Arguments (inferences) in physics

[Pentcho Valev]

The importance of the problem raised in my previous posting can be extracted
from the following quotation (W. H. Newton-Smith, The rationality of
science, Routledge, London, 1981, p. 229):

"A theory ought to be internally consistent. The grounds for including this
factor are a priori. For given a realist construal of theories, our concern
is with verisimilitude, and if a theory is inconsistent it will contain
every sentence of the language, as the following simple argument shows. Let
'q' be an arbitrary sentence of the language and suppose that the theory is
inconsistent. This means that we can derive the sentence 'p and not-p'. From
this 'p' follows. And from 'p' it follows that 'p or q' (if 'p' is true then
'p or q' will be true no matter whether 'q' is true or not). Equally, it
follows from 'p and not-p' that 'not-p'. But 'not-p' together with 'p or q'
entails 'q'. Thus once we admit an inconsistency into our theory we have to
admit everything. And no theory of verisimilitude would be acceptable that
did not give the lowest degree of verisimilitude to a theory which contained
each sentence of the theory's language and its negation."

Newton-Smith derives an apocalyptic picture from the coexistence of the
proposition p and its negation not-p inside the theory. As if by magic, the
contradictory couple transforms the theoretical construction into a useless
totality composed of all sentences of the language and their negations.
Obviously a theory undergoing a degeneration of this kind should be
rejected. Yet this conclusion is not incontestable. The statements in the
above quotation are acceptable to the logician but not necessarily to the
physicist. The latter may recognize their formal correctness but still may
not worry too much - after all, the conditional "if p then (p or q)",
although formally true, has no physical meaning. Accordingly, the
demonstrated degeneration of the theory into the set of all possible
sentences appears to be nothing more than a curious exercise of formal logic
without any physical significance.

Is that true? Should a physical theory that has been proved to be an
inconsistency be rejected? It can be shown that, if the inconsistent theory
is an axiomatic system, it SHOULD be rejected. It can also be shown that
there are only two ways of converting an axiomatic system into an
inconsistency:  1) through the introduction of a false axiom; 2) through
performing an invalid deductive step.

For instance, Clausius' famous result "Entropy always increases" is in fact
a conclusion Clausius derived from the following two premises:

1. For a closed system, cyclic integral of dQ/T is always smaller than or
equal to zero.

2. Any irreversible process in a closed system has a reversible counterpart
that connects the same initial and final states.

In this case Clausius' deduction is valid so he cannot be accused of
producing an inconsistency "through performing an invalid deductive step".
But are the two premises (axioms) true? By the end of his career Clausius
abandoned both the second premise and the conclusion "Entropy always
increases". Why? Did he notice something wrong about the second premise? For
generations of thermodynamicists this problem did not exist since "Entropy
always increases" had become fashionable and a fashionable science is always
more fruitful than a non-fashionable one. Philosophers of science however
may wish to know if the second premise is true. They may also wish to know
if the first premise is true.

Pentcho Valev




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