RE: Cabbage; No, they are, and that’s why I want to develop it and in rereading the thread I must admit my carelessness here. Given my P and Q I would translate the tautology as (in a rather conversational tone), “Since I woke up on time this morning—and you know, if I wake up on time I have eggs for breakfast—I had eggs for breakfast.”
I never have used that. My background with anything in programming is Commodore BASIC, DOS batch files, Java, C++, assembly language for programming microprocessors, and programs used to design circuits (which are devoid of anything that someone would consider syntax, purely representational of the chips themselves, though you can construct chips like the person seemed to construct an operation in the second example).
As far as “A is A” goes, I’m not sure how you want me to take it. If we were discussing the alphabet, for example, I would take it one way, whereas if we were discussing essences or universals I might take it another. Were we discussing logic, I wouldn’t “take it” at all. Perhaps you can help me here: what does it mean (to you)? And by that I should say, try to answer this in the same way you would answer, “What does ‘I’ll be there at noon’ mean?” For shouldn’t they have the same type of answer if logic means something?
it’s pretty clear that this is exactly what you would be doing if you were to refrain from applying logic (or as you say here, reality) to your sentences.
anyway, i think i am beginning to understand that your definition of “meaning” comes only when specific instances are considered. that is to say, logic is too general to have “meaning” to you. i can agree with that, to an extent.
what i must say in response though, is that logic has meaning independent of english. english, on the other hand does not have meaning independent of logic. consider the sentence:
P & P->Q.
by this sentence, if we take it to be true, we know something about Q. and Q could be anything. to me, that’s the beauty of logic. any propositions you can communicate can be substituted here, and we will have learned something we may not have known about for whatever value we give Q.
suppose now, we consider the sentence:
it is raining, and if it is raining i will not go out.
without our present understanding/definition/whatever of logic, we cannot conclude anything about whether or not i will go out. this comes only when we substitute the propositions stated here (it is raining, i will not go out) into the appropriate rule of logic.
certainly the Ps and Qs have no independent meaning. that’s why they’re variables. but certainly the notion of a proposition (that which is not variable) is fundamental to logic. when we substitute propositions for variables, we achieve (what i take to be) your definition of meaning. to me, the fact that we can make that substitution and achieve meaning is meaningful.
so, to you (it seems to me), logic is the application of reality to language. to me, the beauty of logic is that it is universal for all languages. that set of valid relations between Language and Reality is logic, and its ability to stand alone gives it meaning that language lacks. to me, logic is reality.
it would be clearer to say: “Since I woke up on time this morning—and you know, I always have eggs for breakfast if I wake up on time —I had eggs for breakfast.”
No? This seems to imply that we’ve always implicitly known logic.
And I can see why some might say that. Where did I say it earlier, I can’t find it, but I said something like, “We might as well say reality approxiamtes logic/math”.
the basic tenets of logic are considered implicitly knowable. they can be reduced to nothing further.
i think i would prefer to say that logic/math approximates reality. or seeks to understand it in a finite and concrete way. but it is just a matter of perspective i guess.
Sorry for being kind of dense here, but could you clarify that? It’s just that, going back to my post, I’m not sure what “no” and “they are” are referring to. Anyway, yes, I would interpret P=>Q as “If I wake up on time, then I eat eggs for breakfast” (your interpretation of the tautology is a little different from mine, the thing that’s bothering me about yours is that it seems to have slipped into the past tense: “Since I woke up on time this morning?and you know, if I wake up on time I have eggs for breakfast?I had eggs for breakfast”. And yeah, as Math Geek said, my main concern was the difference between P=>Q and (P & (P=>Q))=>Q.
Anyway, in the mean time, here’s something to try (and I hope that I’ve been understanding you correctly, otherwise this may be a little pointless, but may be a fun little logic game nonetheless). Some of your comments:
I disagree with, and led me to think that the following might be interesting to bring up. The “game”:
What is the negation of the following sentence:
In every culture there is a taboo, such that any person violating that taboo will be punished.
Or, if you prefer to remove all references to “logical operations” (negation), rephrase it as: If the above sentence is false, what do you know must be true as a consequence?
Some questions to think about while doing it:
Is it easier to find the negation to this sentence through an analysis of meaning, or an analysis of syntax? (And, in general, do we always find errors in English sentences through an analysis of meaning, or is it sometimes easier to find them through analysis of syntax?)
