The terms "only" and "only if" mean different things. The first is English and the second is mathematical.
1) "only" in English means "just this and not others", where "others" could be an object or an action, depending on what "only" is connected with. For example, "We only proved that $p$ is prime" means you showed $p$ is prime but not anything further, which I think is synonymous with "We proved only that $p$ is prime"; it is just a stylistic judgment as to which of those you use and I prefer the second version at the moment. However, "We proved that only $p$ is prime" means you proved $p$ is prime while other numbers in the argument (that may possibly have been prime) are not prime.
The classical example, which is not meant as a personal remark, is to insert "only" in front of each word of the sentence "I love you". Each version has a completely different meaning. This is discussed under the topic "Modifiers" on the page
http://www.kaptest.com/GMAT/Learn-and-Discuss/Community/blogs/tag/verbal/
2) In math, "only if" has the same meaning as "implies". (Logically, it's the direction that is not "if", which is how I first was able to remember its meaning.) In terms of Tom's example, "shoes are on ==> socks are on" has the same meaning as "your shoes are on only if your socks are on."
We have two principles dominating our conception of identity, called Leibniz's laws (unfortunately, they are usually referred to as singular, though they are separate laws). One of them is the indiscernibility of identicals, which is represented in symbolic form as
$$\forall x\,\forall y\,\big(x=y\rightarrow \forall F(Fx\leftrightarrow Fy)\big)$$
This principle states the logical identity: For any $x$ and $y$, if $x$ is identical to $y$, then $x$ and $y$ share any of their properties $F$; hence, they are numerically identical, for they count as one.
The other one is the identity of indiscernibles (some refer only to this by Leibniz's law), which is represented in symbolic form as
$$\forall x\,\forall y\,\big(\forall F(Fx\leftrightarrow Fy)\rightarrow x=y\big)$$
This principle states the qualitative identity: For any $x$ and $y$, if $x$ and $y$ share any of their properties $F$, then $x$ is identical to $y$. This is not a logical principle; because $x$ and $y$ may be qualitatively identical, but numerically distinct according to the theory the statement is grounded on, i.e., the theory may not distinguish $x$ and $y$ by their properties, but count them as two.
$(a, b) = (c, d)\rightarrow (a = c\wedge b = d)$ by the definition of ordered pair. In case that $(a = c\wedge b = d)$, what can be said about $a$ is ipso facto about $c$, and so for $b$ and $d$. Hence, we have
$$(a, b) = (c, d)\leftrightarrow (a = c\wedge b = d)$$
as is the case in general for the strictly formal definitions in mathematics. In abstracto, mathematical objects are conferred existence by their properties. Thus, the coincidence of logical and qualitative identities is a rule, rather than exception in mathematics. For example, let us consider an algebraic structure $X$. Suppose $X$ is a group under multiplication. Then, we can list its properties and there is nothing above and over them to its identity. As we acquire more information about $X$'s properties, that it is abelian, that it is finite, and so on, we may make it more concrete. However, at each stage, $X$'s identity is constituted merely by the properties. When a structure $Y$ comes to be identical to $X$ (we leave such algebraic issues as isomorphism aside), $X$ and $Y$ share their identities, numerically and qualitatively.
In the reverse direction, suppose we are given the following (composition) table for a structure $X$:
$\begin{array}{c|c c c}
* & 1 & \omega & \omega^{2}\\
\hline
1 & 1 & \omega & \omega^{2}\\
\omega & \omega & \omega^{2} & 1\\
\omega^{2} & \omega^{2} & 1 & \omega\\
\end{array}$
Then, we can make out that $X$ is a finite abelian group. The more information we acquire, that $w$ is cube root of $1$, the more concrete it becomes, while its identity is always bounded by its properties.
The upshot is that two mathematical objects $x$ and $y$ are always (it may not be of convenience or practically feasible in some cases, though) intersubstitutable whether they are logically or qualitatively identical.
An ordered pair is a mathematical object composed of two mathematical objects $x, y$ together with a relation between them and represented by $(x, y)$. A set $\{\ldots\}$ is another example for mathematical object.
When considering the syntactic notions of formal languages of logic, it may be helpful to draw analogies between them and natural language. Terms function like names, and formulas function like sentences of natural language. A logical name (i.e., what we should understand by "name" that corresponds to "term") is a syntactic element that designates an object. So, a logical name may be a noun by grammar as well as a description (consider a compound term $f(t_{1},\ldots, t_{n})$).
A term can be open or closed in a similar vein as formulas. In an open term, at least one variable occurs, a closed term is either a constant defined in the language or composed of constants. If $a, b$ are variables, then $(a, b)$ is an open term, and $(1, 2)$ is a closed term. Clearly, a closed term designates a specific object. An open term designates an object in a generic sense.
Note that closure difference does not interfere with identity operations. For example, $x + 2$ is an open term and $5$ is a closed term. If $x + 2 = 5$, then we can substitute the terms $x + 2$ and $5$ one for another so long as there is no illegitimate binding of $x$.
In the light of the foregoing discussion, let us examine Suppes' Rule Governing Identities:
$t = t$ is derivable from empty set of premisses as any theorem is; it comes with the logical identity predicate denoted by $=$.
$S$ and $T$ are substitution instances of each other and logically equivalent, because $t_{1}$ and $t_{2}$ are identical. As Suppes notes on p. 103, the use of open formulas is "[t]o avoid lengthy restrictions regarding the presence of quantifiers"; there is nothing deeper about it.
Best Answer
A term is a "name": variables and constants are terms.
In addition, "complex" terms can be manufactured using function symbols.
Example: $n$ is a variable, $0$ is a constant and $+$ is a (binary) function symbol.
Thus, $n,0$ and $n+0$ are terms.
Formulas are statements.
Atomic formulas are the basic building blocks for manufacturing statements.
They are formulas that have no sub-parts that are formulas.
They are manufactured using predicate symbols, like e.g. $\text {Even}(x)$, equality and terms.
Thus, $\text {Even}(n), 0=0$ and $n+0=n$ are atomic formulas.
With connectives and quantifiers we can write more complex formulas, like: $\forall n (n+0=n)$ and $0=0 \to \forall n (n+0=n)$.
Expression can be a "generic" category: it may mean a string of symbols.
We may call well-formed expression a string of symbols that satisfies the rule of the syntax.
If so, it is either a term or a formula.