Predicate and Statement

Question

I was confused on some problems involving predicate.

First, there are many different definitions of predicate, the one I learned is "a predicate in one variable is a mathematical sentence involving a "free variable parameter" which ranged over a well-defined domain of values." Why are there so many different definitions of predicate??

Also, my professor says that the following sentence is a statement, for every x in the set of D, P(x). Where P(x) is a predicate of variable x.

We know that for a predicate to be a statement, we need to quantify all variables, however, in the above statement, we didn't know what set D is. I mean, If we do not know set D, why would the above statement be a statement?

Answer 1

When you are confused in this way you need to take a step back. In mathematics many words are used in different ways in different contexts. For instance, in one topology class I took, a "map" was defined to be a continuous function. Then later in a differential geometry class, "map" was the broad term and a "function" was defined as a map with co-domain $\mathbb{R}$.

This things aren't defined in this way to confuse you, English just only has so many synonyms and so words have to pull multiple duties sometimes.

You are correct that there is something called the predicate calculus, however it isn't really relevant here and so you can forget about it for the moment.

As for "for every x in the set of $D$, $P(x)$", this is an abbreviation for the statement "For every x, if $x\in D$ then $P(x)$". Thus the predicate that you are quantifying the free variables out of is "If $x\in D$ then $P(x)$".

This certainly has $x$ as a free variable, your question is why is $D$ not free too? Basically by stipultion, your professor is declaring that $D$ is a name for a set. Which set? Well that is important for saying whether the statement is true of false, but it is not important for deciding whether or not we have a predicate.

To contrast, you aren't asking why we are allowed to use $P(x)$ here without specifying exactly what it is, but it doesn't matter because we have declared that it is a predicate so everything works out. Similarly we declare that $D$ is a set and the collection of symbols "for every x in the set of $D$, $P(x)$" is a statement, independent of exactly what $D$ and $P(x)$ are.

Answer 2

The language of first-order logic is made of :

• sentential connectives

• quantifiers

• the equality symbol

• countable many (individual) variables : $x_1, x_2, \ldots$

• a set (possibly empty) of constants

• for each positive integer $n$, a set (possibly empty) of predicate variable : $P_1^n, \ldots$ .

Thus we can build a formula like :

$\exists x_2 \forall x_1 \lnot P_1^2(x_1,x_2)$.

In order to give meaning to formulae, we need an interpretation, i.e. we have to choose a domain (usually not-empty) of objects : we will use it for interpreting the quantifiers, and a mapping of the constant symbols on "distinguished" objects of the domain and of the predicate symbols on relations (of suitable arity) on the domain.

Note : you can see here some more details.

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We can consider some formalized language for mathematical theories :

(i) Language of set theory

Equality: $=$; one binary predicate symbol : $\in$; a constant symbol : $\emptyset$.

Thus, using $\in$ in place of $P_1^2$ and introducing the abbreviations : $x \in y$ for $\in(x,y)$ and $x \notin y$ for $\lnot (x \in y)$, the above formula becomes :

$\exists x_2 \forall x_1 (x_1 \notin x_2)$

which is true in the domain of sets, because the domain includes the emptyset.

(ii) Language of elementary number theory

Equality: $=$; one binary predicate symbol : $<$; a constant symbol : $0$.

Thus, using $<$ in place of $P_1^2$ and introducing the abbreviations : $x < y$ for $<(x,y)$ and $x \ge y$ for $\lnot (x < y)$, the above formula becomes :

$\exists x_2 \forall x_1 (x_1 \ge x_2)$

which is true in the domain of natural numbers, because the domain includes the number $0$.

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The bounded quantifiers are not part of the "usual" language of first-order logic.

In the language of set theory

there are two bounded quantifiers: $\forall x \in t$ and $\exists x \in t$. These quantifiers bind the set variable $x$ and contain a term $t$ (which may not mention $x$ but which may have other free variables).

The semantics of these quantifiers is determined by the following rules:

$\exists x \in t\ (\varphi) \Leftrightarrow \exists x ( x \in t \land \varphi)$

<p>$\forall x \in t\ (\varphi) \Leftrightarrow \forall x ( x \in t \rightarrow \varphi)$</p>

Their use is clear; they "bound" or "restrict" the "range of application" of the quantifier to a specific set "named" by the term $t$; of course, $t$ must be a primitive or defined term of the lenguage, like $\omega$ for the set of natural numbers.

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Consider now the expression :

for every $x$ in the set of $D, P(x)$.

We can translate it as : $\forall x \in D (P(x))$.

The question is : what is the meaning of $D$, outside set theory ?

For a formula in first-order language we do not have to specify the domain of the interpretation of the formula: we choose a domain when we interpret it and the quantifiers and variables will receive a meaning through the interpretation.

Consider now the traditional example : "all men are mortal" and translate it in first-order logic in the usual way; we have :

$\forall x (Man(x) \rightarrow Mortal(x))$.

If we note that a subset of the domain of the interpretation is the denotation of a (unary) predicate symbol, we can use the set $Men$ for the intended meaning of the predicate $Man(x)$, and rewrite the above formula as :

$\forall x \in Men (Mortal(x))$.

But in first-order logic there are no variables ranging over the subsets of the domain; thus, $D$ cannot be a variable: it must be an "abbreviation" for some suitable unary predicate.

Thus, outside of set theory, the above formula is of little significance.

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We can find bounded quantifiers used also in semi-formal mathematical contexts, like :

$\forall x \in \mathbb R^+ \exists y (y^2=x)$.

In mathematical analysis, where the domain is the set of real numbers, can be useful to use this symbol to specify different subsets of $\mathbb R$, like in the example above, which is an abbreviation for :

$\forall x (x \ge 0 \rightarrow \exists y (y^2=x))$.

Answer 3

Why are there so many different definitions of predicate??

I would be hard to go wrong just memorizing the definition given by your prof. It's enough to know that there are many subtle variations. At this point in your career, it would be a waste of time to try to determined which definition is the best. After working through several sample problems, it will become second nature.

Also, my professor says that the following sentence is a statement, for every x in the set of D, P(x). Where P(x) is a predicate of variable x.

We know that for a predicate to be a statement, we need to quantify all variables, however, in the above statement, we didn't know what set D is. I mean, If we do not know set D, why would the above statement be a statement?

You should clarify this with your prof, but D may just be his "well-defined domain of values." Or the domain of quantification. If he was writing this using a universal quantifier, he might write:

$\forall x: P(x)$

Some writers may be more explicit, especially if there is more than one domain of quantification under consideration as often happens in mathematics:

$\forall x\in D: P(x)$

Both can be thought of as statements. Opinions may vary, but it is a very minor point.

Answer 4

As we know any indicative sentence that is True or False but not both ,is a statement , so in this case $D$ is a set and we don't know about its content and we say for every $x$ in the set of $D$, $P(x)$ and this point that "we didn't know what set $D$ is" may be cause that our statement be False or maybe be True anyway it's is not against out define !