Can the negation be found in a systematic way, or must we pay attention to what the terms mean? (Now that I think about it, here I may have been missing your point in your second quote above, but what the hell, I’ll leave this here anyway).
Actually, the more I think about it, the more I think I may be missing your point here (with the last question, anyway), but now that I’ve written it, I don’t really want to scrap it now, just in case.
One final comment, however, which I think may be more to the point:
Finally, my point here would be that (P & (P->Q))->Q does have meaning–it relates to all sentences of this structure, and logic tells us that they all are necessarily true.
One way to think of (part of, anyway) logic is the following. Logic partitions all possible propositions into classes with the same logical structure. Logic then provides a map (a truth table) which, given the truth value of the components of the proposition, gives us the truth value of the proposition itself.
The “meaning” of (P & (P->Q))->Q, along with its map (its truth table), is that any sentence with that logical structure is necessarily true (since it’s a tautology). (Note that, by itself, I’ll agree with you that (P & (P->Q))->Q has no meaning–it’s important to also consider what logic tells us about the truth value of the sentence). The meaning of any logical sentence or logical argument (along with their maps to their truth values ) is the information it provides about all other sentences/arguments with the same logical structure–namely, their truth values.
Anyway, I think I’ve rambled on quite enough here, I hope there was something worthwhile and I’m not still missing your point.
Finally (really, this time), regarding the sentence, “In every culture…”; bonus points to anyone who can tell me what common mathematical definition has the same logical structure.
Anyway, the “RE: CAbbage” was meant to respond to Math Geek… oops! Sorry 'bout that. “They” are your concerns. Dontcha love me using terms as variables when I’m complaining that free variables render a sentence meaningless? Ok, so maybe it does go farther than that… but, the irony hasn’t escaped me.
Oh, syntax, definitely. Once I know what the meaning is. And I can’t “check” the negation is appropriate without knowing the meaning. This isn’t quite so true in English where double negatives yield a positive, but in some languages double negatives do not yield a positive (and some would say they don’t in English, either, only that a double negative is bad grammar).
Oh, please don’t get me wrong, I agree that syntax is a part of every language. I don’t want to claim that we can form sentences by taking words and tossing them together willy-nilly. And it does depend, sometimes looking at the syntax is a way of parsing meaning. It might even happen that we cannot come up with a meaningful interpretation of a sentence or claim given such parsing (as often happens in philosophical arguments, for example). But logic is pure syntax, and if it had meaning, all sentences which follow a logical form should be meaningful. Indeed,
I feel that this comes from the tendency to assume a direct relationship between English (or whatever language, I’m not trying to be language-centric here) and logic that isn’t there. That is, structural analysis of English is not the same as structural analysis in logic. Consider thinking that “it takes a hammer to build a house.” But suppose I instead use a brick for a hammer (which is, in principle, possible). Now it seems we really only meant to say it takes a hammering action to build a house. And I would say, of course! But this doesn’t make a brick a hammer.
Similarly, should we use logic in analyzing English (should we use syntax in analyzing meaning) doesn’t mean logic (i.e.- the symbols and rules of) means something. Not that it is incidental, accidental, or anything of the sort. Again, I think we really do use logic for this purpose quite often. But the validity of English is not determined by logic (if anything, many debates about fledgeling logics like deontic or modal logic show that logic is determined by English). The test of the truth of English statements cannot be reduced to syntax, whether we need a syntax there to test for truth in the first place or not (that is, again, words may not be combined willy-nilly). If it were, physics and English would be sister fields of inquiry.
Ah! So close. All possible propositions… that already have their own meaning.
Perhaps you might then say, but logic adds the meaning of truth to this. And I might ask, “Then why would we need to test the conclusions?” (Among other questions which I have raised in the thread so far.) And, note, when don’t we test the conclusions? —when it is only an analysis of the syntax; that is, when we are simply doing logic or math (unless we consider checking our results testing, but I’m not sure I can stomach that).
I want to take the time here to thank people for talking this subject seriously, and not as some inherently self-contradictory attempt at… whatever.
