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Logic/causality puzzle

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Onar Åm

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Logic and causality are intimately related. If we know that A causes B then we can also say that A logically implies B. Thus, A=>B. However, consider the following logical equivalence:

(A=>B)<=>(~B=>~A)

If A causes B then A=>B, but this is equivalent to saying ~B=>~A. Is this then the same as saying ~B causes ~A?

What makes this so tricky is that in order for A to cause B, A must be an event that precedes B in time, but is it meaningful to say that ~B precedes ~A in time? If not, is it still valid to say that ~B *causes* ~A since A causes B?

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Logic and causality are intimately related. If we know that A causes B then we can also say that A logically implies B. Thus, A=>B. However, consider the following logical equivalence:

(A=>B)<=>(~B=>~A)

It is possible that A&C=>~B , and so ~B does not mean ~A. It might also mean A&C.

For example, if I drop an object it hits the floor and makes a noise. But if I drop an object and also put a pillow on the floor, then noise doesn't happen. Just because the noise didn't happen you can't conclude that necessarily no one dropped the object.

But anyway, what is the point of these symbols? What's wrong with English?

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Logic and causality are intimately related. If we know that A causes B then we can also say that A logically implies B. Thus, A=>B. However, consider the following logical equivalence:

(A=>B)<=>(~B=>~A)

If A causes B then A=>B, but this is equivalent to saying ~B=>~A. Is this then the same as saying ~B causes ~A?

What makes this so tricky is that in order for A to cause B, A must be an event that precedes B in time, but is it meaningful to say that ~B precedes ~A in time? If not, is it still valid to say that ~B *causes* ~A since A causes B?

A=>C is just shorthand for ~(A&~C). Nobody would confuse the latter with "causality".

This kind of formal logic has nothing to do with causality. Modal logics attempt to get at some notion of "material implication".

Honestly though, I don't know what kind of meaning "cause" can have outside of a physical theory.

Edited by punk
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We recently had a thread where the topic came up, here. As you can see, the issue of what "⊃" means or is good for is a central issue. In the symbolic formalization of propositions according to certain contemporary views (not universally accepted), "⊃" is used for a farrago of concepts. One of those concepts is causation, which does have a temporal relationship. But it is not the case that every proposition embodying the connective "⊃" expresses the concept of causation. For instance, it can be used in combination with universal quantifiers to express a completely different notion, of conceptual hierarchy. As I understand it, part of the motivation for Generalized Quantifier Theory is to eliminate the ridiculous use of the connective in propositions like ∀x(Man(x)⊃Mortal(x)). The failure to correctly encode time relations is one reason why ⊃ is not the same as "cause"; in fact, it's utterly non-obvious that "⊃" is a valid concept, even if it is a known formal symbol. As you know, "⊃" is entirely optional.

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Logic and causality are intimately related. If we know that A causes B then we can also say that A logically implies B.
Why?

If A implies B, then A could be false and B still be true; in such a situation, it can't be said that A is the cause of B.

In terms of causality (A causing B ), you'd want something that'd say that if A is true then B is true and if A is false then B is false. And given that equivalence is usually used to refer to current states (as opposed to asynchronous states)... I don't think this intimate relation will work out on paper (unless you create your own system :worry::lol:B) )

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I see that it is really, really, really hard to get a straight answer from anyone on this forum. It seems like the question needs to be framed in a specific code of some kind in order not to trigger an array of complaints about the formulation of the question. So let me try again, rephrasing the question as best as I can:

If we know that the event A causes the event B, is it then correct to say that if B fails to occur then this *causes* A not to occur?

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If we know that the event A causes the event B, is it then correct to say that if B fails to occur then this *causes* A not to occur?
No, that is wrong. Also, it would help if you didn't include irrelevant material about connectors like "⊃" which have nothing at all to do with what turned out to be your real question.
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No, that is wrong.

Because?

Also, it would help if you didn't include irrelevant material about connectors like "⊃" which have nothing at all to do with what turned out to be your real question.

