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Here are some "concepts" that are confusing me:

The assertion $x>2$ is true when we replace $x$ by a number greater than $2$($3$ for example) and false if we replace it by $1$. But the assertion $\forall x(x>2)$ is false. In the first case $x$ is free while in the second case $x$ is a bound variable. Now for these two assertions:

If $x>2$ then $x>3$.($x$ is understood to be a real number).

For every $n$ if $n>2$ then $n>3$.

Is the $x$ in the first assertion free while the $n$ in the second assertion bound ? The Handbook of Mathematical Discourse states that in the first assertion $x$ is actually universally quantified like the second assertion. Can someone elaborate on that ?

Also, when proving them, their proofs are exactly the same except in the second one we add "let n be arbitrary"(How to Prove it). So do the two assertions differ in their Logical structure ?

Now for 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.

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I think there are too many questions at once. Some related to each other, others not so much. I wouldn't be surprised to see this question being closed as "Too broad". In fact notice the description below this closing reason: "There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs". –  Git Gud Jun 4 at 17:03
    
@gitgud Its all related to logical forms of definitions and the semantics used to express it? Otherwise what do you suggest I do ? –  Nameless Jun 4 at 17:06
    
I counted fifteen question marks in your question alone. These are too many questions. I would separate the first three questions from the rest, making that one MSE-question. Then from "Now for definitions" to "Also when trying to prove "$x$ is even" what exactly should I do ?" I would make it another MSE-question and the last part yet another MSE-question. You're free to link each question to each other if you feel it's relevant. Do note, however, that all this is just my opinion. The rest of the community may disagree. You're free to do as you wish. –  Git Gud Jun 4 at 17:13

1 Answer 1

up vote 3 down vote accepted

The simple, summary answer to a lot of these questions is that when math definitions and proofs are given in (approximately) plain English, often a universal quantifier is implied rather than expressly stated. That's an elaboration of the statement in the Handbook. You don't have an actual statement if there are free variables, so the binding is implied and reader must mentally supply it. Normally it's a universal quantifier, not an existential one, but context determines.

So, for example, "a number $n$ is even if it is divisible by 2" could be read, literalistically, to mean "There is a number $n$ such that (etc.)," but since what's being presented is a definition of the concept "divisible by 2," obviously that needs to be generally applicable, and so the correct reading is "For any number $n$, $n$ is even if (and only if) it is divisible by 2." Notice that the "only if" was also implied rather than stated.

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So you mean to say that Every variable in a proof or definition is actually implicitly quantified ? Also on a side note if I say $x=2$ this is an assertion(right?) but if I say $f|C=f\cap (CxB)$ this is not an assertion as the equal sign acts as a "define" in this context(right)? –  Nameless Jun 4 at 17:29
    
I wouldn't say absolutely every without spending more time thinking about it. There might be a style of proof or definition that involves formulas with unbound quantifiers. But when you encounter a definition or proof presented in plain English plus the occasional symbol, normally the quantification is implicit if it is not overt. How about this: you should assume so unless there is evidence to the contrary. –  StumpyLeg Jun 4 at 17:35
    
In response to your side note.... in the absence of context, $x=2$ is not an assertion because $x$ is neither a bound variable nor a name for a specific identified object, so that the equation is neither true nor false. The definition, on the other hand, is an assertion (to answer another question you raised in passing). Definitions exemplify a particular kind of truth, namely truth by stipulation, and because they are true, they have a truthvalue, and so they qualify as assertions. –  StumpyLeg Jun 4 at 17:39
    
What do you mean by nor a name for a specific identified object. Also can you give a quick example of when $"="$ is used to name a mathematical object ? –  Nameless Jun 4 at 17:44
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I disagree with Stumpy's take on definitions, but I don't want to elaborate too much on it, so I'll just leave it as a comment here. Definitions are part of the language people use to communicate mathematics (and other things as well). In mathematics, in particular, definitions are just abbreviations (perhaps cognitively more natural) of statements in the (formal) language at hand. For instance saying that $n$ is even is just an abbreviation of $\exists k\in \mathbb Z(n=2k)$, nothing more. –  Git Gud Jun 4 at 17:45

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