Another note. General Relativity utilizes an “extra” dimension in its calculations to explain gravity. The Kaluza-Klein theory utilizes another on top of that to explain electromagnetism. From what I have read those mathematical theories do accurately predict behavior, even if their methods have not yet been able to unite all forces and explain everything (and, indeed, the K-K theory is considered a dead end now, from what I understand). But the question which haunts us is: do these other dimensions exist? If you think math and logic are meaningful, are you compelled to answer affirmatively? Why or why not?
Again… thanks. I don’t want to show a disrespect for logic. But my motivations for this lie in the dismissal of arguments I often see, here and elsewhere, based purely on logic. It has never sat right with me, and this is my attempt to flesh it out.
Lib, you flatter me. You have also had a profound effect on me, not the least for rekindling my interests–and stimulating me to new insights–in formal logic and epistemology. Thanks.
Fatwater, thanks for the kind words, especially since you figure prominently on the list of people to whom I have unfulfilled obligations. Erl is on that list, too, actually, but what I owe him is less obscenely tardy than what I owe you.
Now to the arguments . . .
Lib
“Love” and “the goal of my moral journey” are both fine examples of definitions. Do you find the tautologies of logic to be somehow different in character than what you mean when you say God? How?
I believe you had left the thread in question before I stumbled upon it. I would be interested in your response to the points I raised concerning that modal proof, if you care to revive that discussion. My post can be found here.
Again, I have to ask whether you find this experience with respect to your concept “God” to be somehow different than with respect to other definitional tautologies.
erl
And I’m saying there is no gap.
I’m sorry, but I cannot make those two statements into a coherent expression. I could restate, here, my objection to the phrase “the meaning of logic” (which I maintain is a misapplication of English), but quite apart from that I cannot imagine what theory of meaning you are using to say: The meaning of A is it’s application to X.
A is intended to stand alone from X.
The statement is sytactically correct and semantically free of contradiciton. It is not “wrong” as a statement of English. The truth value of thestatement is “false”, but that fact is independent of any valuation which can be paced on the statement from within the English language.
You do see, I hope, that you are claiming that logic is inherently meaningless because the truth value of logical statements can be found only by reference to elements outside of logic while simultaneously claiming that English is meaningful because the truth value of English statements can be found only by reference ot elements outsode of the English language. What I don’t see is what reasoning you use to justify an alternative (and contradictory) theory of meaning which applies to the structures of logic but to no other languages.
Which tautology? More than one has appeared in this thread. But my point was that rules of syntax exist to assist interpretation of meaning. For instance, “parentheses around a group of symbols establish a precedence for applying operational symbols within the grouping before applying operational symbols outside the parentheses.” Really, I don’t think there’s much controversy in this position. Syntax is designed (or has evolved) in order to foster clarity in communication.
I don’t know. I suspect that it is because you are applying a different standard for meaning to formal logic than you apply to any other language. Why you are doing that, I confess, is a mystery to me.
And given valid logical structure the truth of a conclusion is not guaranteed. Syntactic examination of a logical argument addresses validity, not truth.
I understand that you are making the distinction. I do not understand why you are doing so or what theory of meaning you attempting to explicate. I’ve asked some of these questions already, but let me set them out clearly:
[ul][li]How is meaning determined in an English statement.[/li][li]How is meaning determined in a formal logical statement.[/li][li]How do these two theories of meaning relate to each other, if at all?[/li][li]Are formal logics the only languages which fall under this second theory of meaning?[/li][li]What defining characteristic(s) of a language determine which theory of meaning applies to statements within that language?[/li][/ul]
The proof is invalid in arithmatic because the symbol grouping (X/0) has no meaning in arithmatic. This is equivalent to saying that dicision by 0 is syntactically forbidden in arithmatic.
The conclusion is also false for most number sets, but that fact is not necesarily dependent upon the (in)validity of the proof.
Consider the english statement: Spiritus is a smart guy so whatever he says about logic and meaning must be true. How might one determine the truth value of the sentence element “must be true”?
As to the second: poor logic :D. If logic can be useful in deciding the truth value of some elements in English, then we can conclude that determining the truth value of some English sentences is more difficult for those who do not understand logic–not that one could not learn English without first learning logic.
I might be wrong, but I think you might be getting hung up b some alternative connotations in the word “represent”. Try thinking of it as “Meaning as use . . . symbols that indicate.” Or not. Just a thought.
