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Consider two different ways to formally define a predicate, say the predicate $E(x)$ which is to be interpreted as "$x$ is even."

CASE $~1$

$\forall a:[a \in N \implies [E(a) \iff \exists b:[b \in N \land a=2b]]]$

CASE $~2$

$\forall a:[E(a) \iff a\in N \land \exists b:[b \in N \land a=2b]]$

There is a subtle difference in outcomes. $E(1)$ will be FALSE in both cases. $E(-1)$, however, will be UNDEFINED in case 1, but FALSE in case 2.

Which method, if either, is to be recommended?

Dan Christensen
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  • This is an interesting question. How are handling the domain of discourse? Are you making it an inherent part of what a predicate is? Also, how are you going to use this notion of predicate? In some cases, you can just define a predicate as a set (Jech does this for relations in the introduction, but I don't have that book with me at the moment). – Greg Nisbet May 09 '22 at 05:42
  • I think we'd need more context to decide which method is "to be recommended". If your informal goal is to define a predicate on just natural numbers, why do you want a symbol with a larger domain? Who are you communicating with with these expressions? What do you want them to think when they see "$E(-1)$"? – Karl May 09 '22 at 05:55
  • You can see Suppes, page 18, for conditional definitions. – Mauro ALLEGRANZA May 09 '22 at 08:49
  • As per your final comment, the issue is "semantical": in standard semantics for FOL we want that a sentence (i.e. a formula with no free occurrences of variables) has a definite truth value, and thus the "undefined" case must be avoided. If we want to be consistent with usual mathematical practices according to which $-2$ is Even, we have to remove the clauses $a,b \in \text N$ from both, with the result that the two will be the same. – Mauro ALLEGRANZA May 09 '22 at 13:18
  • If we move to a still larger domain (e.g. rationals) we have to find a way to manage "undefined"; we may adopt the convention to "define" it as False when the argument is not an integer. In this case, we have to define also Odd as False when the argument is not an integer, with the consequence that Odd is not equal to not-Even. – Mauro ALLEGRANZA May 09 '22 at 13:40
  • @MauroALLEGRANZA If "undefined" is to be avoided, how about the definition being simply inapplicable for anything but natural numbers? – Dan Christensen May 09 '22 at 15:48
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    The question at https://math.stackexchange.com/questions/687238/what-is-a-formal-definition-of-predicate-logic in no way similar to my question. – Dan Christensen May 10 '22 at 01:40
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    To elaborate on my earlier comment: this question seems to assume that formally defining a predicate requires expressing the definition as an axiom in a logic where quantifiers range over a domain larger than that of the desired definition. This usually isn't necessary, because the "informal" definition is precise enough for practical use and because computer-verified math uses type-aware languages. So what are you trying to do? Are you asking how we imagine translating natural definitions into first-order set theory (in order to justify applying theorems from logic to everyday math)? – Karl May 10 '22 at 19:54

2 Answers2

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CASE $~1$

$\forall a:[a \in \mathbb N \implies [E(a) \iff \exists b:[b \in \mathbb N \land a=2b]]]$

Definition 1 accurately and legitimately defines the even natural numbers.

CASE $~2$

$\forall a:[E(a) \iff a\in \mathbb N \land \exists b:[b \in \mathbb N \land a=2b]]$

For Definition 2 on the other hand, it is unclear what the domain of discourse $U$ is. In any case, this definition says that the set of even objects is $$U\cap2\mathbb N.$$ This definition is accurate precisely when $$U\cap (2\mathbb Z{\setminus}2\mathbb N)=\emptyset,$$ that is, when the domain of discourse contains no non-natural even number.

ADDENDUM

Not sure about your use of $U$ for CASE 2.

The difference between your two definitions depends on what the universe of discourse $U$ is.

To wit: let $U$ be $\mathbb Z$ (i.e., all the integers) and notice that while Definition 1 leaves ‘$-6$’ undefined (which is fine), Definition 2 nonsensically claims that $\mathbf{-6}$ is non-even.

I was also more interested in the correct way to specify the unary $E$ predicate, rather than the subset of multiples of $2$ in $\mathbb N.$

The above example shows that Definition 2 is incorrect when we consider every possible multiple of $2;$ on the universe $\mathbb Z^+$ though, Definition 2 is legitimate.

Which method, if either, is to be recommended?

On the other hand, Definition 1 is always consistent with the standard interpretation, even as we modify its domain of discourse.

ryang
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  • Agree with you on case 1. Not sure about your use of $U$ for case 2. I was also more interested in the correct way to specify the unary $E$ predicate, rather than the subset of multiples of 2 in $N$. – Dan Christensen May 25 '22 at 00:29
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It may be helpful to look into the statements in terms of restricted quantifiers. Restricted quantification is used when the range of a quantified variable is desired to be restricted to a part of the domain of discourse. Usual notations for the general case are (simpler notations are also used):

$(\forall x)_{R(x)}\phi(x)$, which abbreviates $\forall x(R(x)\rightarrow\phi(x))$

and

$(\exists x)_{R(x)}\phi(x)$, which abbreviates $\exists x(R(x)\wedge\phi(x))$,

where $R(x)$ is the restricting predicate and $\phi(x)$ is any formula in the language.

In the context of arithmetic and set theory, they are called bounded quantifiers. For example:

$(\exists n)_{n<t} \phi(n)$

$(\forall x)_{x\in S} \phi(x)$

By restricted quantifiers, we can rewrite the statements as:

  1. $(\forall n)_{n\in\mathbb{N}}(E(n) \leftrightarrow (\exists k)_{k\in\mathbb{N}}(n = 2k))$
  2. $\forall x(E(x)\leftrightarrow x\in\mathbb{N}\wedge(\exists k)_{k\in\mathbb{N}}(x = 2k))$

In the first case, the universal set for the interpretation of the variable is prespecified as natural numbers.

In the second case, the the universal set for the interpretation of the variable is unrestricted (possibly, real numbers). If the set is specified externally, then $x\in\mathbb{N}$ is rendered superfluous (as well as the restricted quantification of $k$). However, that does not mean that such a definition is inferior with respect to the other, it depends on the practical choice to be made according to the context.

Tankut Beygu
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  • The "trouble" in the 2nd case is that $E(-1)$ would be false since $-1\notin N$. I think it is better that it be undefined. – Dan Christensen May 25 '22 at 00:16
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    It is the main task of a formal definition to take in the right samples as true and leave out the wrong ones as false. If the suggestion is that -1 is even not substitutible for x prior to being true or false, as it were, like a syntax error, see my answer. – Tankut Beygu May 25 '22 at 10:29