "I wish John liked me". In a different context, I might be watching a spy movie and the see the hero utter the words "John has an ego", and in response, a voice activated door opens. In that context, my best guess as to the meanings of the words are closer to "I want the door to open" and don't have anything directly to do with someone named John, or his ego. To suggest we can understand the meaning of such a sentence totally out of context, or that the words only have one "real" meaning, would be pointless. So how, given this complex landscape of language, could we assign a truth value to the statement? We basically can't. At least at not without making a huge set of assumptions about the sender and the receiver and the entire context that caused the words to be produced. And even if we make all these assumptions, the odds that our understanding being very accurate, is poor at best. The only way to get a truth value from a sentence, is to make up ARBITRARY rules of truth that roughly fits the default or standard meaning of the words. Not the real meaning when used, but the sort of meaning which gets taught in class when attempting to teach a language or the sort of meaning found in a dictionary. And only when that meaning is extremely simple and generally clear, does it work at all - such as when we say "John has two peaches". When then use logic to verify that "John has two peaches" -> "John has some fruit". But we had to make many assumptions about the meaning, that often can't be made in real language use. Only in an highly artificial interpretation of language (one that can never be used in the real world by real people unless they want to be considered idiots) can such logic be evaluated. And only in that highly artificial interpretation of meaning, does the truth of X -> Y imply Y includes X become even close to valid. Ok, so writing the above has allowed me to think about this some more. Let me now respond again to what you wrote above: > Nevertheless, the net effect is as I have described above: some > (or all) of the meaning of X is preserved in the meaning of Y, > with nothing being added -- regardless of what interpretation > (meaning) we may subsequently assign to the abstract logical > forms of X and of Y. Well, the meaning is not "preserved" as much as it was arbitrary _assigned_. That is, for someone to declare that X -> Y, they first has to arbitrary assign a meaning, and in stating X -> Y, they are not talking about meaning the words might have had when spoken, but simply making a statement about what sort of meaning they choose to assign the words. So what it tells us, is not so much about some real meaning of the sentence, but instead, the highly artificial meaning assigned for the purpose of attribution truth to the statements int he first place, all in the context of discussing the truth values of natural language statements. > > When we limit the discussion to simple binary logic, then all this > > becomes trivial. It's well know exactly what X -> Y means by it's > > truth table so there is nothing there that needs 1000's of years of > > debate to understand or explain. > > Statements don't have truth tables -- formulas (abstract forms) with > variable elements have truth tables. It's important not to confuse > a statement X with the form of a statement X. Right, I was talking about the truth table for the -> operator, not the truth table for the statements X and Y. > Failure to maintain the distinction has contributed to the confusion > that has prevailed in this matter. > > > What people don't agree on, or fully understand, is meaning. But that > > has almost nothing to do with simple truth and logic. > > I agree that the relation of implication that holds (or fails to hold) > between statements X and Y bears only a remote relationship to their > truth or falsehood. I guess I think meaning has little to do with logic. The only way we can work with logic, (or math in general), is to define highly simple, and highly artificial meaning to our words, and once we have done that, and agreed to what we that assignment is, we can then have a worthwhile discussion about the logic of the statements. And such understanding of logic, has little use in natural language, except, when we are speaking a highly simplified, and highly artificial form of language (which we do at times when we need to communicate something that is highly structured and artificial - like math, and science, and engineering, etc). Outside of these highly artificial and structured usage of lagniappe, logic just doesn't have much of anything to do with meaning. -- Curt Welch http://CurtWelch.Com/ curt(a)kcwc.com http://NewsReader.Com/
From: herbzet on 23 Jul 2010 19:23 Curt Welch wrote: [...] Good post -- I'll respond in more detail later. -- hz
From: Vesa Monisto on 23 Jul 2010 20:46 "Curt Welch" <curt(a)kcwc.com> > ... > Outside of these highly artificial and structured usage of lagniappe, > logic just doesn't have much of anything to do with meaning. Jigs, tools, programs and more abstract *means* to handle meanings do not care much to which meanings they are used. Just therefore means are very usable. OTOH, without meanings there were no means. V.M. (Wittgenstein: "Meaning is in us(ag)e" ... of means? !)
