Proof of logic

[drkitten]

I didn't get the memo where we totally removed the semantics and the study of arguments from logic.

It might have been before your time; it was some time in the 1910s.

So I will just state what I want from my logic.

Cool. We serve all kinds in here.

I want my logic to formalize correct arguments.

Excellent. We have just what you want in stock -- I recommend first-order predicate calculus. It has exactly the property you need; it's "sound." It's also a nice, friendly, easy to use logic with wide application -- a good entry-level logic. I doubt you'll go very far wrong with that.

However, some of our (ahem) more specialized clientele have more, um, sophisticated needs into which "soundness" doesn't enter. A colleague of mine, for example, is interested in an automated hypothesis generator that will scan databases and propose hypothesis for investigation, and he's not as interested in "soundness" as he is in other aspects such as plausibility. Of course, he's also an expert user and is likely to tinker with the product himself as soon as he gets it home. Whatever. As I said, we serve all kinds here.

 

[Mangafranga]

But logic doesn't "allow" that.

Different people in this thread are using "logic" differently. Some people are using "logic" to just refer to the formalization of a syntax. Others are using it to refer to the study of inference.

I actually use it both ways. I have no problem with referring to a logic, meaning a formalization, and still holding that a logic can be correct or not in its ability to formalize correct inferences. Perhaps in this thread this isn't appropriate. I will now use "Logic" to mean the study of valid arguments. I will now use "logic" to mean a formalization.

But it's not that the logic is wrong; the logic shows you how the statement is invalid. Validity relates to form, soundness relates to content.

No, the hypothetical logic I referred to actually claims that form is valid.

It might have been before your time; it was some time in the 1910s.

You should send the memo out to all the philosophers of Logic, then. Perhaps you should also send the memo out to to the 1910 Russell. This would save him the effort of all the justification he wrote for the inferences in Principia.

Excellent. We have just what you want in stock -- I recommend first-order predicate calculus. It has exactly the property you need; it's "sound." It's also a nice, friendly, easy to use logic with wide application -- a good entry-level logic. I doubt you'll go very far wrong with that.

But there is that damn classical conditional, it just makes no sense. "If p, then q" is false only when the antecedent is true, and the consequent false. OK I buy that it is false in this case, but is this the whole story? What if the antecedent and consequent have no relevance to each other? And what about when the antecedent is false? Oh wait wait, sorry, I can't actually talk about that can I. "logic" can only mean a formalization, it can never make sense to talk about it's correctnesses. No-one ever talks about correct or incorrect inferences.

However, some of our (ahem) more specialized clientele have more, um, sophisticated needs into which "soundness" doesn't enter. A colleague of mine, for example, is interested in an automated hypothesis generator that will scan databases and propose hypothesis for investigation, and he's not as interested in "soundness" as he is in other aspects such as plausibility. Of course, he's also an expert user and is likely to tinker with the product himself as soon as he gets it home. Whatever. As I said, we serve all kinds here.

And a professor who is giving a talk this weekend, directly before I give my talk (ok ok I am just a student, but this is just as good as an appeal as yours) holds that there are multiple correct logics. He is a pluralist about Logic. Another Logician at the workshop holds that the concept of a second order logic doesn't make sense. I'll be sure to give them the bad news that they are both out of a job.

Finally I suggest that in the context of this thread, it is not unreasonable to think that "logic" should refer to the study of valid arguments. Further I contend that there is an objective measure of validity.

 

[drkitten]

You should send the memo out to all the philosophers of Logic, then.

Most of the professionals already read it.

Perhaps you should also send the memo out to to the 1910 Russell.

I don't need to. He was one of the principel authors of the memo.

.But there is that damn classical conditional, it just makes no sense. "If p, then q" is false only when the antecedent is true, and the consequent false.

And that's part of the problem. That statement you make is, by 19th century standards, or late 20th century standards, silly. The idea that any conditional statement whose antecedent is false is true "just makes no sense" (and it's one of the hardest ideas to get across to first year students); it makes counterfactual reasoning and contingency planning impossible to formalize. FOPC's treatment of counterfactuals is considered to be one of its major weaknesses, and that's one reason that a lot of people prefer to use a different system in logical environments where "dirty" data is a real possibility.

