If A also B, A also B
Or rather,
If A also B also A also B
I’m trying not to make modus ponems temporal or spatial, because it becomes even more obvious that purpose is an unsupported imbedded concept. The point here is that Modus Ponems ACTIVELY turns the universal into the existential without supporting the claim.
Is this considered the only way to find validity?
You misunderstand modus ponens. Implication does not turn a universal into an existential.
A -> B
The above implication is independent of the actual existence of A or B. Consider, for instance, if it is a unicorn, then it has a magical spiral horn. That implication has no consequence for the existence of either unicorns or magical spiral horns.
Now, if a unicorn does exist, a fact about which modus ponens remains utterly silent, then the implication does force us to conclude that a magical spiral horn exists. That existence, however, does not derive from modus ponens. The implication merely describes the consequence.
Similarly, if I say “God exists because I have felt Him in my heart”, it would be incorrect to say that a phenomenological event has turned God from a Universal to an existential.
Maybe I can understand what you’re saying better if you can provide me with imbedded concepts that are different than the ones I used. If you cannot, I’ll try to parse what you’re saying “as is”.
I consider it a given that:
then = temporal
therefore = temporal or spatial
So, you have two types of modus ponens
Temporal-Spatial
or
Temporal-Temporal
By using the concept “also”, I find that this isn’t as much an issue.
Like with temporal, you have the problem of something not existing, which is declared consequently as existing, because the duration of the “if” clause isn’t restricted.
The “if, then, therefore” kind of modus ponens faces these sorts of issues quickly – where the implication is being forced by an unsupported, imbedded operator.
No: there is no temporal or spatial aspect to conditionals. You’re misinterpreting the use of the word “then”.
What do you mean by ‘embedded concepts’, and how does it matter?
“then” is definately a temporal term.
This then That “This first, That follows”
Time is implied for an anthropomorphic concept like “follows”.
When using therfore in the spatial sense, you have
X (is) there for (the purpose of) Y
purpose is imbedded without support.
Therefor in the temporal sense is “consequentally”
Let’s look at the temporal-spatial mix, and you’ll see what I’m saying…
If X, consequentially Y, X is there for the purpose of Y
It doesn’t follow for example that a person who died 100,000 years ago is there or even was there for the purpose of me – this is unsupported assertation. It’s imbedded in Modus Ponens.
Maybe it’ll help you if I say that I’m trying to flush the “folk psychology” out of Modus Ponens. maybe it’s already been done, and you know this… so far, I haven’t figured it out.
As Hansel posted, there is neither a temporal nor a spatial element of modus ponens. The “embedde” meaning to which you object exist only in your misreadings. Consider:
[ul][li]If Socrates is dead, then he is not breathing.[/ul][/li]The implication holds, but it is not correct to say that because of the implication Socrates died before he stopped breathing.
Please do. It would be refreshing to see you actually respond to what I have actually said.
If there is a particular element in my language that you have difficulty understanding, please say so. I will be happy to clarify.
If you are expecting me, however, to supply different concepts that are “embedded” in modus ponenes, then I am afraid you are going to be disappointed. Logical implication is the only concept carried by modus ponens.
“Then” is a temporal term only in common usage. While you’re correctly identifying an ordering between A and B in “if A then B”, the ordering is not temporal, it’s strictly logical, stating only that A is a necessary condition for B, not that A precedes B in any real-world sense. Whatever temporal meaning you can glean from examples is circumstantial.
This can’t be argued from examples or from common usage. We’re talking here about the definition of conditionals in symbolic logic.
Not quite. Actually, it’s backwards.
While we can say that “A necessitates B”, it is not corrct to say that “A is a necessary condition for B”. Rather, A is a sufficient condition for B. Or, alternatively, B is a necessary condition for A.
In your characterization of modus ponens, “If A also B also A also B”, you have three different “alsos”. The first is material implication, the second is conjunction, and the third is provability. We might write
(A -> B) & A |- B
Where “|-” is my attempt at an ASCII turnstile.
There are some problems with material implication, and various attempts to make it match the conditional of our informal discourse. Essentially, these all amount to a different notion of semantics than truth tables (material implication is the only two argument boolean function that comes close to our idea of the conditional). These include strict implication, conditional logics, intuitionist logic, relevant logic.
