Proper Notation for Predicate Variable Not In Universe? Let's say I have the simple statement Q(x): x + 1 > 2x.
For a universe of all integers, I can easily compute any truth value. If my universe of discourse changes to, say, x | x < 1, what is the truth value for Q(1) now? Do I simply say it is undefined for the universe? What is the proper/standard terminology?
Thanks for any/all help provided!
 A: It depends on what you take your formal language to be. If you take the language to include a constant symbol for $1$ (along with $+$, $\times$, and $<$), say, then any model in this language must interpret the constant symbol '$1$'. In the universe of discourse $D = \{x \in \mathbb{R} \mid x < 1 \}$ (I'm assuming that's what you mean by '$x < 1$'), the interpretation of '$1$' will have to be some element in $D$, and hence won't be $1$ itself. So model with $D$ as its universe of discourse will make $Q(1)$ true/false, depending on which non-one element the model assigns to the constant '$1$' and depending on which functions/relations it assigns to the symbols '$+$', '$\times$', and '$<$'.
On the other hand, if '$1$' is not a constant, then $Q(1)$ won't even be a sentence in your language, so its truth value isn't something you can discuss in the language you're working in. It may be that you can define the constant '$1$' in the language you're working with, in which case the use of '$1$' is just a useful abbreviation for a sentence not involving '$1$'.
