The Critical Rationalist                       Vol. 03  No. 01
ISSN: 1393-3809                                    [DRAFT: 11-Feb-1998]


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2.10 Why Logical Content is Not Atomic

(27) Lying behind the view that scientific theories are infinitely varied in their consequences there is, I suspect, an atomistic view of content: the idea that there exist minimal independent morsels of information from which the contents of all more informative statements and theories are compounded by finite or infinite conjunction. The reader is warned not to be misled by the use of the word `bit' in information theory to express what sounds like exactly this idea. It is not the same idea. Bits are not minimal, except in the logically insignificant sense of being representable by expressions of minimal length. Indeed, the atomistic thesis as it stands is untenable. For the only possible candidates for the role of atomic contents would be irreducible statements, and no axiomatizable theory with infinite content can be built entirely from irreducibles. This is easily shown. For a set of k irreducibles generates a theory with tex2html_wrap_inline1361 distinct consequences; and an axiomatizable theory that is logically equivalent to the conjunction of denumerably many irreducible statements is, by finitude, equivalent to the conjunction of some finite subset of them, so that we return to the previous case. A simple example is supplied by the calculus described in 2.6, elementary logic with identity as the only relation: for each positive i the statement tex2html_wrap_inline1227 is irreducible, but the conjunction of all these statements yields a theory tex2html_wrap_inline1287 (`the number of objects is not finite', or `the universe is infinite') that is not finitely axiomatizable.

(28) The objection may be strengthened by noting that in many calculi there exist no irreducible theories at all. This follows from the result of Mostowski already cited. An example is provided by ordinary classical sentential calculus with denumerably many sentence letters. In such calculi, of course, all non-tautological theories have infinite content.



next 2.11 A More Model-theoretic Approach
previous 2.9 An Example (Kepler's Laws)
contents

The Critical Rationalist                       Vol. 03  No. 01
ISSN: 1393-3809                                    [DRAFT: 11-Feb-1998]


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TCR Issue Timestamp: Fri Mar 27 14:21:33 GMT 1998

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