[FOM] A textbook on logic with natural deduction
paul at personalit.net
Thu Dec 10 13:22:55 EST 2009
> 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.
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
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