In Shakespeare’s famous St. Crispin’s Day speech, Westmoreland laments that the English don’t have more troops on the battlefield, to which Henry IV counters

Basically, the arguemnt goes, the fewer men on England’s side, the better. Formall, it can be phrased as

where A = “we will win the battle”, B = “we will lose the battle”, and C = “It is better to have fewer men”.

Of course, the conclusion is incorrect. Westmoreland is right: The more troops you have on the battlefield, the better. But the King’s argument looks logically sound. What’s the fallacy, here?

It isn’t a logical fallacy, as such - it’s just a skewed set of priorities. Westmoreland’s priority is to field an effective fighting force, and win - the King’s isn’t. He just wants to fight an “honorable” battle, and the degree of honor (in his mind) isn’t contingent upon the outcome. The King doesn’t care about winning, and his statement is correct from his own set of priorities.

I don’t think it’s a formal logical fallacy per se, just ignoring the fact that C influences the likelihood of the A and B outcomes. If the A or B outcome was going to be the same regardless of how many men you put on the field, Henry IV is correct.

Either we will win the battle or lose it. (Trivially true)
If we win the battle, it would be better to have done so with fewer men.
If we lose the battle, it would be better to have done so with fewer men.
It would be better to fight the battle with fewer men.

The missing bit of logic is that Westmoreland would like to win the battle, and that England is more likely to win with more men rather than with fewer. Henry’s argument doesn’t address that point, except insofar as he might be implying that the outcome of the battle will be determined by fate/God regardless of the troop numbers. If you are indifferent to whether you win or lose or believe the outcome to be fixed in advance, then Henry is quite correct - better to have fewer men.

It seems to me that the argument is based on the idea that the outcome of the battle is predetermined, and not affected by how many people are involved in it.

I don’t think you can point to an error of formal logic. Rather, the error lies in the weakness of the formal system, since it can only express things that are true, or things that are false. It does not deal with probabilities or change.

Of course, battles are not predetermined, and are in fact affected by how many people are involved. But that doesn’t change the truth of the statement “Either we will win the battle, or we will not”.

Basically, it looks like all three of the premises (A or B, If A then C, and If B then C) are all individually true, and if those premises are true, then under the standard rules of logic, the conclusion C should be true, too.

No, I don’t think A or B is true. It is not true that they will win the battle, and it is not true that they will lose it. It is true that they might win the battle and it is true that they might lose it.

Neither A nor B say “they will win the battle” or the opposite. It says they “will either win or lose”. “A or B” does not mean “A” and it does not mean “B”.

The problem, as others have pointed out, is that the word “better” is subjective. Winning and having less honor is better than losing with more honor, to Westmoreland. Not so for the king.

I’m not sure I understand what you mean by this. How are you parsing the statement “they will win or lose the battle” into a formula with the structure A or B? What is A and what is B?

No, but if both A and B are false then so is A or B.

For the King, fewer troops was better, for Westmoreland it wasn’t. But the question is what premise does Westmoreland deny in that argument.

Losing, yet with more troops, you may be in a better position to negotiate and therefore get better results. Even winning, more troops can lead to a more decisive position.

But as an example of the puzzling part of that speech; the first item in this grid is the %, the second, the objective score, obviously war isn’t all or even mostly objective but just for the argument
% - Score
MORE L 20% -10 W 80% 5
LESS L 90% -5 W 10% 10

The overall EV is higher for MORE (20% * -10 + 80% * 5 = -2 + 4 = 2, 90% -5 + 10% * 10 = -4.5 + 1 = -3.5), yet within the fixed set of given a win, or given a loss, the LESS choice is always better.

A = they will win the battle
B = they will lose the battle

If “A OR B” is FALSE, then it must be that “(NOT A) AND (NOT B)” is true, which would mean they neither lost nor won the battle. It could be argued that you can only win or lose is a false dichotomy, but that’s a stretch.

