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ahuff44 comments on 2014 Less Wrong Census/Survey - Less Wrong
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I would say that "x=3" has a function from values of x to truth values, as does "if x = 3, then x² = 9" (a constant function to the value "true").
Sure, that's one way to look at it. And a function from values of x to truth values is not itself a truth value. You may say that a constant function from values of x to the value True is not itself a truth value either, but it's much closer (after all, you know which one it would be if it were one), so it's a minor shift to your way of looking at it to get what I said.
Now consider ‘If x² = 9, then x = 3’. A lot of people would naturally want to label that False (at least if they remember about negative numbers). As a function from values of x to truth values, this is not constant (and in fact it assigns True to every real value of x except one), so this is not even the same way of looking at things as in my previous paragraph. But it's common.
So if you want implication between non-truth-values to be a truth value consistently, then this is how I would do it.
That depends on the domain of x. That and the universal quantifier over its domain are typically omitted when they are clear from the context.
Yes, if we're talking only about positive numbers, then ‘If x² = 9, then x = 3’ is true.