Rules of Logic.

[IoftheBholder]

Originally posted by Ottman Out
In general, how important is it to name specific fallacies in arguments?

Perhaps the broader question is, "How important is it to state the logical transformations or the rules applied in an argument?" Let's explore with regard to Argument A:

Argument A (partial)

Premise 1) Socrates was a boot.
Premise 2) All boots are made of jelly.
---------
Conclusion: Socrates is made of jelly.

How do we know this argument is valid? We might say that it makes sense or it's "logical". But as we move to formal logic (or logics, as the case may be), we state the rules that are known to lead to a valid conclusion (or the fallacies that lead to invalid conclusions). For Argument A, there is a rule that explains how we get to the conclusion.

Chain Rule: If A is B and B is C, then A is C.

Therefore, we might clarify the argument with the following:

Conclusion: "Socrates is made of jelly." By reason of the Chain Rule.

Just as fallacies (invalid rules) have names, so too do the accepted rules of logic. Whether we need to call out the names is a matter of context (as described in earlier posts). In the context of formal logic, explicit reference to the rules may be required. In a common argument, it would probably obscure the soundess of the argument by focusing too much on the validity. Instead, we assume everyone is familiar with the accepted rules of logic.

[Clutch]

Originally posted by IoftheBholder
Just as fallacies (invalid rules) have names, so too do the accepted rules of logic. Whether we need to call out the names is a matter of context (as described in earlier posts). In the context of formal logic, explicit reference to the rules may be required. In a common argument, it would probably obscure the soundess of the argument by focusing too much on the validity. Instead, we assume everyone is familiar with the accepted rules of logic. Right. There is an asymmetry between naming fallacies and naming rules, however. It's pragmatic -- in the sense of linguistic pragmatics. Paul Grice called this the Maxim of Quantity: Give no more and no less information than is needed. It's familiar, moreover, that we need to be particularly clear and particularly careful when matters of disagreement arise. If we all agree on an argument's validity, then no elucidation of the relevant rules is required; but if I take issue with your argument, the requisite standard of precision is, plausibly, immediately raised. Now I need to make good on my rejection of your argument, and you (if you're not convinced) need to defend its virtues.

In treating my interlocutor's argument as valid, then, I typically don't need to limn the rules in virtue of which it is valid; my reaching this judgement won't be an "issue", and won't require explanation. But in calling my interlocutor's argument fallacious, I typically create an "issue", and am obliged to explain my judgement.

[Muad'Dib]

Originally posted by Clutch
The most basic rules (axioms) for Aristotelian logic and Propositional logic over the long years were threefold. (The way of expressing them varies from logic to logic, but the idea is the same.)

[b] The Law of Identity: P if and only if P

The Law of Non-Contradiction: Not both P and not-P

The Law of Excluded Middle: Either P or not-P I couldn't help but notice that Non-Contradiction and Excluded Middle are derivable from each other via DeMorgan's rules. Are all three laws stated for historical reasons, or are there some situations where you can't distribute a negation across a conjunction or disjunction, or not have ~~P = P?

Thanks,
Muad'Dib

[wade-w]

As far as I know the normal way to prove DeMorgan's Laws is by reductio ad absurdum. But this technique is valid only if the Law of Non-Contradiction and the Law of the Excluded Middle are both taken to be true. Now, if you could find a proof of DeMorgan's Laws that uses only one of the axioms in question then you could do what you propose and derive the other.

[wade-w]

The other way that DeMorgan's Laws are typically proven is via brute force using truth tables. But don't even truth tables assume all of the above axioms?

[seebs]

Just a side note:

While I think laws 2 and 3 are orthogonal, you can compress them:

4. P exclusive-or not-P.

[Muad'Dib]

Originally posted by wade-w
The other way that DeMorgan's Laws are typically proven is via brute force using truth tables. But don't even truth tables assume all of the above axioms? Comparing truth tables would have to use a fourth axiom, since there's nothing in the first three that tells us what to do with two possibly distinct statements P and Q. That's why I was assuming that the three laws were informal/intuitive statements; under the intuition that "and", "or" and "not" behave normally, DeMorgan's rules are immediate. I was curious about whether there's a time when they don't work the way they usually do.

For a complete formalization that doesn't involve truth tables at all, one could use something like Gries & Schneider's* axioms, where equivalence is taken to be invariant under substitution:

1. ((p = q) = r) = (p = (q = r)) [Associativity of equivalence]

2. p = q = q = p [Symmetry of equivalence]

3. true = q = q [Identity of equivalence]

In this system (p = p), ~(p ^ ~p), and (p v ~p) [the three laws from above] are all theorems, as are DeMorgan's rules. The proofs, however, are rather unpleasant (though I could write some of them out if you're interested). For pedagogical reasons I think it's better to stick with truth tables.

* A Logical Approach to Discrete Math, Springer-Verlag '93

[Muad'Dib]

I was too hasty last night: in Gries and Schneider's system, (P = P) and ~(P ^ ~P) are in fact theorems, but (P v ~P) is actually one of the axioms used to define "or".

That's the root of my original question about the laws of thought; it's clear there have to be other axioms in the system (specifically, formal definitions of "and", "or" and "not", as well as one or more that explain how to deal with distinct statements), and I was thrown off by the intimate relation between 2 and 3. It's more a question of aesthetics than anything, I think. :)

[Hugo Holbling]

This discussion has been split from the intro thread (http://www.iidb.org/vbb/showthread.php?s=&threadid=67057&perpage=25&pagenumber=1) on logic.

[Clutch]

Originally posted by Muad'Dib
I was too hasty last night: in Gries and Schneider's system, (P = P) and ~(P ^ ~P) are in fact theorems, but (P v ~P) is actually one of the axioms used to define "or".

That's the root of my original question about the laws of thought; it's clear there have to be other axioms in the system (specifically, formal definitions of "and", "or" and "not", as well as one or more that explain how to deal with distinct statements), and I was thrown off by the intimate relation between 2 and 3. It's more a question of aesthetics than anything, I think. :) A nice way of putting it. Also, many folks have found Gentzen-style introduction and elimination rules for the connectives to illuminate the commitments of a logic particularly clearly.