Yes, MAth Geek… so you’re saying that
(P & (P -> Q)) -> Q
is only true so long as you don’t [have and not have breakfast]?
Interesting.
Yeesh. I don’t mind the occasional spelling mistake, but I should at least form meaningful sentances. That should read “Just as the truth of (P & (P->Q))->Q…”
Bertrand Russell.
Bertrand Russell.
The statement 2+2=4 does not follow from your previous argument.
erislover: That’s a slight rephrasing, but essentially correct, yes.
You aren’t exactly blinding me with the light of your reason either, bub.
So? What do you mean by valid?
One definition would simply be that something is valid when it obeys certain principles of logic. Okay, fine – but now your argument is circular.
Does the concept of validity involve being an accurate representation of the way the world works? Let’s say it does – but then we can’t just make the claim that obeying logical principles makes a statement valid.
Either your definition is circular, in which case no one is particularly interested, or you’re making a fundamental claim about the nature of reality by linking properties together with a single word, in which case I’d like to see an actual argument in favor of the position.
It’s a simpsons quote.
And I’m right you know.
But it’s true, and the premises are false. Relevance is irrelevant here.
btw, logic has changed quite a bit since the days of Russell. You may wish to check with a current textbook–I recommend Mendelson’s book, or Barwise & Etchmenedy for a slightly less advanced exposition.
The statement does not logically follow though, so its not a valid argument.
Validity has nothing to do with relevance. If the premises are false, then an argument is automatically valid. If the premises are true, then the argument is valid whenever that implies that the conclusion must be true. Look it up in any modern logic textbook if you don’t believe me.
There was a controversy in the early 20th century brought about by some people who wanted to bring in rules of relevance, but they never came up with a very good system to replace the standard predicate calculus. And that’s the one we use.
What does a hammer mean?
Hmm… I was actually expecting something different here. I agree with Eris that logic is meaningless, but for a different reason. Logic is meaningless because its primary Axiom, A=A, is a tautology: it is true regardless of the values of its variables.
Moreover, it seems this thread is destined to branch out all over the place. An argument is valid if all its premises follow logically from one another. It is sound if all its premises are true.
Not all tautologies are meaningless, and the axiom of reflexivity is nowhere near a primary axiom: in fact, it doesn’t even exist in first-order predicate calculus.
It depends on whose head it’s buried in.
Cervaise: ah, but logic is more than a tool. It’s also a language for describing the world around us. Thus, I would argue that “What is the meaning of logic?” is as valid a question as “What is the meaning of art?” or “What is the meaning of science?”, as both art and science are also in part languages for describing reality.
Libertarian: out of curiousity, in what sense do you consider “A=A” (or any other axiom) to be true? Logically? Empirically? Something else? (I’m guessing that you’ve thought about this question before, since I’ve seen you post about epistemology on other occasions…)
Of course every tautology is meaningless. It is proved by everything.
It is a mistake to assume that first order logic is, merely because of its name, the basis for all other logic. In fact, it is derived from more general symbolic logics of the broader logical calculus. It’s not the case that other logics are built from the fifteen axioms and two rules of inference of first order logic, but rather merely that those axioms and rules are constituent formulas built from the manipulation of variables by operators that are derived from higher logic’s wiffs.
Keep in mind that propositional logic, with two fewer axioms, is simpler even than first order predicate.
Math Geek wrote:
In general, I subscribe to the Deflationary Theory of truth. Of course, you would expect that that is my theory of choice since I believe that the universe is an illusion. And as I said elsewhere, that’s why logic models the universe so well. It, too, is a house of cards.
“In the Euclidean plane, the angles of every triangle add up to 180 degrees”
That’s a tautology. Is it meaningless?
First-order predicate logic is, IMHO, more properly viewed as a subset of higher-order logics. But that’s not my area of expertise. I need to do more reading on that.
Math Geek
I guess really I should answer your question more directly. A = A is true if A = A is true. It’s an anaphoric dependence.
Ultrafilter
You’re right that first order logic is a subset of higher logic. That’s what I meant to say, only you said it more clearly.
And yes, “triangle = 180 degrees” is meaningless. Another name for tautologies is “definitions”. And the obvious problem with them is their recursion. The words that comprise them themselves have definitions, which have words that have definitions, which have words that have definitions, and so on ad infinitum. Eventually, it circles back to itself. Logic rests on air. There is no foundation, i.e., nothing absolute upon which all of it stands.
That’s no definition. It’s a theorem, following from other theorems about parallel lines. It’s not hard to prove, but it’s not trivial to your average high-school geometry student.
In fact, a definition has no truth value. Definitions are equivalent to function letters in the predicate calculus. They’re just names.
Personally, I think that it might be interesting to define the information content of a theorem as the length of its shortest proof, and see where you go with that.
Of course. But that’s my point. The meaning of logic isn’t there; its the meaning of the sentences we use. But those had meaning already. They don’t gain meaning by showing that there is an analogous construction in logic.
Everyone
My motivation for this attack is founded on my continued rereading of Wittgenstein. Specifically, his Tractutus Logico-Philosophicus. Not the best book to read in search of an exhaustive treatment of meaning, but logic itself is the target of this work, and it has influenced my thoughts appreciably.
It is a difficult book to read, but no one should ever be satisfied by reading summaries of the man’s work. Much controversy still exists regarding the points he made. Similarly, one wouldn’t go to a Catholic priest to get word on the final interpretation of the Bible… you’d read it yourself. However, barring that… [ultrafilter, this quote is especially relevant to your point about some tautologies not being meaningless]
http://plato.stanford.edu/entries/wittgenstein/
I am not trying to rehash his arguments; however, I may use some quotes from sources about this work (and from the work itself) to aid my points. No sense in reinventing the wheel.
To continue…
What am I describing here:
(P & (P->Q))->Q
? English is a language for describing the real world. In fact, we teach it by pointing to the real world, and pointing to behaviors of other being (people) in this world (among other ways which are outside the scope of this thread). The reference is implicit. What do we point to for the logical statement? And if two English sentences have the same logical statement, are they the same sentence? Why not?