Semantics and Logical structure in Definitions

Continuation of Free and bound variables in "if" statements

definitions:

A number is even if it is divisible by $2$.

The number is even if it is divisible by $2$.

Is the usage of the and a different in mathematical definitions(sorry for the poor example above, if you have any better example that would be appreciated). Also aren't mathematical definitions assertions ?

A number $n$ is even if it is divisible by $2$.

Every number $n$ is even if it is divisible by $2$.

I know that a definition is NOT an assertion and thus cannot be true or false(Right ?). The definition just describes a property(ies) of some mathematical object.

Now for the above two definitons is $n$ a free variable in the first but bound variable in the second ? Also the second definition can be expressed as $\forall n (n\ is\ even\ \leftrightarrow n\ is\ divisible\ by\ 2)$. Doesn't that mean that a definition in a sense is true or false since it can be expressed by a logical symbols ?

Another definition:

Suppose $n$ is an integer. Then $n$ is even if it is divisible by $2$.

Where is the location of Suppose $n$ is an integer(a precondition) in the logical structure of a definition(after or before the biarrow) ? $\forall n (n\ is\ an\ integer \rightarrow n\ is\ even\ \leftrightarrow n\ is\ divisible\ by\ 2)$ or $\forall n (n\ is\ an\ integer \land n\ is\ even\ \leftrightarrow n\ is\ divisible\ by\ 2)$ or $\forall n ( n\ is\ even\ \leftrightarrow n\ is\ an\ integer \land n\ is\ divisible\ by\ 2)$

Also when trying to prove "$x$ is even" what exactly should I do ?

Another definition:

Suppose $R$ is a partial order on a set $A$, $B ⊆ A$, and $b ∈ B$. Then $b$ is called an $R$-smallest element of B (or just a smallest element if $R$ is clear from the context) if $∀x ∈ B(bRx)$.

How do I express this definiton in logical symbols(are $A$,$b$,$B$, $R$ free or bound?)?What should I do if I want to prove that $z$ is a $H$ smallest element of $M$ ?

Suppose $f : A → B$ and $C ⊆ A$. The set $f ∩ (C × B)$, which is a relation from $C$ to $B$, is called the restriction of f to C, and is sometimes denoted $f|C$. In other words, $f|C = f∩(C × B)$.

What is the location of $f|C$ in the logical structure ?

Also, it would be great to give a list of books that could clarify misconceptions of these type.

Some theory

See George Tourlakis, Lectures in Logic and Set Theory. Volume 1 : Mathematical Logic (2003), I.7. Defined Symbols, page 112-on :

We start with a theory $$\Gamma$$, spoken in some basic formal language $$L$$. As the development of $$\Gamma$$ proceeds, gradually and continuously we extend $$L$$ into languages $$L_n$$, for $$n ≥ 0$$ (we have set $$L_0 = L$$). The theory itself is being extended by stages, as a sequence $$\Gamma_n$$, $$n ≥ 0$$. [...] At that same stage we also add to $$\Gamma_n$$ the defining nonlogical axiom of the new symbol in question.

Specifically, if $$\varphi(x_1, ..., x_n)$$ is some formula, we then can introduce a new predicate symbol “$$P$$” that stands for $$\varphi$$.

This entails adding $$P$$ to $$L_k$$ as a newn-ary predicate symbol, and adding

$$P(x_1, ... x_n) ↔ \varphi(x_1, ..., x_n)$$

to $$\Gamma_k$$ as the defining axiom for $$P$$.

Thus, we can start with the formula $$\varphi(n) := \exists y(n = 2 \times y)$$ and extend the "basic" language with the new predicate $$Even(n)$$ and the theory with the defining axiom :

$$Even(n) \leftrightarrow \exists y(n = 2 \times y)$$.

This is "implicitly" universally quantified, i.e. : $$\forall n [Even(n) \leftrightarrow \exists y(n = 2 \times y) ]$$.

Comment

About the natural language, my "feeling" is that :

a number $$n$$ is even if it is divisible by $$2$$

is a "correct" form for a definition; "a number $$n$$" must be interpreted as "a number $$n$$ whatver" i.e. as having $$n$$ universally quantified.