The only reason that the FOPC formalism -- and it's not a "classical" formalism; it follows "classical" logic by something like two and a half millenia -- is used is because it's mathematically convenient and makes -> usefully different from = or <->. And it happens to be "correct" in the sense of "sound"; as long as we agree on what the if/then construction means, we will never be led into a contradiction. Now, if you want to propose a different interpretation for that symbol, you're welcome to. But assigning meanings to symbols is not a property of the logic, but a property of the interpretation. (See the Lowenheim-Skolem theorem for examples of that.)

Oh wait wait, sorry, I can't actually talk about that can I. "logic" can only mean a formalization, it can never make sense to talk about it's correctnesses.

Well, yoiu obviously can't talk about that. Professional logicians can and do talk about correctness, but they recognize is as a property that does not reside solely in the "logic."

No-one ever talks about correct or incorrect inferences.

Bullfrog. Logicians talk about that all the time. But the correctness of a particular logic can only be established against particular models, which is

And a professor who is giving a talk this weekend, directly before I give my talk (ok ok I am just a student, but this is just as good as an appeal as yours) holds that there are multiple correct logics.

He's right, although he probably used more correct terminology than you did.

Another Logician at the workshop holds that the concept of a second order logic doesn't make sense.

He, on the other hand, sounds like a quack. Or more likely, you didn't understand him.

Finally I suggest that in the context of this thread, it is not unreasonable to think that "logic" should refer to the study of valid arguments.

Suggest away. We are under no compulsion to pay attention to your suggestions borne of ignorance.

Further I contend that there is an objective measure of validity.

Contend away. You're still wrong.

 

[Mangafranga]

I am going to just sum up my position. If we understand "logic" as a just a way to manipulate symbols (i.e. all we can talk about is a particular logic), then there is no answer to the OP, and indeed the question doesn't even make sense. If we understand "logic" as being the study of valid (correct, good, truth preserving or whatever you prefer) arguments, there is an answer. We can give an account of what makes a valid argument (and there are multiple answers to this). So we can tell a story about truth preservation. We can say that a valid argument is one in which it can't be such that the premises are true, and the conclusion false. We can concede that the methodology is similar to science, in that it is hard to confirm a theory, but it can be falsified. We can at least know that some arguments are invalid (e.g. the fallacy I mentioned, or a more interesting case is the failure of "(p and (q or r)) <-> ((p and q) or (p and r))" in quantum mechanics).

 

[JetLeg]

You said : If we understand "logic" as being the study of valid (correct, good, truth preserving or whatever you prefer) arguments, there is an answer. We can give an account of what makes a valid argument (and there are multiple answers to this).

If we understand this as logic, what answer can there be?

In order to prove that some arguments are valid and some are not, you need to pressupose a way of proving that. But that would be presupposing what you need to prove.

You need to assume an epistemology to prove one, and you can't do it -> so there is no way to prefer one epistemology to another.

 

[Mangafranga]

In order to prove that some arguments are valid and some are not, you need to pressupose a way of proving that. But that would be presupposing what you need to prove.

You said before that you can see a hammer, but you can't see logic. Well, compare the inferences to the world. Has it ever been that, for instance, the premises of the argument "p or q, ~p, so q" have been true, while the conclusion has been false. The system of proof here is simply looking at the world to see if there has been a case where this argument hasn't held. If not, we accept the argument as valid (and just like in science we accept that it may be falsified in the future, but so what).

 

[RandFan]

But it all ends up with logic being just a premise which you take. What if this premise is wrong?

Nonsensical but I get the gist of what you are asking. I'm capable of abstract thought. I've read Carroll many times and sometime in my twenties I actually got it. Hint: the adventures of Alice in Wonderland and Through The Looking Glass are not simply children's stores.

Ok, I'll bite, what if the premise is wrong? Can you live your life in any other way than that which is logical?

Now for the fun. If the premise is wrong and we assume your premise, how can we consider and answer your question?

Wait, I've got it!

The answer is @x!$^vstoeo.

Did that answer your question? Did you find it sufficiently illogical?

Jet, you can toss out logic if you want but I'm affraid you will have to sit in a corner and drool or somethin akin to that. When you come here and EMPLOY LOGIC in an attempt to question logic I can only want to smak you in the head.

