[extropy-chat] Non-classic logics

[Ian Goddard]

--- Technotranscendence <neptune at superlink.net> wrote:

> > Last semester I took a course in metalogic
> > (Phil 470) covering the semantics of classic
> > logic (propositional and first-order predicate).
> 
> I found Raymond M. Smullyan's _First-Order Logic_ --
> available in an inexpensive Dover reprint -- to be 
> a great introduction to first-order logic.  


 I'll see if I can find that text. I just placed an
order for Beall & van Fraassen's text that you
recommended. Thanks too for the links. Stanford's
Encyclopedia o Phil is an invaluable resource!


> It was especially helpful with the tableaux
> method that Priest, Beall, and Bas C. van Fraassen 
> rely heavily on.  (In fact, it's not too inaccurate 
> to say a lot of Priest's exposition is relating how 
> different logics have different tableaux rules.  At 
> least, that's one device he uses throughout his 
> book to relate the differences.)


 Do you mean proof trees? In my first logic course we
used a method of evaluating statements known as a
"model (or toy) universe" and the professor referred
"tableaux" while teaching it. That method involved
semantics by relating the meaning of statements to
entities in a domain. 

But that method there was far short of the semantics
covered in Gamut vol I, which presents a formal study
of classical-logic semantics in a metalogic framework.
[1] In first-order predicate logic this involves (a)
interpretation by substitution and (b) interpretation
by means of assignments, which defines the means of
preserving the compositionality of meaning in
predicate logic.

The latter reduces to Tarski's truth definition (where
'V' denotes "value of," 'm' denotes "based on our
model," 'g' assignment, 'E' existential quantifier,
'@' a statement, '1' true, 'e' set membership, and 'D'
domain of our model):

Vm,g(Ex@) = 1 iff there is a d e D such that
Vm,g[x/d](@) = 1. 

In English: The valuation based on our model and given
assignments of "there exists some x such that @" is
true if and only if there is some thing d that is a
member of the Domain of our model such that the
valuation based on our model and assignment of x to d
of "@" is true. 
 
There we see the compositionality of meaning preserved
for the existential quantifier, such that we can say:
if V(@) = 1, then V(Ex@) = 1 too. In short, V(@) =
V(Ex@).



> > During the last two weeks we skimmed the
> > surface of propositional modal, or intensional,
> > logic, which is a nonclassic logic. I want to
> > study further than that.
> 
> The Beall and van Fraassen book focus to a large
> extent on modal logic -- as the title reveals.  If 
> you're interested in one philosopher's view of 
> modality, you might want to read Alvin Plantinga's
> _Essays in the Metaphysics of Modality_.  If you
> want to learn more about second- and higher-order 
> logic in the context of metamathematics, there's 
> Stewart Shapiro's _Foundations without 
> Foundationalism: A Case for Second-Order Logic_.


 Yeah, I've yet to really study second-order logic.
Gamut only devotes a cursory section to it without
exercises (unlike first-order that has many
exercises). I find that I don't really get a memorable
understanding of some logico-mathematic routine unless
I can do it in exercises many times. 

Thanks for all the great pointers. I think you'd
really like Gamut, both vol I and II. I've got a
collection of other logic books, but nothing comes
close to Gamut for the reasons I stated. [2]  

_________________________________________________
[1]
http://www.press.uchicago.edu/cgi-bin/hfs.cgi/00/7087.ctl
http://www.press.uchicago.edu/cgi-bin/hfs.cgi/00/7088.ctl

[2]
http://www.lucifer.com/pipermail/extropy-chat/2005-March/014601.html

~Ian

http://iangoddard.net

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