It would also be useful if you would not use strange symbols that disappear from the text. It makes it really hard to follow.

Also, it was not irrelevant because it is certainly true that "if ~B then ~A" follows from "if A then B." The question then is, how do we resolve the apparent discrepancy between the two? In other words why is the TRUTH of ~A dependent on the truth of ~B if there is no causal relationship?

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I see that it is really, really, really hard to get a straight answer from anyone on this forum.

If it helps, I'll tell you the "code". Make your questions relevant to the answer you seek and avoid making them unnecessarily complex. Think accuracy and simplicity.

[Edit - Corrected second sentence - RB]

Edited by RationalBiker
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Because?
Because of the relationship between time and causation: for X to cause Y, Y must precede Y in time. That's in the nature of causation.
It would also be useful if you would not use strange symbols that disappear from the text. It makes it really hard to follow.
I take it that you don't see the logical connectives. Well, I'll take the blame for that. A certain countably infinite member complained bitterly that I didn't use the standard symbols. My reason was that I believed that some people did not have their computers set up in such a way that they could see special symbols, but because of my subsequent interaction with him (indeed, pretty much everybody), I was of the belief that these days, anybody can read the characters we put in our posts. But apparently my original belief was correct. I wonder how widespread this problem is.
Also, it was not irrelevant because it is certainly true that "if ~B then ~A" follows from "if A then B." The question then is, how do we resolve the apparent discrepancy between the two? In other words why is the TRUTH of ~A dependent on the truth of ~B if there is no causal relationship?
You're confusion a particular program regarding natural language, formal logic, and philosophy. Let me assume that you're read the threads I referenced, then as you will no doubt appreciate from that discussion, it is false that "if ~B then ~A" follows from "if A then B". This is a classic case of a floating abstraction. What do you mean by "if A then B"?

Let me take this opportunity to point out that man is not born with a priori knowledge of the laws of formal logic which are taught in typical undergraduate courses in symbolic logic. Formal logics are the highly developed product of intensive reasoning (and unreasoning). Your false presumption that ^B=>^A "follows" from "A=>B" is a case in point. Natural languages and human reasoning does not work on the basis of meaningless symbols like "A", "B", but symbolic logic does not work on the basic of "if" and "then". You should pick context: are you asking a question about the meaning of "if" and "then", or about symbolic systems?

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The conditional connective does not imply causality by any means. It merely says that when the first connective is true the second connective is also true. This has nothing to do with cause and effect; each may be caused by a third event or by separate events that merely work out in such a way that the conditional is true. It is the case that A->(Bv~B ) but that does not mean that the arbitrary A caused that tautology (Bv~B ) to be true, it merely means that whenever A is true (Bv~B ) is also true.

In terms of your question, if A being true causes B to be true and B is only true when A is true (I don't know of any connectives for that) then it would be the case that ~B would imply ~A, which again does not mean ~B caused ~A, merely that when ~B is true so is ~A. So it is certainly not the case that we can say that ~B causes ~A in this case.

You have a very skewed understanding of causality, and formal logic.

Edited by Ogg
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Because of the relationship between time and causation: for X to cause Y, Y must precede Y in time. That's in the nature of causation.

Fair enough. The question then is what the relationship between causality and logical implication is. Because if the truth of A implies the truth of B then the non-truth of B implies the non-truth of A. So how can it be that you have a logical implication that goes both ways but not causality?

You're confusion a particular program regarding natural language, formal logic, and philosophy. Let me assume that you're read the threads I referenced, then as you will no doubt appreciate from that discussion, it is false that "if ~B then ~A" follows from "if A then B". This is a classic case of a floating abstraction. What do you mean by "if A then B"?

By "if A then B" I mean that the truth of A implies the truth of B. That is, if A is determined to be true then always, always, always B will be found to be true.

