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Deriving Facts from Counterfactuals

In this section we show that we can derive facts that do not contain counterfactual implications from counterfactuals.

We take as our theory, the axioms in Section 5, save for the last axiom tex2html_wrap_inline1714 . From the counterfactual,

we derive that, tex2html_wrap_inline1714 .


Proof: We first note that in A, the co-ordinates have the values, tex2html_wrap_inline1776 . We expand the definition of a counterfactual being true, namely,

to get, by replacing the function c everywhere its value at that argument,

This is a meta-theoretic statement, so we translate this into second order logic, by quantifying over predicates, functions and objects in the standard waygif.

where tex2html_wrap_inline1780 is the tuple of all constant symbols in the language, and tex2html_wrap_inline1782 is a tuple of variables, equal in type and arity to the constants in tex2html_wrap_inline1780 , we write variables in Z' as tex2html_wrap_inline1788 etc.

This second order sentence captures the truth of the counterfactual. We can conjoin this with the other axioms in the theory A, and ask whether it implies, tex2html_wrap_inline1714 in second order logic.

It does, as can be seen from instantiating the universally quantified variables, with their corresponding constants, except for tex2html_wrap_inline1794 and tex2html_wrap_inline1788 which are instantiated by the predicate P true of the smallest set of situations satisfying the formulas below, and the partial function, tex2html_wrap_inline1800 tex2html_wrap_inline1802 tex2html_wrap_inline1804 , defined from P,

The result of instantiating these terms, and noting that the left hand side is derivable from the theory A gives the sentence.

However, we also have tex2html_wrap_inline1810 , and the axiom,

Instantiating this with tex2html_wrap_inline1454 and simplifying gives tex2html_wrap_inline1814 . As we know by domain closure that

we have

as required. tex2html_wrap_inline1718

John McCarthy
Wed Jul 12 14:10:43 PDT 2000