[FOM] A textbook on logic with natural deduction

[paul@personalit.net]

>
> One thing I especially liked about Lemmon's book is that one could  
> see clearly what depends on what, in contrast with the book from with  
> which I began the study of logic, Quine's Methods of Logic, in many  
> other ways a masterpiece. For example, one can see clearly what in  
> the rules forces the equivalence of 'if p then q' and 'not both p and  
> not-q' and thus the truth-functional reading of the conditional.

Charles,

For a similar reason -- that it displays constructive relationships 
between truth functors -- I value Hilbert and Bernays's axiomatization 
of propositional logic in the Second Edition of Grundlagen der Mathematik. 

This is comprised by the following schemas, found on p. 66 (using '&' 
for conjunction, 'v' for disjunction, '>' for implication, '<>' for 
mutual implication, and '~' for negation):

I.  Formulas of implication.
(1)  A > (B > A)
(2)  [A > (A > B)] > (A > B)
(3)  (A > B) > [(B > C) > (A > C)]

II.  Formulas of conjunction.
(1)  (A & B) > A
(2)  (A & B) > B
(3)  (A > B) > [(A > C) > (A > (B & C))]

III.  Formulas of disjunction.
(1)  A > (A v B)
(2)  B > (A v B)
(3)  (A > C) > [(B > C) > (( A v B) > C)]

IV.  Formulas of equivalence.
(1)  (A <> B) > (A > B)
(2)  (A <> B) > (B > A)
(3)  (A > B)  > [(B > A) > (A <> B)]

V.  Formulas of negation.
(1)  (A > B) > (~B > ~A)
(2)  A > ~~A
(3)  ~~A > A

Each schema can be regarded as the conditionalization of a rule of 
natural deduction.  Positive propositional logic is comprised by groups 
(I-IV).  Intuitionistic logic is (I-IV) plus V.(1-2) Groups (I-V) in 
toto comprise classical propositional logic. 

The Hilbert Bernays Project is drafting a new English translation of 
this book:

http://www.ags.uni-sb.de/~cp/p/hilbertbernays/goal.htm

Cheers,

-paul