The Problem of Deduction?

[ainwood]

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[Ayatollah So]

if you accept A but you don't accept B, then it can't be true you accept A -> B.

I think that's right. But I also think Fifty is right about "intuition" (though that isn't the word I'd use, it's close enough).

The best (i.e. least crazy) interpretation of Tortoise is that he doesn't really accept, or probably even understand, A -> B. He says he understands and accepts it, but that doesn't settle the matter. People say they understand things, that they don't really understand, frequently enough.

But why is this the best interpretation of Tortoise? Because our norm (intuition if you will) of modus ponens is really really strong. The idea that Tortoise truly understands the statements but rejects modus ponens is just too crazy! The "He doesn't understand "if ... then" " hypothesis will always be superior.

You can try to play one norm or "logical intuition" against another. (That's how fallacies, which appear valid at first, are revealed as fallacies.) You can use some norms to buttress others, and construct systematic treatments that rely at the basic level on only a few simple norms. But you can never banish norms altogether, without defeating the point: i.e. bringing all inference to a halt, which would be truly crazy.

 

[greenpeace]

I'm sorry if this is completely irrelevant, as I didn't read the entire thread but:
Isn't it true that there is no reason other than the fact that we assume that the statement is true? What I mean is that given those first two statements we accept the third as true because it is true by definition, there is no reason other than that. I mean an analogy would be:
If you have a green and red apple then you have a blue apple. We accept this is true without need to proove it is true. In fact:
A
A->B
~B
Is just as true (without the definition being that it equals B), its just that it is much more useless in practicle reality. The same goes for all postulates. I can't proove the parallel postulate, I just accept it as fact. Furthermore I can accept anything as true, like if all apples are blue than there are 10 green apples. Its impossible to proove that statement false without using a postulate, which in turn can't be prooven true (it can only be accepted true).

 

[brennan]

The best (i.e. least crazy) interpretation of Tortoise is that he doesn't really accept, or probably even understand, A -> B. He says he understands and accepts it, but that doesn't settle the matter. People say they understand things, that they don't really understand, frequently enough.

That's what I said.

But why is this the best interpretation of Tortoise? Because our norm (intuition if you will) of modus ponens is really really strong. The idea that Tortoise truly understands the statements but rejects modus ponens is just too crazy! The "He doesn't understand "if ... then" " hypothesis will always be superior..

Right, he isn't really 'accepting' the propositions.

You can try to play one norm or "logical intuition" against another. (That's how fallacies, which appear valid at first, are revealed as fallacies.) You can use some norms to buttress others, and construct systematic treatments that rely at the basic level on only a few simple norms. But you can never banish norms altogether, without defeating the point: i.e. bringing all inference to a halt, which would be truly crazy.

Right. This shows that logical constructions are not self evident but rather, are arrived at by a process of elimination (they are synthetical rather than analytical - have I got that the right way around?). We know A -> B is valid because it has been repeatedly shown to work, not because of 'intuition'. Any intuition we have regarding this arises because it is advantagous for our brains to be wired up in this fashion, it would not be a good survival trait to jump to illogical conclusions all the time. Not that this doesn't happen.

Greenpeace. Of course you can say that owning ten red apples implies that you have ten blue apples, but you can be shown to be wrong. Logical truisms are not self-evident, they are 'proven' by experience.

 

[warpus]

But "-->" doesn't mean "if you accept the antecedent then you must accept the consequent". It means "if the antecedent is true than the consequent is true". While the incredible obviousness of the inference makes it seem to be a matter of "meaning" or "definition" for us normal people, it really isn't. Drawing the person a truth table means absolutely nothing, because a truth table just expresses the same propositions in a different way.

--> means whatever we defined it to mean, much like AND, OR, NAND, or any other logical connector.

You don't seem to get that the formal definitions of logical connectors sometimes differ from the every-day English use of the same word..