I’ve discussed a similar fallacy before in our debate on Newcomb’s Paradox (e.g., here). Moving from “A or B; if A, then C; if B, then C; therefore, C” is perfectly valid (in conventional logic). However, the situation in the OP is presumably better analyzed with some sort of modality. So, letting X mean “X is guaranteed to hold in the future”, A mean “We win the battle”, etc., what we actually have is “(A or B); if A, then C; if B, then C; therefore C” which is invalid; it could be made valid if we were to assume that (A or B) implied (A or B), but that would be a silly thing to assume; indeed, this example shows that it is erroneous conflation to do so. [The more reasonable thing to assume is that distributes across AND, giving the minimal normal modal logic K, on top of which further standard assumptions can be laid as suitable].

Yes, but if both A and B are false, that means that the English will neither win nor not win the battle. They can’t both be false. One of them is true; we just don’t know in advance which one.

Thank you for the abstract value-table: That isolates the fundamental problem here without quibbling over Westmoreland’s and the King’s differing priorities. So, if one is playing a game with that payoff table, how does the King’s argument for optimal strategy fail?

Either/or means “might”, in the sense that you’re using it. Try this:
“The result of the battle will be a win or a loss” - R is W or R is L. Now you’re saying that “R is W” (and the other one) is false, but it isn’t. It’s undetermined. My point is that nowhere in the proof does it conclude what R is. You want the proof to read “R may be W. R may be L. R is not [anything else].” That’s the same thing as “R: W or L”, as is stated in the proof.

That is, to make it clear, we should NOT consider it to be true that “If A, then C” (if tomorrow has us winning the battle, then we are currently in a position as that we should want less men); rather, it is only true that “If A, then C” (if tomorrow is currently guaranteed to have us winning the battle, then we are currently in a position as that we should want less men). The fallacy lies in sliding these modal operators around through the slipperiness of natural language so as to make it look like they aren’t even there, thus giving the argument an apparent logical form which is not appropriately understood as its actual one.

This is only the case if you assume the outcome is predetermined (which you claimed it is not). If it is not predetermined, then the opposite of “the English will win the battle” is not “the English will lose the battle”, but “the English might lose the battle”.

R can have three values: “win”, “loss”, and “yet to be determined” (unless you assume the outcome is predetermined). The proof only works if we know it to be either “win” or “loss”, but we dont, or at least Henry didn’t.

Logic aside, this was the battle of Agincourt, where the French lost partly because they had too many men on the battlefield (they were overcrowded and getting in each other’s way). Also, Henry didn’t have the option of having more men, so he was putting the best face on the situation that he was presented with.

Right, parsing “will” as “necessarily will”. And those pesky "necessarily"s fuck up the argument in the OP by being in the wrong places for the argument to have the form it depends on looking like it does.

Conditionalizing on Win, it is better to have Less. Conditionalizing on Lose, it is better to have Less. However, conditionalizing on (Win or Lose), which is the same as the “unconditional” a priori evaluation, it is better to have More; the third is not simply determined by the first two, because there is non-independence between “Less vs. More” and “Win vs. Lose”.

“If A then C” and “If B then C” are faulty premises. “If necessarily A, then C” or “Conditionalizing on A, then C” are accurate premises.

Therefore his expected value is just using the FEWER row of that grid

LESS L 90% -5 W 10% 10 == -3.5 expected value score.

But in the aggregate, its smarter to use MORE troops because the expected value of that is 2 (20% * -10 == -2, + 80% * 5 == 4, the sum is 2). So our intuitive expectations between probability and logic seem to break down, or something like that.

The King’s argument says using fewer troops is better. If better is quantified by our score grid, the King’s premises are still true, and the conclusion follows, but yet the conclusion doesn’t match up to our calculations, using MORE troops is better in this case.

Of course, we calculated our answer in a different way than the King’s logic uses. But it seems to me to be like a lot of paradoxes that rely on the gap between intuitions and formalism.