I would prefer to "read" :

every number $$n$$ is even if it is divisible by $$2$$

as meaning : "for every number $$n$$, $$n$$ is even if it is divisible by $$2$$", which amount to the same statement.

Question (1) :

when trying to prove "$$x$$ is even" what exactly should I do ?

$$Even(n)$$ is defined as : $$\exists y(n = 2 \times y)$$; thus, proving that e.g. $$6 = 2 \times 3$$, by rule of logic we can derive : $$\exists y(6 = 2 \times y)$$, which - by definition - is :

$$Even(6)$$.

Question (2) :

suppose $$R$$ is a partial order on a set $$A, B⊆A$$, and $$b∈B$$. Then $$b$$ is called an $$R$$-smallest element of $$B$$ (or just a smallest element if $$R$$ is clear from the context) if $$∀x∈B(bRx)$$. How do I express this definiton in logical symbols(are $$A,b,B,R$$ free or bound?)?

See Tourlakis, cit, page 114 :

In this case we have to add a new $$n$$-ary function symbol $$f$$ into $$L_k$$ by a definition. That is, we add $$f$$ to $$L_k$$ and also add the following formula to $$\Gamma_k$$ as a new nonlogical axiom:

$$y = f(y_1, ... y_n) ↔ \varphi(y, y_1, ... y_n)$$

provided we have a proof in $$\Gamma_k$$ of the formula : $$(∃!y) \varphi(y, y_1, ... y_n)$$.

In our example, assuming for simplicity that we have already introduced with a definition the symbol $$PO(x,y)$$ as an abbreviation for "$$x$$ is a Partial Order on $$y$$", we can introduce the following symbol $$min(x,y,z)$$ as an abbreviation for "the smallest element of $$z$$ for the relation $$x$$ on $$y$$" by the definition :

$$\forall R \forall A \forall B \forall b \quad [ b = min(R,A,B) \leftrightarrow (R \subseteq A \times A \land PO(R,A) \land B \subseteq A \land b \in B \land \forall x (x\in B \rightarrow bRx)) ]$$

Now, according to this definition, the answer to the question :

what should I do if I want to prove that $$z$$ is a $$H$$-smallest element of $$M$$ ?

is straightforward. You have to prove that, for $$z \in M$$ :

$$\forall x (x\in M \rightarrow zHx)$$.

• Great Answer but here is a couple of points I don't understand: 1) $R⊆A×A∧PO(R,A)∧B⊆A∧b∈B→∀x(x∈B→bRx)$ OR $(R⊆A×A∧PO(R,A)∧B⊆A∧b∈B)→∀x(x∈B→bRx)$ 2)why $→∀x(x∈B→bRx)$ not $\leftrightarrow ∀x(x∈B→bRx)$. I thought if in definitions meant if and only if Jun 5 '14 at 13:06
• @Nameless - you are right for 1) : it is better to group between parentheses all the conjuncts [but note that $(A \land B) \rightarrow C$ is equivalent to : $A \rightarrow (B \rightarrow C)$]. Regarding 2) : NO, the $\leftrightarrow$ of the definition is that between the definiendum (i.e.$b=min(R,A,B)$) and the definiens (i.e.the "big" sub-formula on the right). Jun 5 '14 at 13:14
• Ok. Lets say we have this Definition: Every integer is even if it is divisible by 2. is same as If n is an integer then n is even if it is divisible by 2. means $\forall n (Even(n)\leftrightarrow$ (**n is integer** $\land \exists k(n=2k)))$ (right ?) is it the same as $\forall n (Even(n)\leftrightarrow$ (**n is integer** $\rightarrow \exists k(n=2k)))$ ?? Jun 5 '14 at 13:28
• Also why $→∀x(x∈B→bRx)$ and not $\land ∀x(x∈B→bRx)$. Just to clarify it: you put the precondition before the $\rightarrow$ because these are always given, because without them the definition will not apply to the object we want. And the thing we want to prove after the $\rightarrow$. In this case this ∀n(Even(n)↔ (n is integer →∃k(n=2k))) is right while the other one wrong(is this correct ?). Right ? Jun 5 '14 at 13:34
• 1) Why did you change ∀x(x∈B→bRx) to ∀x(x∈B∧bRx). The first one is right when we have a universal quantifier ? Jun 5 '14 at 14:31