Paradoxes of Material Implication - Simplification

Simplification

The classical paradox formulas are closely tied to the formula,

the principle of Simplification, which can be derived from the paradox formulas rather easily (e.g. from (1) by Importation). In addition, there are serious problems with trying to use material implication as representing the English "if ... then ...". For example, the following are valid inferences:

but mapping these back to English sentences using "if" gives paradoxes. The first might be read "If John is in London then he is in England, and if he is in Paris then he is in France. Therefore, it is either true that (a) if John is in London then he is in France, or (b) that if he is in Paris then he is in England." Using material implication, if John really is in London, then (since he is not in Paris) (b) is true; whereas if he is in Paris, then (a) is true. Since he cannot be in both places, the conclusion that at least one of (a) or (b) is true is valid.

But this does not match how "if ... then ..." is used in natural language: the most likely scenario in which one would say "If John is in London then he is in England" is if one does not know where John is, but nonetheless knows that if he is in London, he is in England. Under this interpretation, both premises are true, but both clauses of the conclusion are false.

The second example can be read "If both switch A and switch B are closed, then the light is on. Therefore, it is either true that if switch A is closed, the light is on, or that if switch B is closed, the light is on." Here, the most likely natural-language interpretation of the "if ... then ..." statements would be "whenever switch A is closed, the light is on", and "whenever switch B is closed, the light is on". Again, under this interpretation both clauses of the conclusion may be false (for instance in a series circuit, with a light that only comes on when both switches are closed).