Let me take this opportunity to point out that man is not born with a priori knowledge of the laws of formal logic which are taught in typical undergraduate courses in symbolic logic. Formal logics are the highly developed product of intensive reasoning (and unreasoning). Your false presumption that ^B=>^A "follows" from "A=>B" is a case in point. Natural languages and human reasoning does not work on the basis of meaningless symbols like "A", "B", but symbolic logic does not work on the basic of "if" and "then". You should pick context: are you asking a question about the meaning of "if" and "then", or about symbolic systems?
Are you saying that there is no way to abstract out the general pattern of logic from many instances of reasoning? The way I see it formal logic is merely a way of putting logic into a clearly defined language. I refuse to accept that logic can only be achieved through mumbling and floating thoughts in the head. Logic is the abstraction of identity in reality, the laws of logic are then simply the laws of identity.

Formal-logical implication is always ultimately, though usually not directly, reducible to identity and causality; and there are many different kinds of ways to reduce to facts the various different kinds of implication.

Yes, obviously this is the case. (Well, not *obvious* given that there are plenty of people who refuses to see this) The question is *how* logical implication relates to causality. I gave a specific example namely A=>B is logically equivalent to ~B=>~A. I fail to see why it is a crime against nature to abstract out the laws of logic and put them in linguistic form like this, but for some reason I cannot express this perfectly non-ambiguous statement in formal logic without ensuring that I never get a straight answer. It's very frustrating.

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As other have said, logical symbols do not necessarilly indicate a causal relationship.

For example (assume that the premises are true):

1) Everyone who owns a moped enjoys sushi.

2) John owns a moped.

/:. Therefore (by modes ponens) John enjoys sushi.

The a above does not imply that owning a moped is the CAUSE of enjoying sushi.

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As other have said, logical symbols do not necessarilly indicate a causal relationship.

For example (assume that the premises are true):

1) Everyone who owns a moped enjoys sushi.

2) John owns a moped.

/:. Therefore (by modes ponens) John enjoys sushi.

The a above does not imply that owning a moped is the CAUSE of enjoying sushi.

This is true, but, for the third time, what if you have the situation that A *causes* B. Then obviously you have that A *implies* B. What then about ~B *implies* ~A which is the logical equivalent of A implies B? What is the *causal* status of this statement? That is, if you know for a fact that A causes B, and you also know for a fact that you don't have B (~B ). From this you logically infer that you don't have A either (~A). In other words: A causes B. Not B, hence not A.

(Edited for spelling)

Edited by Onar Åm
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I see that it is really, really, really hard to get a straight answer from anyone on this forum.
I never get a straight answer. It's very frustrating.
Maybe you just don't understand the straight answers you've been given.

If we know that A causes B then we can also say that A logically implies B.
I disagree. Can you prove that claim?
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I would challenge even that, Laszlo. If A causes B then we can say that A implies B will occur, but not logically implies--which is a different concept. For A to logically imply B means that the very meanings of A and B require that, where A is the case so is B.

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The question then is what the relationship between causality and logical implication is. Because if the truth of A implies the truth of B then the non-truth of B implies the non-truth of A. So how can it be that you have a logical implication that goes both ways but not causality?
To prevent any further misunderstanding, let's be clear on what "logical implication" means. If you want to mean something else, you can say what that other thing is. "Logical implication" refers to a particular connective in symbolic logic, symbolized as "=>" among other things. Then, the relation ship between logical implication and causation is "statistical tendency". The connective is never obligatory, but if you use it, it can be used to express "identity", especially in connection with explicit universal quantifiers ("All men are mortals"). It can express correlation which is due to some other causal principle ("If it's snowing, it's cold" but note that B sorta causes A and not the other way around). An example of the causal use where the antecedent causes the conclusion would be "If John forgets Sally's birthday tomorrow, she will divorce him" (although one could quibble over whether that would really be the cause).

My suggestion is to never look at floating abstractions again, and instead look at real examples of "logical implication" (i.e. uses of the connective "->").