There are also logics where modus ponens (for whatever sort of conditional there is) does not hold for one reason or another. These include some modal logics, some conditional logics, first degree entailment, many-valued logic, and fuzzy logic.
Finally, another problem with modus ponens is that if we want to work backward using the rule as the last step in a proof to conclude B, we have to guess a suitable formula A. This means that automating deduction is not syntax-directed.
Graham Priest has a neat little book entitled An Introduction to Non-Classical Logic that is about the various attempts to fix up the classical conditional.
Whoops, you’re correct. “If A then B” means the B is a necessary condition for A, not the other way around, as I put it.
I thought this was an excellent counter-example that caused me to certainly re-think the necessary temporal usage of “then”. The only question in my mind from this point, is that it doesn’t have the form of modus ponens that I was encountering in my research… namely
If X then Y, X therfore Y.
It seems that you used only the anticedent, and what struck me was that there seems to be something forcing an existential quality into the consequent, that’s imbedded - to force the implication in an unsupported way. In my general thinking (correct me if I’m wrong), the comma symbolizes whatever “therefore” represents… which oddly enough struck me as funny in the way of “Just because you say it twice doesn’t make it any more true!”
Now as Newton suggested, there are three seperate underlying operations ocurring for the three equal symbols (material implication, conjunction, provability). The symbols “became” equal when I used “also” in place of “therefore” and “then”, which to me were spatial and temporal respectively.
Then I ran it for the comma as well.
So there definately is an imbedded operation occurring here that the symbols themselves don’t refer to… probably the same thing that bothers me about “tautology” as I’ve encountered it.
“a=a” rather than “a=a=”
I’m assuming that this recommended book will help answer this query. It certainly heartens me to think that this isn’t considered the ONLY method of asserting proof (aparently modus tallon relies upon modus ponens) - to hear that it doesn’t work in other argument forms that are themselves consistent.
That’s because I was not making an argument (no conclusion) just showing that the structure of implication carries no temporal component.
I have no idea what you are getting at, here. If I complete the syllogism to conclude that Socrates is not breathing what exactly is the existential that you feel has materialized out of the modus ponens?
Well, to enhance slightly N-M’s explanation, your characterization is also inaccurate in yuor placement of “If”. You wrote:
[ul][li]If A also B also A also B[/ul][/li]But the structure of a modus ponens is either:
[ul][li]A (implication) B (conjunction) A (QED) B[/ul][/li]or, if the truth requirement for all premises is expressed rather than implied
[ul][li]If {A (implication) B} (conjunction) If {A} (QED) B[/ul][/li]
More importantly, though, the three symbols are not equal as expressed formally. It was only your chosen parsing into natural language that created a possibility for confusion of meaning.
No, the comma is simply the appropriate English grammar punctuation for a dependent introductory phrase of more than a couple words. It is an element of the expression of modus ponens in natural language. It has no significance in the formal structure.
No. I still am unclear what “imbedding” consumes your thought. Please be so kind as to provide some explicit examples. I suspect that you are confusing the consequence of binding a variable with the structure of modux ponens, but since you have yet to introduce the classical quantifiers I am hesitant to push that conclusion.
Almost forgot:
“=” standardly represents the two place (polyadic, requiring two arguments) predicate “identity”. It is syntactically incorrect to write “a=a=”.
If it helps make things clearer, tautology is often expressed as implication, too. A->A.
I think the problem, olanv, is that you’re trying to interpret the standard logical terms as if they meant just what they came from etymologically. It’s true that the meaning “at a later time” for “then” is older that the logical sense of the word. It’s also true that logical meaning for the word “if” is not its original meaning. It probably once meant something like “suppose that” “on the condition that.” It’s also true that “and” once meant something like “next to, facing, on the edge of” if you were to trace its etymology far back enough. It’s furthermore true that “therefore” meant “because of that thing” at one point. So you’re trying to say that
IF ( (IF A THEN B) AND A) THEREFORE B.
should be interpreted as implicitly containing the etymological roots of all those words, some of which are temporal and spacial.