 

[drkitten]

I am going to just sum up my position. If we understand "logic" as a just a way to manipulate symbols (i.e. all we can talk about is a particular logic), then there is no answer to the OP, and indeed the question doesn't even make sense.

I'm glad you've summed it up, because it makes it much easier to show that you are wrong.

If we treat logic as a way of manipulating symbols, then there are several ways that we can "prove" logic to be correct (depending upon how you define correct), and most of them were explored during the first half of the twentieth century.

It is, for example, possible to prove that manipulating symbols according to the rules of FOPC cannot produce a statement of the form (P ^ ~P). Since the conventional interpretation of this statement is "P and not-P," a contradiction, this means that following the rules of FOPC will not introduce contradictions into an otherwise sensible model of the universe. (This was Goedel's Completeness Theorem).

It's also possible to prove that in any model M in which P is derivable using FOPC, P is also a consequence. (that is, if M|-P, M|=P). That's also the "soundness" property defined purely in terms of symbol manipulation.

For example, we may have an unsound logic that is nevertheless sound over some models but not all, and as long as we choose our model appropriately, the logic "works."

If we understand "logic" as being the study of valid (correct, good, truth preserving or whatever you prefer) arguments

... then we understand logic incorrectly. This understanding was abandoned nearly a hundred years ago.

We can give an account of what makes a valid argument

Yes, but we can do that in the standard formalism as well, and we can go well beyond and discuss under what models a given argument is or is not valid. Because validity is not just a property of the argument, but of the world model to which it applies.

 

[cyborg]

Well Dr K has covered it quite well but the basic point remains:

Semantics comes after syntax.

 

[Mangafranga]

So would someone like to explain what the arguments over K, S4, S5 are about? And would someone like to explain what Putnam was doing when he argued that logic is an empirical science?

 

[drkitten]

So would someone like to explain what the arguments over K, S4, S5 are about?

In a nutshell, they're about whether the formal syntax of the logic is appropriate to represent the domains in the real world about which we wish to reason.

For example : K is a very simple form of "modal" logic (and a very weak one). By "modal" logic, I mean a logic that purports to capture the distinctions between necessity and contingency; it is necessary that a full coffee cup is full, because that's what the phrase "a full coffee cup" means. It is merely contingent that my coffee cup is full right now; sometimes it's full, sometimes it's empty, it just depends on when you catch me.
(Also note that it's not just a question of form. It is not, for example, necessary that a small elephant be small, because something that is small for an elephant is still pretty darn big. Philosophers have gotten their panties tied in knots for centuries over this one.)

And, in fact, K, although a provably "sound" logic and provably "complete" in some sense doesn't really capture what most people think "necessary" means. That's why people argue about which logic to use. In the words of a web page I will cite in a bit, "Consider the English version of LMLp: It is necessarily possible that it is necessary that p. It is unclear what, if anything, that statement means." If you don't have a clear intuition about what that means, it's hard to say that any meaning assigned to it is definitively wrong. "One reason S5 is the most popular logic is that all statements with nested modalities (such as LMLp) reduce to statements with a single modal operator. [...] In S5, you can simply delete all the modal operators except the last one. This means that LMLp is equivalent to Lp. Lp has an easy (and meaningful) translation. In S4.2, all strings of modal operators are equivalent to one of the following strings: L, M, ML, LM."

Now, I would argue that S5 does not capture the nuances of the real world; there's a difference between something that is necessary, something that is possible, and something that is possibly necessary. So I prefer S4.2 as a better model of the unbounded world than S5. But there are also a lot of restricted environments where S5 better captures what's going on.

The site goes on to say : "In fact, I see the lack of meaning to be the great strength of logic and math. The reason mathematic representations of the world and formal logical representations of thought have been so effective is that we are free to reinterpret the semantic content of the relevant symbols as appropriate for a given context." I find it hard to improve on that phrasing.

Logic is a tool. If you only know one logic, you only have one tool.