Are you saying that there is no way to abstract out the general pattern of logic from many instances of reasoning?
No, I'm saying that there is no single way to abstract out a pattern from all forms of valid reasoning, especially if you're going to use FOP logic. If you take reasoning to be primary and extract symbolic logic from that, there's no problem. You will get the connective "&", come hell or high water. But you will not get the connective "=>" used the way it is in typical logic 101 class fashion; instead, you'll get something like generalized quantifier theory. You will also get a separate concept of "causation", although this is as much a part of logic as the concepts "individual", "number", "identity", "existent", "consciousness" and "comparison" (inter alios). The error arises in accepting the modern logicians' package deal, when it comes to the nature of logic (as being completely separate from man's cognition), and assuming that there is -- a priori -- such a thing as "=>". What is the proof that "=>" is a valid expression of something in logic? Why should I grant that "=>" is valid? What is it valid for?
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The connective is never obligatory, but if you use it, it can be used to express "identity", especially in connection with explicit universal quantifiers ("All men are mortals").

I'm sure you weren't trying to make the rookie mistake, but properly speaking this is an expression of predication, not identity.

An example of the causal use where the antecedent causes the conclusion would be "If John forgets Sally's birthday tomorrow, she will divorce him" (although one could quibble over whether that would really be the cause).
Ah, is it called "quibbling" when you don't want something disputed? :rolleyes:

Here is why I like more categorical truths, like "If a man is a bachelor, then he is an unmarried male." There's no need (or way) to quibble over the truth of the implication.

No, I'm saying that there is no single way to abstract out a pattern from all forms of valid reasoning, especially if you're going to use FOP logic. If you take reasoning to be primary and extract symbolic logic from that, there's no problem. You will get the connective "&", come hell or high water. But you will not get the connective "=>" used the way it is in typical logic 101 class fashion; instead, you'll get something like generalized quantifier theory.

Will you get '~'? If so, by definition you get '=>'.

The error arises in accepting the modern logicians' package deal, when it comes to the nature of logic (as being completely separate from man's cognition), and assuming that there is -- a priori -- such a thing as "=>".

The a priori, as modern logicians and philosophers argue, is part of man's cognition. It's the very framework and basic axioms, like consciousness, will, identity, etc.

What is the proof that "=>" is a valid expression of something in logic? Why should I grant that "=>" is valid? What is it valid for?

It provides the very meaning of something being valid.

[Edit for typo]

Edited by aleph_0
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I'm sure you weren't trying to make the rookie mistake, but properly speaking this is an expression of predication, not identity.
Uh, no, the statement "All men are mortals" is a recognition of an aspect of man's identity. You do recall, I presume, that "identity" doesn't mean "equivalence".
Here is why I like more categorical truths, like "If a man is a bachelor, then he is an unmarried male." There's no need (or way) to quibble over the truth of the implication.
Hey, are you familiar with Katz & Fodor's paper on this from the 60's? If so, then you know why that's a lousy example. But I take it that your real point was to mention so-called "analytic truths" as another class of uses of the connective. Well, since there aren't any analytic truths, there's no reason to distinguish mindless tautologies such as "All married men are married men" from synthetic truths like "All men are mortal". We have to decide what this little chat is about. Onar seems to be interested in symbolism and reasoning, so your examples are off topic (you may supply your intended formal statement, to get yourself on topic).
The a priori, as modern logicians and philosophers argue, is part of man's cognition. It's the very framework and basic axioms, like consciousness, will, identity, etc.
Yes, yes, we all know what these primacy of consciousness jerks have done to modern philosophy and logic. We're talking about actual reasoning and cognition, not the made-up nonsense of POC philosophers. This forum presumes Objectivism, so I can only conclude that you forgot that we're not assuming the nonsense assumed over at the Eye Heart Kant club. To show that "=>" is valid, you have to prove that it has real referents. And since you know very well that "=>" is just syntactic sugar, I don't know why you're bothering to try to defend it.
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