MHB Can Predicate Properties Be Neither True nor False?

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The discussion centers on the nature of predicates and their truth values, questioning whether a predicate can be neither true nor false. It is established that a predicate's truth set consists of values that make it true, and that predicates have finite variables, leading to the conclusion that certain options in the posed questions are false. The truth set of a predicate can indeed be empty, supporting the notion that a predicate can lack true values. However, participants express concerns about the precision of the questions, particularly regarding the definitions of "always" and the role of variables. Overall, the conversation highlights the complexities and ambiguities in understanding predicate properties.
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I'm not very sure whether a predicate can be neither true or false, and I haven't seen any example so far.

The second choice is false because it is the truth set that is the set of all values which make the predicate true.
A predicate has finite variables, so the third choice is false too.

I believe the truth set of a predicate can be empty, so the last choice is true.
 
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I agree with your choices, but I find the questions not precise enough. According to Wikipedia, a predicate on a set $X$ is a function from $X$ to $\{\text{true}, \text{false}\}$. With this definition it is not clear what "always" means in question 1 and what it means for a predicate to contain variables in question 3.
 

Thread 'Formal derivation of statement from Peano Arithmetic system'

I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...
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