User:Ryguasu/Logic, Math, etc.

Here's my question and Larry's answer

Larry,

I've been reading about the "Problem of Induction" lately, and I'm now left with the impression that philosophers have put a lot more effort into figuring out the circumstances under which inductive reasoning is justified than those under which deductive reasoning is. In particular, while it seems very popular to be skeptical of the soundness of induction, deduction seems, for the most part, unquestionably acceptable in almost every case. (Provided fallacies are avoided, of course.) I'm wondering if, to help even the score, you could suggest any philosophers who either try to problematize deductive reasoning, or others who make a serious attempt to justify it. Surely this must keep someone up late at night.

The most interesting things I've found along these lines are:

1. Lewis Carroll's "What the Tortoise Said to Achilles" (in Godel, Escher, Bach)
2. Hofstadter's justification of the rules of first order logic (also somewhere in Godel, Escher, Bach). It's something along the lines of, "Don't these rules sound like what a sane person must believe? If you are sane, and you intuitively believe them, do you really need to question them?"

Obviously I could find someone willing to argue about deduction if I go far enough afield, say, into postmodern literary theory. But that stuff doesn't strike me as terribly serious or compelling.

--Ryguasu 04:33 Nov 7, 2002 (UTC)

One of the best things on that question is my dissertation.  ;-) Seriously, what my dissertation was about was the problem that the justification of induction and of deduction have in common.

The leading view on the question you mention is that there's nothing wrong with the circularity involved in deductive justifications of deduction. The modern locus classicus of this view is Nelson Goodman's Fact, Fiction, and Forecast. Also, Susan Haack wrote an article called "The Justification of Deduction" in Mind. For the underlying issues, you could always go here and use your browser's search function to find "deduction" on the Chapter 3 page. --Larry Sanger

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Logic pages:

• logic ("classical logic" redirects here)
• laws of logic
• college logic
• logical argument (argument vs explanation?)
• argument (disambiguation/dictionary page)
• logical calculus (the one with GEB-like 1st order axioms)
• philosophical argument (?)
• good argument (redundant?)
• vacuous truth
• logical equivalence
• truth table
• contradiction
• negation

There are too many pages with an introductory feel, each probably a little contradictory with respect to others. Especially unhelpful (and some people link to it for "elementary logic" or some such) is college logic.

3 views of logic:

• logic as grammar to generate an infinite set of valid argument forms (not necessarily all, thanks to Goedel?)
• an isomorphism: propositions <-> symbols, "logical words" <-> logical operators, truth values <-> 1,0
• an isomorphism: propositions <-> a set of potential world states

The first looks much less useful for the "Relationship between first-order logic and thought" page.

Math pages:

• universal quantification

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Rambling formerly on talk:Vacuous Truth:

Note to self: it would probably be best to explain the two ways in which first-order logic is usually "grounded": truth tables (that sort of match our intuition) and in a-proposition-is-its-extension set theory. Perhaps a "How First-order logic relates to thought" article is in order? --Ryguasu

Note to self: "Most logical systems agree that S must be either true of false, otherwise boolean algebra would suffer from a lack of closure." Fair enough, but this does not explain why anyone should care about boolean algebra or closure, or that logic can be formalized without boolean algebra. How can I fix this?--Ryguasu

...

I usually see "vacuous truth" in the context of logical implication. But I guess it results whenever you have:

• a mental model of something
• a corresponding, isomorphic formal model
• the freedom, in the mental model, to leave truth values undefined
• the necessity, in the formal model, to define truth values

Some other examples would be good. What other operators/connectives/etc. generate vacuous truth?

I guess the concept "vacuous falsity" also exist, though, for some reason, vacuous truth is more prevelant.

...connections to the concept of closure

...like 0!=1

--Ryguasu

That 0!=1 is actually quite logical. (I assume you mean 0 factorial) -- Tarquin