And on the other hand, even without appealing to induction, a contradiction can also be derived in the form of a conditional sorites argument by deriving \(F \tau)\) as the result of repeatedly applying modus ponens to the series \(F 0), F 0) \rightarrow F(0'), F(0') \rightarrow F(0"), \ldots\) obtained from (ii) by universal instantiation.
Abelard spends a great deal of effort to explore the complexities of the theory of topical inference, especially charting the precise relations among conditional sentences, arguments, and what he calls 'argumentation' (roughly what follows from conceded premisses).
Further information about indicative conditionals, especially arguments pro and con the truth functionality of indicative conditionals, can be found in section two of Edgington (2006).
For example, the pf code can be kept as-is and it suffices to add this before the code shown: The conditional checks for arguments and, when present, replaces the input channel with one for the named files.
"I haven't heard a single argument that conditional access for something like the iPlayer would not be a good thing to try.
Of course \(\mathsf{PA}^F\) is still inconsistent in virtue of the conditional form of the argument.
Hook will reply to the above argument about conditional commands that we need to appeal to pragmatics.
Logic is seen as applicable to natural language, in particular the sorites paradox, but the conditional form of the argument is seen as proceeding from a faulty premise.
In fact, one can also evaluate the truth conditions of conditionals under this point of view by saying that a conditional is true if an argument from the antecedent and suitable co-tenable premises to the conditional's conclusion exists.