Yep. I will consider this an answer to “How is meaning determined in an English statement.”. Now if I could just understand why you feel compelled to add, "except when you are using a language of formal logic. . ."
No, it cannot be. But I think you make a mistake to reference “truth” here. I think you pobably meant to focus on “valid”.
Of course, the statement isn’t true for valifity, either, though it would be if you had left it at “the content of free variables”. The context of the sentences within a logical proof, with regards to tests of validity, is the set of assumptions for that proof and the axioms of the particular formal logic.
The answer to that question, of course, is trivially, “yes” for some meaings of “crying”. The question, “what does this crying mean to the person on the film” is more problematic. I’m not sure what relevance one can glean from that and apply to questions of meaning in other languages.
As is the case for logical propositions. Though I think a more precise phrasing (for both languages) would reference “knowledge of the truth values of the sentence’s constructive elements” rather than “appeal to the content of free(ish) variables.”
In English, if you know that the person on the film was truly sad at the time of filming, then many questions about “what does the crying mean” can be answered. It doesn’t necessarily mater whether the person was a method actor recalling a past emotional experience to produce tears: the meaning of those tears are the same.
In logic, evaluting the truth of P->Q requires only the truth values for P and Q, not any appeal to the content of P and Q.
I disagree, but I have yet to hear your theory of meaning in formal logics. Clearly free variables are used in logic. Clearly they indicate an idea which can be successfully interpreted by someone reading a logical proof. So, under “meaning is use” they would have meaning.
Math geek already provided one, though I am not sure you understood it. “A is A” is an English sentence which contains only free variables and a relationship. “P->Q” is a logical statement containing only free variables and a relationship. The two sentences are analagous.
It indicates a relationship.
You use it to indicate a relationship.
The truth of said relationship, of course, depends upon the truth values of the constituent elements, which depends on the context(s) in which it is applied.
BTW, if you want another example, how about: something exists.
“I’ll be there at noon” indicates a relationship between a specific individual, a specific place, and a specific time.
You use it to indicate said relationship.
I’m not sure that I agree that the relationship between meanings of these 2 English sentences carries necessary implications for the meaning of (IN, damn it–don’t you remember writing: *Of course a language doesn’t have meaning. *) a language other than English.
Again, the same is true for formal logics. You again seem to be confusing truth with validity. Validity in formal logic can be determined solely from syntax and context. Truth cannot be.
I don’t know.
No.
Because the map is not the country. (Didn’t we dance that tune already? ;))
No, seriously, what would it mean for these dimensions to be real? Are length, breadth, and depth real?
Math is a language. Unlike the natural languages, it’s very well defined: all representations have very precise meanings. That precision is what makes math powerful, but the natural languages can be just as powerful, if we’re careful to use them precisely.
All languages describe the universe we live in. It doesn’t really matter if we describe a phenomenon with an equation or a sentence, as long as the description is accurate.
If the behavior of the universe can be accurately described by a model with n dimensions, great! Until we find phenomena that are incompatible with the model (and it’s important that we look for unexpected and unexplained phenomena), it’s as real and valid as any other model that explains things.
Oh, I see it. What you’re not seeing is that without reality there could be no English; i.e.—a tautologous dictionary is not enough, English is underdetermined. Logic holds in a vacuum, there would just be nothing to apply it to.
Same theory of meaning the whole time. Logic never appeals to reality for truth.