From: herbzet on 25 Jul 2010 00:06 Curt Welch wrote: > herbzet wrote: > > Curt Welch wrote: > > > Simple logic is not a good tool for understanding meaning. > > > > But it is the reverse that is the case. > > > > Though we may have some doubt about what precisely it means to assert > > "John has a big ego", it is indubitable that this proposition includes, > > and hence implies, the proposition that "John has an ego" -- regardless > > of whether or not John has a big ego, or indeed whether he has an ego at > > all. > > Well, yes and no. I agree with the idea you are suggesting, but meaning is > actually always more complex than that. Meaning to me is defined in two > ways. It's defined first by the physical chain of events which caused a > specific language stament to be produced (the senders meaning), and it is > defined by the physical effect it has on the universe (the receivers > meaning). > > There is no (absolute) "generic" meaning to a statement such as "John has > an ego". Each time the words are generated by someone, it has a different > meaning - the meaning which is associated with why that person spoke those > words. And each time someone reads the words, it creates meaning in the > receiver. That meaning will never be the same twice even for the same > person (even though it will likely be very similar for consecutive times). > > We can talk about the average meaning in a society - that is, what the most > likely average expected meaning of someone speaking those words in that > language to be. But in truth, meaning is so highly complex, and so > different from person to person, that there's little truth in talking about > expected meaning. > > If I hear a teenage girl say "John has an ego" in a conversation she is > having with another teenage girl that seems to be talk about boyfriends, my > brain will make all sorts of educated guesses about what the meaning of the > sender was. I might guess the real meaning might be closer to "I wish John > liked me". > > In a different context, I might be watching a spy movie and the see the > hero utter the words "John has an ego", and in response, a voice activated > door opens. In that context, my best guess as to the meanings of the words > are closer to "I want the door to open" and don't have anything directly to > do with someone named John, or his ego. Heh, good illustration of your point. > To suggest we can understand the meaning of such a sentence totally out of > context, or that the words only have one "real" meaning, would be > pointless. > > So how, given this complex landscape of language, could we assign a truth > value to the statement? We basically can't. At least at not without > making a huge set of assumptions about the sender and the receiver and the > entire context that caused the words to be produced. And even if we make > all these assumptions, the odds that our understanding being very accurate, > is poor at best. > > The only way to get a truth value from a sentence, is to make up ARBITRARY > rules of truth that roughly fits the default or standard meaning of the > words. Not the real meaning when used, but the sort of meaning which gets > taught in class when attempting to teach a language or the sort of meaning > found in a dictionary. And only when that meaning is extremely simple and > generally clear, does it work at all - such as when we say "John has two > peaches". > > When then use logic to verify that "John has two peaches" -> "John has some > fruit". Right. > But we had to make many assumptions about the meaning, that often > can't be made in real language use. Right. Logic cannot be applied without the meaning of the sentences, at least in part, being arrived at beforehand, somehow or other. As I mentioned before, logic presupposes meanings, truth, falsehood, are somehow already present, at least in principle. > Only in an highly artificial > interpretation of language (one that can never be used in the real world by > real people unless they want to be considered idiots) can such logic be > evaluated. And only in that highly artificial interpretation of meaning, > does the truth of X -> Y imply Y includes X become even close to valid. > > Ok, so writing the above has allowed me to think about this some more. Let > me now respond again to what you wrote above: > > > Nevertheless, the net effect is as I have described above: some > > (or all) of the meaning of X is preserved in the meaning of Y, > > with nothing being added -- regardless of what interpretation > > (meaning) we may subsequently assign to the abstract logical > > forms of X and of Y. > > Well, the meaning is not "preserved" as much as it was arbitrary > _assigned_. > > That is, for someone to declare that X -> Y, they first has to arbitrary > assign a meaning, and in stating X -> Y, they are not talking about meaning > the words might have had when spoken, but simply making a statement about > what sort of meaning they choose to assign the words. So what it tells us, > is not so much about some real meaning of the sentence, but instead, the > highly artificial meaning assigned for the purpose of attribution truth to > the statements int he first place, all in the context of discussing the > truth values of natural language statements. In the context of discussing logic, no doubt it is convenient and customary to just