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L. Wittgenstein, Tractatus Logico-Philosophicus, 5.122: "If p follows
from q, the sense of 'p' is contained in the sense of 'q'.

A few paragraphs earlier, in 5.101, Wittgenstein implicitly vindicates
the validity of the following arguments:

From 'p and not-p' it follows that 'q'.

From 'p' it follows that 'p or q'.

Philosophers and logicians often compare 5.122 and 5.101 and each time
they do so they cannot make out how the logico-philosophical wisdom
can be so deep. At such moments they clearly see that the sense of 'q'
is contained in the sense of 'p and not-p' and that the sense of 'p or
q' is contained in the sense of 'p'. Then they apply part of the
logico-philosophical wisdom in their analysis of inconsistent science:

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."

Finally, some philosophers (e.g. Feyerabend) reject this standard
defence of consistency (something must be wrong with a logico-
philosophical wisdom that says that from an inconsistency everything
follows) and solemnly declare that inconsistent theories bring
progress in science:

http://homepage.mac.com/otaviobueno/.Public/InconsistenNotHell.pdf
Why Inconsistency Is Not Hell
Making Room for Inconsistency in Science
Otavio Bueno
"On most accounts of belief change, inconsistent belief systems are an
"epistemic hell" to be avoided at all costs (see, e.g., Gärdenfors
1988, p. 51). From a normative point of view, we can perhaps
understand why this is the case. The underlying logic of most theories
of belief change is classical, and classical logic is explosive, that
is, everything follows from a contradic-tion. And a belief system from
which everything follows should definitely be avoided. It is certainly
of not much use if one wants to determine what one should believe and
what one should do.....I argue that Levi's own approach to belief
change could become stronger by making room for inconsistency, without
giving up the crucial features of his pragmatism. The crucial move is
to change the underlying logic to a paraconsistent one, which, in
consistent contexts, yields exactly the same results as the classical
approach (see, e.g.,da Costa and Bueno 1998). Once there is room for
inconsistency, there is also room for informativeness in inconsistent
belief systems.....Pursuing inconsistent systems is sometimes the only
way to obtain new information, particularly information that conflicts
with deeply entrenched theories (see Lakatos 1978 and Feyerabend
1988).....The considerations above indicate that, properly
conceptualized, inconsistent belief systems, whether about empirical
or nonempirical domains, can provide invaluable information. Even
though an inconsistent system may only be a first, but important, step
toward a consistent successor, the point still remains that such a
system is significant.....Suppose, however, that the only way of
obtaining certain bits of information is by deliberately expanding
into inconsistency. Bohr's case discussed above illustrates this
situation - as well as the earlier formulations of the calculus and
the conjunction of quantum mechanics and relativity theory. To obtain
the relevant information, in each of these cases, it looks as though
one is forced to expand into inconsistency. Now, as long as the
underlying logic is paraconsistent, there need not be anything
unacceptable here."

Pentcho Valev
pvalev@yahoo.com