Denying the Antecedent?

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.
ergo
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
~p
ergo
~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?

Approach One

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. 

Approach Two

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.

Approach Three

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.
ergo
Jack does not deserve a traffic citation.

Related articles

Could There Have Been Nothing at All?

The Lewis-Stalnaker Analysis of Counterfactuals: What about Sprigge? - The Campaign for Better Citation and Credit-Giving Practices in Philosophy

On Hating and Despising Philosophy

Comments

[Joe]

"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?"

I am not sure why you would be loath to say the argument is invalid. Just because the conclusion and all the premises are true that does not mean the argument is valid.

Interestingly, enough I think premise 1.5 could be false. For example even if there were just one man, and he was not an angel there would still be no need for government.

"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 no government is necessary

the resulting argument is valid."

I think people do this quite a bit, but I can't think of examples right now. On LSAT tests they usually have a critical reasoning question that involves denying the antecedent. I suspect some people never get that one right.

[BV]

You must not read this blog, else you would not have suggested that I think that the validity of an argument follows from its having true premises and conclusion.

To evaluate a stretch of argumentative prose, one must do two things. First, identify the argument. Second, identify the logical form of the argument you have identified.

Now someone who gives the (1)-(3) argument is charitably interpreted as arguing something like the following:

Whether or not government is needed depends on whether or not men are morally perfect. If they are, then they do not need government. If they are not, then they do. Now men are not morally perfect. Therefore, they need government.

This argument is plainly valid.

So the original argument cannot be reasonably dismissed as denying the antecedent; it is more reasonably understood as an enthymeme which, when the tacit premises are added, is valid.

Compare the last argument I cited, which cannot be given a charitable reading.

You're right that there is no need for government on Robinson Crusoe's island. But of course Madison was not talking about that situation, nor was he talking about a situation in which a few people who love one another live together.

To understand my analysis you must realize that a one and the same sentence can be used to express different propositions, and that an argument is a sequence of propositions, not of sentences.

[Joe]

I see what you mean. Sometimes the hidden assumption is also true, so plugging it in is not a problem. Some things can be assumed and need not be stated.