The site, by the way, is http://dtww.blogspot.com/2005/03/logic-is-for-tricking-people.html

 

[Mangafranga]

From Logic and Ontology, SEP

There are several quite different topics put under the heading of ‘logic’ in contemporary philosophy, and it is controversial how they relate to each other. 2.1. Different conceptions of logic

On the one hand, logic is the study of certain mathematical properties of artificial, formal languages.

Seems to be how you are using it. Let me see, my conception covered?

A second discipline, also called ‘logic’, deals with certain valid inferences and good reasoning based on them.

Yep, so this makes the first part of your statement

... then we understand logic incorrectly.

seem rather dubious. Perhaps your second part

This understanding was abandoned nearly a hundred years ago.

is supported

A second discipline, also called ‘logic’, deals with certain valid inferences and good reasoning based on them.
...snip to the end of this paragraph...
The central notion of logic in this sense is the notion of logical consequence. How this notion should be understood more precisely is presently widely debated, and a survey of these debates can be found in the entry on logical consequence.

And perusing the bibliography of this entry finds plenty of recent work. So this statement of yours seems incorrect.

 

[Loss Leader]

But it all ends up with logic being just a premise which you take. What if this premise is wrong?

"Your system could be wrong therefore all systems of belief are equal and my wackdoodle beliefs are just as valid as yours."

 

[Yiab]

Semantics comes after syntax.

In the idealized realm of abstraction known as mathematical logic (equally, philosophical logic), yes. In reality, of course, the human species developed syntax and semantics together and only learned fairly recently that they are seperable.

Actually, if we look at the way logical languages are chosen for study, we often start with a particular semantics in mind and develop a syntax we hope will reflect it. We then explore what the limits of expression are within that syntax.

For example, Euclid developed geometry with some very specific semantic choices in mind for his terminology ("point" and "line" meaning, well, point and line respectively), but he also realized that these semantic choices are independant of what he was proving. Later, we found that if we took the axioms of projective geometry and decided "point" would refer to a line and "line" would refer to a point, it would still make sense.

In this sense, syntax is developed after semantics but is independant thereof.

 

[JetLeg]

Do you all agree on the way of prooving logic, as Magafranga said ? :

You said before that you can see a hammer, but you can't see logic. Well, compare the inferences to the world. Has it ever been that, for instance, the premises of the argument "p or q, ~p, so q" have been true, while the conclusion has been false. The system of proof here is simply looking at the world to see if there has been a case where this argument hasn't held. If not, we accept the argument as valid (and just like in science we accept that it may be falsified in the future, but so what).

 

[JetLeg]

You said before that you can see a hammer, but you can't see logic. Well, compare the inferences to the world. Has it ever been that, for instance, the premises of the argument "p or q, ~p, so q" have been true, while the conclusion has been false. The system of proof here is simply looking at the world to see if there has been a case where this argument hasn't held. If not, we accept the argument as valid (and just like in science we accept that it may be falsified in the future, but so what).

And how do you proove that a specific fallacy in logic is a fallacy?

For example, ad hominem - how do you prove that is a fallacy?

 

[slingblade]

And how do you proove that a specific fallacy in logic is a fallacy?

For example, ad hominem - how do you prove that is a fallacy?

That's easy. Let's assume I smoke. Smoking is bad for one's health. This statement is true regardless of who utters it. Therefore, it is fallacious to say that my premise has no merit because I smoke. It has merit, regardless of the speaker's habits.

That's how you prove (one O) an ad hom.

 

[Robin]

And how do you proove that a specific fallacy in logic is a fallacy?

For example, ad hominem - how do you prove that is a fallacy?

If you are talking about proof, then you must be talking about formal logic, there is no concept of ad hominem in formal logic.

In fact you could put phrase ad hom so that it would be formally valid, so "atheists are evil, Richard Dawkins is an atheist, therefore Richard Dawkins is evil" would be formally valid.

 

[Robin]

As for the question in the OP, yes you can prove the rules of logic. Pick up a good text book on Discrete Mathematics and you will find that all the rules of logic for any given system require and have proofs.

But you can't prove the axioms. They are, well, axiomatic.

 

[RandFan]

And how do you proove that a specific fallacy in logic is a fallacy?

For example, ad hominem - how do you prove that is a fallacy?

Do you accept mathmatical proof?

If I say that I have two apples and I give you one and that leaves one would you accept that 2-1=1?