Your specific questions:[ul][li]How is meaning determined in an English statement? Primarily through how the statement, or the terms of the statement (in the case of a previously unencountered statement), are used otherwise. English does not succumb to a first cause, and is not tautologous, because to teach use is to point to reality, not back to English again. In this way, many words and phrases in English represent, though there are terms which do not represent anything (pure use-words). Their use is still taught by an appeal to reality, however, in the form of an appeal to context. []How is meaning determined in a formal logical statement. The same way as English. But when we leave the tautologous constraints of logic, our truth is no longer assured (as it was before) without the appeal to reality. In such a case, it is not clear we are really dealing with logic anymore, though the symbols are the same, and their manipulation is the same, suddenly the truth of the statement is underdetermined (that is, we have to appeal to reality). But if we have to appeal to reality for the truth instead of it being guaranteed, we are not operating within the scope of logic, but English. Whether we choose to speak English or not is quite irrelevent; the meaning came from reality (this is English’s scope).[]How do these two theories of meaning relate to each other, if at all? There is and cannot be a formal relationship. There is, at best, an analogy. []Are formal logics the only languages which fall under this second theory of meaning? Anything that is completely determined would, yes. By that I mean, a language is completely determined if the truth of the statements expressed in it can always be found (if found at all) within the confines of the language itself, and, if the truth cannot be found within the language, there is nowhere else to go to look for it (in the case of paradoxes and incompleteness, for example). []What defining characteristic(s) of a language determine which theory of meaning applies to statements within that language? I am not proposing two theories of meaning, just one. The defining characteristic which determines if something is meaningful is that it appeals to reality as you find it. If you find that reality is completely determined (that is, you are an extreme rationalist) then logic is meaningful under my understanding of “meaning”. I am not an extreme rationalist (I would imagine most people are not).[/ul][/li] The Vorlon Ambassador’s Aide
I would say: all languages can be used to describe a universe. I would say that is pretty much the definition of a language.
As I read it, this is, in fact, the scope of the Tractutus. But this is also why the Tractutus is so dogmatic, for it proposes a symmetry of meaning between objective reality (and if you’ve read it then you know what I mean by “objective”: composed of objects), and meaningful statements in any language, including logic. That is, the relationship between a meaningful logical sentence, a meaningful English sentence, and the situation those describe, is ontological equivalence. The. World. Is. A. Logical. Picture. Of. Facts. There is no core distinction between the three possible modes of expression (factual reality, linguistic reality, logical reality). Logic does not represent.
I break from the Tractutus where linguistic reality’s truth is determined by reality, and logical reality stands entirely distinct. The edge between English implication and factual existence is blurred; the edge between logical implication and anything is quite clear. When we are making our analogy, we seek to understand when to apply logical expressions in certain cases. This makes logic meaningful, but it also de-formalizes it and forces it to be underdetermined. As I then say, we might as well be speaking English, for in any case we must appeal to reality for truth. The power logic had to hand us truth automatically is gone. Our desire to create truth out of thin air might put it back, though.
I am unsatisfied with this answer for numerous reasons, not the least of which becasue it does not address the confusion which I expressed while pretending that I suffered a confusion which I did not express.
Logic’s truth is not found in the real world, it is found in itself. All meaning comes from reality, and though we get meaningful logic statements by applying them to reality, the truth as described is then no longer contained in logic itself. I went into more detail above.
The meaning of any language is its application to reality. Yes. Logic is not created for the sole purposes of application to reality. Yes. The two sentences are not contradictory. Logic is intended to be seperate from reality. If it wasn’t, the declaration of truth from within logic itself would be superfluous. The only way it isn’t superfluous (without accepting some kind of extreme rationalism) is to note that, hey, maybe logic stands appart from reality, even if we may apply it to reality by means of analogies.
Logic does not declare truth internally. Logic only derives truth from asserted or externally observed truths.
Nor could there be any language capable of communication, including logic.
Axioms and definitions must be expressed with references external to the logical structure they define.
Logical validity holds in a vacuum. So does English validity. If you are going to hinge “meaning” upon truth, then this distinction is quite important.
Why? Truth values in logic and English are both asserted through external reference or bald assumption. Why do you find this in scope for one language while out of scope for another?
Neither of those conditions hold for logic.
Causal relationships exist.
P -> Q
Are either of those statements meaningful?
Logic never had such a power, nor does it pretend to such power. The loss you find so meaningful does not exist (which makes it meaningless, right?).
As far as logic is concerned, all truth is asserted. There is no qualifier for, “Hey, this one really happened.” And indeed, whether we logically derive the truth of a conclusion or not doesn’t make such things true. In fact we do return to reality to test them.
As they stand alone, not that I can see. I can imagine a series of contexts in which they would be meaningful, yes.
I disagree. But this is because I differentiate formal logic and spoken language. If I thought that English could really exist as a tautology then I might agree. An answer to, “How do you know that color is green?” might be: “I have learned English.” An answer to, “How do you know that has a color?” is not, “I have learned English.” It is also not, “I have learned logic.” It is also not, “I know English and logic.” Under the framework of the Tractutus it is, of course, but that’s from the Tractutus’s symmetry between expression and fact. It is the rationalist, idealist dream-book.
