# [FOM] A textbook on logic with natural deduction

paul@personalit.net paul at personalit.net
Thu Dec 10 13:22:55 EST 2009

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> 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

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