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Possible deduction systems.

MathematicalPhysics Wizard
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Posted 02/25/09 - 10:20 PM: Subject: Possible deduction systems.	#1
How many possible deduction systems are possible, and what is the minimal system? Iv'e looked at some book, and Lukasweicz (if I have written his name correctly) has a similar system as Hilbert's one, I guess the minimal system must contain at least 3 axioms, we need modus tollens somewhere, and thus we need the axiom (x->y)->(~y->~x). But what with the other axioms is x->(y->x) essential, we can't deduce anything without it can we?
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et cetera
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Posted 02/26/09 - 12:09 AM:	#2
maybe your thinking of meta-logic? ignoring quantification and modality all we really need are variables and the operators and, or, and not. Everything else necessarily follows .
Any necessary truth, whether a priori or a posteriori, could not have turned out otherwise. -- Saul Kripke Meaning is what essence becomes when it is divorced from the object of reference and wedded to the word. -- Quine A possible world is given by the descriptive conditions we associate with it - Kripke

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Posted 02/27/09 - 01:15 AM:	#3
No, that's not what I meant. I mean there seems to be quite a lot of deduction systems, shouldn't there be a limit to the number of deduction systems? And what tautologies (or how many of them) are necessary in order for the system to be complete? Rather simple questions or so I think.
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Posted 02/27/09 - 04:02 AM:	#4
The Bearded Monkey wrote: How many possible deduction systems are possible, and what is the minimal system? Iv'e looked at some book, and Lukasweicz (if I have written his name correctly) has a similar system as Hilbert's one, I guess the minimal system must contain at least 3 axioms, we need modus tollens somewhere, and thus we need the axiom (x->y)->(~y->~x). But what with the other axioms is x->(y->x) essential, we can't deduce anything without it can we? This is not, as it stands, an interesting question, not even when you restrict the systems to ones that generate classical logic (taken as a consequence relation). It is made interesting by restricting the axiomatization in certain ways, e.g. by requiring that the axioms be organic in the sense of Church 56 (p. 138, viewable on google books). So do you have an interesting question?

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Posted 02/27/09 - 06:04 AM:	#5
What is the name of the textbook?
"What kind of a bomb is this? The exploding kind." Peter Sellers

muxol
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Posted 02/27/09 - 08:06 AM:	#6
http://books.google.com/books?id=J...thematical+logic#PPA138,M1

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Posted 02/27/09 - 08:13 AM:	#7
I'll read it someday, but back to my question why isn't it interesting to know what tautologies are more essential than others, I mean if one tautology is not necessary and we can replace it by another one I think it's interesting to know which can deduced from others by instantiation and which combination gives us to a dead end.
"What kind of a bomb is this? The exploding kind." Peter Sellers

muxol
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Posted 02/27/09 - 08:39 AM:	#8
there are infinitely many ways to weaken a system of classical logic. E.g. you can replace p -> .q -> p with p -> p or with p -> q v q -> p or... Now, in all cases you won't get a logic that is closed under substitution, so maybe a more interesting question is what interesting weakenings there are that yield logics closed under substitution. But actually there are a lot of interesting logics that are not closed under substitution (e.g. certain tense logics, constructive logics (e.g. with constructible falsity), etc.) The point is that your question is far too vague and open-ended. And what does 'minimal' mean in this context? Obviously if you take away ALL axioms then the empty logic is THE minimal logic. Also your comment "...we can't deduce anything without it can we?" is OBVIOUSLY answered in the negative, unless the axiom you mention is the ONLY axiom in the system. (Well, strictly speaking we may have a non-empty consequence relation even though the set of theorems is empty, so in that sense we may be able to deduce things from non-empty sets of premises, even though we can't deduce anything from the empty set.)

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Posted 02/27/09 - 08:41 AM:	#9
The Bearded Monkey wrote: I'll read it someday... You were directed to a single small paragraph of the page which gives the definition of an axiom being organic, in a single sentence. Why don't you read it now and call it a day?

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Posted 02/28/09 - 11:12 PM:	#10
I like to read from start to finish and not excerpts of the text, cause I will lose my sense of direction and lose the plot, so to speak.
"What kind of a bomb is this? The exploding kind." Peter Sellers

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