assume that the meaning of a statement has been fixed already, and bears some relation to its logico-grammatical structure. This part your post has been mostly devoted to an interesting discussion of how meaning, and hence truth-value, is assigned to sentences in the first place, which is more properly the topic of this thread. As I mentioned previously, the broader question of this thread, of how truth or falsity is assigned to sentences in the first place, has not been my concern here. I wasn't really following the thread at all until I saw Wolf K's post, which I read because I generally find his posts illuminating. I seized upon Wolf K's invocation of "logical truth" to ride my favorite hobby-horse about the nature of logical implication; Wolf has since dropped out of the conversation, no doubt concluding that I am irrepairably confused -- story of my life, but I digress. [...] > I guess I think meaning has little to do with logic. The only way we can > work with logic, (or math in general), is to define highly simple, and > highly artificial meaning to our words, and once we have done that, and > agreed to what we that assignment is, we can then have a worthwhile > discussion about the logic of the statements. Right, and this is utterly commonplace, especially in the construction of foundations of mathematics, a project that has strongly informed logic since Frege, at least. > And such understanding of logic, has little use in natural language, > except, when we are speaking a highly simplified, and highly artificial > form of language (which we do at times when we need to communicate > something that is highly structured and artificial - like math, and > science, and engineering, etc). I agree that logic is a feeble instrument when applied to most natural language transactions. One need merely try to apply formal logic to today's editorial in the Daily Rag to see how little logic figures in ordinary discussions. I don't mean that in a derisory way -- logic just doesn't apply much in a typical editorial -- editorials, and most natural language conversations, are not meant to be proofs. > Outside of these highly artificial and structured usage of lagniappe, logic > just doesn't have much of anything to do with meaning. Ack! I disagree! That's just what I've *not* been saying! What, then, does logic have to do with? -- hz
From: herbzet on 25 Jul 2010 00:08
Curt Welch wrote: > herbzet wrote: > > Curt Welch wrote: > > > herbzet wrote: > > > > "The "logical" transforms of a sentence X to a sentence Y then > > > > are those transforms which preserve some (or all) of the meaning > > > > of X in the meaning of Y, without adding anything extra." > > > > > > The word "meaning" doesn't fit there in my view. > > > > You are not alone in that -- in modern logic we are used to > > abstracting away the meaning of propositions and dealing > > with their abstract forms -- that is, only the meanings > > of the "logical" elements of the propositions are retained > > for the purpose of analysis. > > Right. > > > Nevertheless, the net effect is as I have described above: some > > (or all) of the meaning of X is preserved in the meaning of Y, > > with nothing being added -- regardless of what interpretation > > (meaning) we may subsequently assign to the abstract logical > > forms of X and of Y. > > Ah, ok, you aren't just being sloppy, you are actually trying to say > something about the meaning once you have abstracted a truth value from the > statements and concluded that X -> Y. Something like that. What I'm saying that in the logical analysis of X -> Y, the truth-value of X and the truth value of Y are not needed and do not commonly figure in the analysis. What I'm saying is that the analysis consists, ultimately, of showing the inclusion of the meaning of Y in the meaning of X. If this inclusion exists, then the compound sentence X -> Y is held to be true, a logical truth, regardless of the truth-values of X and of Y. This is because if it is the case that asserting X is tantamount to asserting Y (along with possibly asserting some other stuff also) then if it happens that everything asserted in asserting X is true, then Y, being part of what is asserted by X, will be true as well. If our analysis proceeds by dropping the actual subjects and predicates of X and Y, replacing them with placeholding symbols, leaving only the "logical form" of X and of Y, then upon restoration of actual subjects and predicates to produce actual propositions X' and Y' -- *any* propositions X' and Y' -- it will be found, magically, astonishingly, that regardless of the truth or falsehood of the resulting propositions X' and Y', the meaning of X' will include the meaning of Y' -- and thus the compound proposition X' -> Y' will be accounted true and called "a logical truth". > Ok, yeah, I think I agree with your point though I might have to think a > bit more. That conclusion however includes a big assumption. That is, the > assumption that the truth value was correctly abstracted from the meaning > (which it probably never can be). Um, not clear here. As I point out above, the truth-values of X and of Y don't usually figure into the analysis. I *am* saying that the recognition of the inclusion of meaning between X and Y *does* lead to the assignment of "true" to the assertion "X -> Y". So, yes and no. -- hz |