No it didn't, arms production would be consider a part of the overall output by definition.
If you are talking solely about consumer goods than that is another matter, however, then you have to provide some data which shows that the living standard, on the average, during the Depression was better than an American during the war.
Uggh... I hate teaching economics, yet I hate hearing poor economic reasoning even more.
Okay let's start with something simple. Opportunity cost is what you give up when you get something. This is incredibly important.
GDP is a measure of the economy. It is the measure of the price of all final goods and services sold in the economy. It is an okay measure of the economy. Personally I don't like it that much, as it is very keynesian and focused on spending, but it is fine.
GDP = Consumer Spending + Government Spending + Investment+ Exports - Imports
Y= C+I+G+X-M
When there is a dip in GDP, Keynesians will try to get gdp back up by increasing governmnet spending to make up for the "slack" in the economy. Simple right. No.
Governmnet is financed by two ways. Taxes, and Bonds(i.e future taxes). By taking money out of the C and I, you are reducing GDP in other sectors. That money would have spent elsewhere.
So how does Governmnet spending increase GDP, well I am glad you asked. Through fiscal multipliers. For example one dollar spent by the government becomes 1.5 dollars, which increases GDP. However it depends on what you spend it on.
Of course it can work in the opposite direction. Government Spending can crowd out Private investment and actually decrease GDP. For example in the case of military spending, 1 dollar can becoming 80 cents and decrease overall GDP.
While yes output has risen, the opportunity cost was a better economic standpoint, albeit Let Hitler overun Europe, let the soviets fight them and not care that millions of people in Europe and Asia were starving and being oppressed by pretty evil motherfuckers. So while the cost was worth it, there were tradeoffs. War is expensive, and don't let anyone fool you otherwise.
Note: This is a vast simplification as I do not want to confuse anyone with too much technical jargon and it 3 in the morning where I live