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Can a predicate in logic operate on something undefined ? Is $P(x)$ true or false for $x$ undefined, where $P$ is a predicate ?

To be more concrete:

Is $x \le 5$ true or false for $x$ undefined ?

Does $\{i \in \mathbb Z \mid P(i)\}$ contain negative integers or zero, if $P$ is a predicate with domain the positive integers ? For example $P(i)=\log(i) \ge 5$.

Just to be sure: a predicate is a function that results in either true or false ? (Please confirm).

Mikkel
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  • In FOL, no, in Fuzzy logic, you can assign probabilities. At least in the def. given in: http://en.wikipedia.org/wiki/Predicate_(logic) – user99680 Jun 28 '14 at 19:11
  • No, a predicate is almost the opposite of what you say. A predicate with free variables doesn't have a truth value because it's not even a statement. – Git Gud Jun 28 '14 at 19:15

1 Answers1

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To be precise, you can only define logical truth for first order sentences. That is, a formula which is closed for all its variables. In your case, you can't even ask whether $x\leq 5$ is true or false. On the other hand, if $x$ is any specific element in you domain of interpretation, then $P(x)$ acts like a function from the domain to $\{T,F\}$.

Kaa1el
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  • The set that made me wonder is this $${x \mid \log(x) \ge 1}$$. Is it implicit here that this rule only applies to positive real numbers ? Since for negative real numbers and zero the rule is undefined ? The rule has an implicit domain ? What does it mean for a formula to be closed for all its variables ? – Mikkel Jun 28 '14 at 19:17
  • If you just write $\log$, then this is merely a symbol written on the board. To give it a meaning, you need to specify a domain of discourse, like $\mathbb{R}$ or $\mathbb{C}$, and say "let $\log$ be the function defined as the inverse of exponential function, blah blah blah". There is no implicit domain behind all symbols, only conventions. A closed formula $\varphi$ means each variable occurred in it is bounded by a quantifier with the same variable. e.g., $\forall x(P(x))$ is closed, but $\forall x(P(y))$ is not closed. – Kaa1el Jun 28 '14 at 19:25
  • For $\log$, if you are in the domain $\mathbb{R}$, then for non positive real numbers $x$, $\log(x)$ is undefined. If you are in the domain $\mathbb{C}$, then only $\log(0)$ is undefined. – Kaa1el Jun 28 '14 at 19:28
  • More general (though this is only part of the problem), you should always be careful with tha classbuilder syntax ${,x\mid \ldots,}$ as opposed to $\langle ,x\in A\mid \ldots,}$ (which is backed by the Axiom scheme of Comprehension). – Hagen von Eitzen Jun 28 '14 at 19:56
  • @HagenvonEitzen Could you explain further what do you mean to be careful with the class-builder syntax? – Eric Sep 25 '16 at 13:48