While I say there is no bridge, you say there is no gap. How do you avoid collapsing into the meaninglessness of everything in the Tractutus? What premise (or premises) do you not accept? (If you recall, and this is a “just wondering” thing).
Now you have confuised me. You object to the way logic deals with truth because it is:[ul]
[li]found in itself. [/li][li][declared] from within logic itself [/li][li]is guaranteed[/li][li][logic] hand us truth automatically [/li]but
[li]all truth is asserted [from outside of logic][/li][li]whether we logically derive the truth of a conclusion or not doesn’t make such things true[/li][li]we do return to reality to test them[/li][/ul]
The last 3, I believe, would hold for you if we substitute the word “English” for “logic”.
Really? “Causal relationships exist.” does not appeal to reality as we find it? The grouping of words is not consistent with how the terms are used otherwise?
I’m afraid I am finding your view of “meaningful” ot be less and less consistent/useful/relevant to my own experiences.
No shit???
Why didn’t I think of that.
:smack:
Perhaps you would be so kind as to supply a counter example in English (or logic, though by what follows it apperars that English is where you disagree).
Why in the world would that make you more likely to agree that truth values in English are, "asserted through external reference or bald assumption."
And this make English meaningful and logic meaningless because . . . ?
Because, like Wittgenstein himself, I reject the theory of language found in Tractutus pretty much en toto.
Meaning is usage.
"Causal relationships exist."
and P -> Q
both have meaning to me. Both statements consist of symbols whose patterns of use I recognize and whose semantic content I can interpret.
all truth is asserted [from outside of logic] ?? There are ways logic deals with asserted truth. Since an expression in logic is not identical with corresponding statements in English, the truth doesn’t come from anywhere. It is simply asserted. We have reasons for putting it there. Logic does not care about those reasons. Those reasons are also not translated into the logical form. The truth of a free variable is simply asserted in a way that logic allows assertions to be.
whether we logically derive the truth of a conclusion or not doesn’t make such things true Yeah, but maybe I should clarify. The truth that logic declares is simply logical truth. The application of logic to English statements is also logically true, if one allows for this sort of mapping (and I would imagine so). This doesn’t make the English statement true; what makes English statements true are a correspondence to reality.
Consider a theory developed to explain particle interactions. This theory is intensely mathematical. While the theory is worked out, the mathematician/physicist finds a solution to some equation. It is true that this is the solution to the equation, that is mathematical truth. This truth does not guarantee a successful description of particle interactions. The gap looms before us in an instant.
Of course. Which is why I said, we aren’t dealing with logic anymore when we are doing such things, we are using an analogy. We might as well be using English, whose words’ meaning are only ultimately determined by reality.
Because I’m saying the truth of English propositions do not come from assertion, but from correspondence with reality. Again, the example of mathematics above should shed light on this. Whether or not the solution to a differential equation can be applied as successfully as what led to its construction is irrelevent to the question of the truth of the solution. In English, there are no such statements that are true regardless of the circumstances of their use.
In a large number of cases, yes. But the opposite is not necessarily true: because something is used does not make it meaningful.
But, hey, let’s run with it, Spiritus.
Meaning is use for “P->Q”. How do you use this statement? Does it mean the same thing when doing pure logic work as when trying to analyze a specific application of it to a series of statements?
Me: Just because we find meaning in one use doesn’t mean it has to be there (or be the same) meaning in another. I mean, that’s like saying all games have to have something in common else we wouldn’t call them all “games”, right?
erl
Are you of the opinion that all statements in mathematics or formal logic must be translated into English before being “tested” for correspondence to reality?
What???
You listed some characteristics about truth and logic which you argue make logic meaningless.
I note that those chartacteristics also hold for English.
You respond with the above.
What???
Um, that would be in the external reference, Bob.
If we are talking about a language construct, then yes, it does.
Meaning is use, use meaning. That is all
Ye know on earth, and all ye need to know.
To indicate that a causal relationship exists.
I’m not sure I understand your second case. If I am analyzing a specific series of formal statemnets, how is that different from doing “pure logic work”?
Certainly it does not have to be the same, but if a language construct is used in a manner which is communicable then it has meaning.
What they have in common is that we call them games.
