The fundamental problem of logic

[Pentcho Valev]

In a popular textbook logic is defined as "the study of valid arguments.
It is a systematic attempt to distinguish valid arguments from invalid
arguments." Since the premises in an argument can be combined in a
conjunction, logic can be redefined as "the study of valid conditionals.
It is a systematic attempt to distinguish valid conditionals from
invalid conditionals". Of course this definition can be questioned etc.
but one thing is sure: distinguishing valid arguments (conditionals)
from invalid arguments (conditionals) is extremely important. And if
some snag has been paralysing the procedure for a long time, the problem
can reasonably be called "the fundamental problem of logic".

Two examples of physical argument. 

Carnot:
Premise 1: Heat is an indestructible substance (calorique).
Premise 2: Perpetuum mobile of the first kind is impossible.
Conclusion: All reversible machines working between the same two
temperatures have the same efficiencies.

Clausius:
Premise 1: For a closed system, cyclic integral of dQ/T is smaller than
or equal to zero.
Premise 2: Any irreversible process in a closed system has a reversible
alternative connecting the same initial and final states.
Conclusion: Entropy never decreases.

The traditional logical tools offered in elementary courses (which
physicists are expected to use as they deal with arguments like the
above ones) are based on Wittgenstein's idea that logical propositions
are analytical and tautological. For instance,

A -> (A or B) 

where -> is the sign of the conditional. In this example the conditional
is a tautology indeed, but do conditionals of this kind have anything to
do with conditionals in physics? If the answer is no the problem is
really "fundamental". Logical tools offered to physicists are
inapplicable.

The problem can by no means be mitigated by the existence of
non-traditional approaches supposed to be more relevant. The unrivalled
supremacy of Wittgenstein's approach suggests that the essence of the
problem remains unknown.

Pentcho Valev
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