 

[kuukkeli]

You can see how this would keep going on and on and on. So what's going on here?

To me it seems that the Tortoise is cheating. After each iteration it's changing its mind while presenting an entirely new hypothetical.

After Tortoise accepts (C) If A and B are true, Z must be true it's already accepted the original hypothetical but instead of admitting that it asks Achillles to write down a different one. Sadly Achilles is too dumb to notice this and plays along.

I'm not sure if deduction can be proven but I'm sure that the Tortoise was unable to break it. It doesn't even try to find a flaw but floods Achilles with infinite number of meaningless hypotheticals.

 

[Perfection]

Here's my thoughts:

I think propositional logic can sort of be treated like a game, where various absolute rules dictate behavior.

In Monopoly you could land on unowned Baltic Avenue (price $60) after Passing Go and say to the banker, "Give me $140 and Baltic" while one might question that because you're asking for the banker to do a complex transaction to which you'd have to cite multiple rules and mathematics, but for a single transaction (like $200 for Passing Go) any questioning beyond the asking to cite a rule would be absurd (like why must I follow this rule). When you play a game you agree to unquestioningly follow the rules.

In the same way, while it's okay to question a logical leap (like Fifty's difficult proof example) the Tortoise is questioning a basic rule of propositional logic. When "x implies y" and "x" we can write down "y", that's just an absolute rule here that can't be questioned.

This raises a new issue, because rules are treated as dogmatic truths to a game that cannot ever be violated, we have no way to prove a game models reality or has any bearing whatsoever to real life. The rules of monpoly do not well model the real estate industry. One can question how much propositional logic models reality, but you can't question its rules, they are defined and to play the propositional logic game must be followed to strictly.

 

[Ayatollah So]

[...] When you play a game you agree to unquestioningly follow the rules.

[...]

This raises a new issue, because rules are treated as dogmatic truths to a game that cannot ever be violated, we have no way to prove a game models reality or has any bearing whatsoever to real life. The rules of monpoly do not well model the real estate industry. One can question how much propositional logic models reality, but you can't question its rules, they are defined and to play the propositional logic game must be followed to strictly.

That's a fine analogy for a formal system of propositional logic, but it's not fine for everyday arguments in natural language. Which is where the Tortoise and Achilles story is set. And rightly so because that's where, so to speak, the real action is. At some point, whether we use a formal model (interpreting real life arguments in its terms) or not (just winging it), we have to decide which real life arguments to accept.

 

[Fifty]

So I was thinkin' some more about this, and perhaps part of the hangup is that we have the wrong view of validity. Quite a few logicians maintain that validity isn't a matter of truth preservation, but rather about the preservation of semantic content across the premise/conclusion divide (because keeping it about truth makes some obviously valid arguments not really valid or invalid at all). And, obviously, semantic stuff is all a conventional part of natural language, that can't really be reduced in any substantive way. So that seems like a more intuitive way to understand the tortoise's problem, rather than the truth thing which brings in all that messiness about rule-following with jibes with natural language in the way Ayatollah mentioned.

So yeah, its just more evidence for the intuition business I guess!

 

[Perfection]

That's a fine analogy for a formal system of propositional logic, but it's not fine for everyday arguments in natural language.

I'd say that natural language contains a fairly formal system of propositional logic (which is why propositional logic can often be written in natural language quite successfully)! Rules seem to pervade natural language just like they do formal logic.

It might be a rule in our (English speaking) heads that if "x being true means that y is true" and "x is true" requires "y is true". Certainly there would be a lot of intuitive machinery behind the scenes to make the rule definitions, but I'd say it shares a lot with formal systems. If someone doesn't agree with these rules, then they are arguing against the common definition of English speakers and they demonstrate that they do not understand the usage of the words in question.

 

[warpus]

One of the problems is that human languages are not context-free..

 

[Perfection]

I don't see that being a big issue, we're pretty sucessful at determining the meaning of words in a given context.