While traipsing through the Superstition foothills Sunday morning in search of further footnotes to Plato, I happened to think of James Madison and Federalist #51 wherein we read, "If men were angels, no government would be necessary." My next thought was: "Men are not angels." But I realized it could be the formal fallacy of Denying the Antecedent were I to conclude to the truth, "Some government is necessary." (I hope you agree with me that that is a truth.)
The first premise is a counterfactual conditional, indeed, what I call a per impossibile counterfactual. To keep things simple, however, we trade the subjunctive in for the indicative. Let this be the argument under consideration:
1. If men are angels, then no government is necessary.
2. Men are not angels.
3. Some government is necessary.
A prima vista, we have here an instance of the invalid argument-form, Denying the Antecedent:
If p, then q
But I am loath to say that the argument (as opposed to the just-depicted argument-form) is invalid. It strikes me as valid. But how could it be valid?
One could take the (1)-(3) argument to be an enthymeme where the following is the tacit premise:
1.5 If no government is necessary, then men are angels.
Add (1.5) to the premises of the original argument and the conclusion follows by modus tollendo tollens.
Might it be that 'if ___ then ___' sentences in English sometimes express biconditional propositions? Clearly, if we replace (1) with
1* Men are angels if and only if no government is necessary
the resulting argument is valid.
One might take the (1)-(3) argument as inductive. Now every inductive argument is invalid in the technical sense of 'invalid' in play here. So if there are good inductive arguments, then there are good invalid arguments. Right? If the (1)-(3) argument is inductive, then I think we should say it is a very strong inductive argument. It would then be right churlish and cyberpunkish to snort, "You're denying the antecedent!"
The question arises: are there any good examples from real argumentative life (as opposed to logic text books) of Denying the Antecedent? I mean, nobody or hardly anybody argues like this:
If Jack ran a red light, then Jack deserves a traffic citation.
Jack did not run a red light.
Jack does not deserve a traffic citation.