The standard reply to this would be to say that words mean just what they do now, and to claim that they mean something that they meant previously (and in this case it’s long ago, at least hundreds and perhaps thousands of years ago) is to mistake etymology for definition. The standard counter-reply to the standard reply is that people who use the words “if,” “then,” “and,” and “therefore” only think that they’re not using the etymological roots of these words. They are actually implicitly using temporal and spacial metaphors, but they refuse to acknowledge that they are doing so. I presume that that’s what you mean by calling these “embedded concepts.”
I guess the reply to this is that if you take the etymologies for the words in any statement, you can discover a lot of old metaphors that the people who make this statement would claim are not part of what they are saying. Indeed, you could take the words in your posts, research their etymology, and discover a lot of old metaphors in them. Where’s the limit here? How far back does one have to trace the etymology of one’s words?
In a way, that is the entire point of symbolic logic.
It allows us to emply pure relations of our own devising, without the linguistic complications arising from the use of new words. I’ve long said that it is a rookie mistake to “translate” the symbols into English before interpreting them. People resist this notion, but that’s symply because we are more facile with words
If instead of symbols, we used strictly defined neologisms, I suspect their objections would vanish. It is quite common in science to use invented terms that mean “something akin to, but in many ways distinct from” a common concept. Quantum mechanics, for example, should never be mistaken for the analogies we use to describe it to beginners, and real understanding only comes when you are no longer completely dependent on the analogies.
Otherwise, it would be impossible for us to formulate any new concept or principle. Physics would be inseparable from “good and evil spirits”. “Maxwell’s demon” and “Schroedinger’s cat” would necessarily incorporate some actual properties of demons or cats, rather than being exemplars of a principle.
I’ve been trying to think of how to answer this question, but I don’t think it really even makes sense from my standpoint. I’m a strict formalist, meaning that I believe that math is basically a set of rules for playing with marks on paper. So modus ponens is just a rule saying that if I have one sequence of symbols, I can write down another sequence of symbols. The fact that it preserves some property called “truth” is interesting, but hardly mystical. I could have some rule that preserves “superitude”, and that would be just as interesting to me.
Well, in any sort of situation where you’re using math, all you’re really doing is manipulating numbers according to a certain set of rules. However, you as the person overseeing or performing the calculations realizes (hopefully) that each number represents something in the real world that you’re trying to find out.
Some people think of it in terms of the numbers on the page, some people think of it in terms of the real world.
Yeah, this discussion is interesting because it pulls in Mathematicians, Philosophers, and Computer Geeks and all of them have a different way of representing the same concepts. Makes it a little harder to communicate, but it seems like we’;ve been managing well enough.
I’ve always been under the impression that IF, THEN, OR, AND, NOR, NAND, XOR, and whatever the hell else logical operators that various people in various professions will use are simply technical terms. In Symbolic Logic, they aren’t even words but merely linking symbols that have a very specific meaning. In any case, their normal, everyday usage should be banished from your mind when you’re working a logical problem.
dakravel writes:
> I’ve always been under the impression that IF, THEN, OR, AND, NOR, NAND,
> XOR, and whatever the hell else logical operators that various people in
> various professions will use are simply technical terms.
Please note that when I wrote “The standard counter-reply to the standard reply is that people who use the words ‘if,’ ‘then,’ ‘and,’ and ‘therefore’ only think that they’re not using the etymological roots of these words,” I was not agreeing with that statement. I was merely going through the various arguments back and forth in order to discover if there’s anything worthwhile in olanv’s OP. Everyone agrees that most people who use logical operators professionally (philosophical logicians, mathematical logicians, computer scientists, etc.) think that these logical operators are simple technical terms with no necessary connection to their etymological roots. I was trying to look at this from olanv’s point of view. I think that olanv would claim that most people who use words like “if,” “then,” “and,” and “therefore” do have some perception of the temporal and spacial metaphors that are found in these words’ etymologies. My reply to this point was that if you insisted on looking at the metaphors in the etymologies of every word you use, you would find a huge amount of dead metaphors which are (in olanv’s theory) “embedded” in these words and thus still at least unconsciously at the back of the mind of anyone who uses these words. But there really is no reason to think that most people are in any sense still using these metaphors. If you did think that they were, why would you only pick out a few of the dead metaphors in all the word we use?