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I simplify the original example (40% down the page). All times are stated in 24-hour format.

[Premise] 1. Necessarily, if [R.] I had started at 5   then  [P.] I started before 6.
[Premise] 2. If it were the case that P   then  [Q.] I would have arrived before 12.
[Conclusion] 3. Therefore, if it were the case that R,   then Q.

...Scenario: ... Nevertheless, 2 may be true and the conclusion false.
[4.] Now suppose that IF I had started at 5 (which is before 6) then I would have been too sleepy to remember [...] the shortcut, then I would not have arrived before 12.
Thus, [5.] while it's true that had I started before 6, I would have arrived before 12,
[6.] it's false that had I started at 5, I would have arrived before 12.
This exception proves that this form of argument is invalid.

If 5 is true, then how can 6 be true (Notice that 6 includes 'it's false that')? To wit: If you started before 6 and if the only other choice was to start at 5, then you must have started at 5?
I symbolise 4-6:

[4.1.] R ⟶ I would forget the shortcut. ⟶ ~Q.

[5.1.] P ⟶ Q.

[6.1.] ~(R ⟶ Q)

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To avoid answering several of these, I am going to lay out my favorite theory behind them. Not everything in this answer is fully relevant to this question, but to the others like it here. This is officially redundant. Someone here already said this, but they used heavy notation that seems unnecessarily precise, and off-putting.

Because of the way English usually works, we are tempted to hear the subjunctive statement as an indicative one -- saying that for any time before six, if I started then, I would most likely arrive before noon.

But since this is a subjunctive implication, it means there is an untested or disputed theory behind it, which I currently accept and am basing my deduction upon. That theory is incomplete and might be wrong. If I find some aspect that shoots down my theory, then the statement loses its meaning.

In this case, the idea that I drive equally well at all times seems to be part of the theory. If it fails, the implication does not have any force.

So in this case, if I left at 5:50, I might arrive before noon, because that time before six is one where my theory still holds. But if I left at 5, I would navigate worse. So I have made an even more drastic correction than I would have needed to (strengthening the antecedent), and the correction itself has caused other unstated preconditions to break down.

So this seems to disprove the statement, but in fact it only disproves the unstated theory behind it. In a more complex case, more than one thing might have gone wrong to make the assertion fail. If I have misidentified what antecedent is too weak, strengthening a different one can make it seem like I am being careful, when, in fact, I am just hiding productive lines of inquiry.

When working with subjunctive statements, in order to make real deductions, one needs to know the 'range of all possible worlds under consideration' basically, to know the theory under which the implication would be valid. The problem is that folks usually cannot articulate those theories.

One formal way of dealing with this is to consider 'modal' statements made in the subjunctive as always true. There is bound to be some bizarre theory on the basis of which they would be true. But then we need to require removing (or 'relativizing') the subjunctive before we make deductions from it.

We can truly remove the subjunctive only by turning it into a complex implication "my theory => S", and since we cannot articulate "my theory" by mind-reading the source of the statement, we have to guess as to the completeness of our understanding of the theory.

This approach is called "modal suspension", because proceeding forward from the proposition is "suspended" by the "mood", awaiting further analysis. And it implies that your confidence in the statement depends on whether the theory has been largely identified and satisfied, or whether you believe the source based on relevant experience.

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This is basically a "tl;dr" version of jobermark's answer (which I upvoted).

English and the language of formal logic are not the same, and not everything which can be expressed in English can be completely captured by formal logic. In this case (and as jobermark mentioned, in similar cases) there is a disconnect between the IF-THEN of logic and the similar but different if-then of natural English.

The IF-THEN of logic describes an entailment relationship. The if-then of English expresses a chain of hypotheticals. There is overlap between the two usages, which allows us in many cases to translate between one and another, but if we lose the distinction, we fall into traps like this one.

  • It is true there is a disconnect, but that is not the OP's main problem. The if-then of logic captures mostly the indicative mood, and the hoops one has to jump through to represent other moods are far worse than the default disconnect implicit in the indicative mood. – jobermark Nov 3 '14 at 20:01

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