In the book im self studying when the author talks about ordered pair equality he writes $(a,b) = (c,d) \to (a = c \wedge b = d)$
and asserts that the converse is a truth of logic i.e $(a=c \wedge b=d) \to (a,b)=(c,d)$
while thinking about this I thought about what rule of logical identity is he referring to, and how could we use it as a rule of inference in derivations? Going by Suppes introduction to logic book (as that is my basic background so far) It seems clear to me that he is going by what he writes as the principle of extensionality:
if $x=y$ then whatever is true of x is also true of y and vice verse
corresponding to this the Rule of Inference needed is what he writes as:
Rule Governing Identites:
If $S$ is an open formula, From $S$ and $t_1 = t_2$, or From S and $t_2=t_1$, we may derive $T$, provided $T$ results from $S$ by replacing one or more occurrences of $t_1$ in $S$ by $t_2$. Moreover the identity $t=t$ is derivable from the empty set of premises
Thus in a derivation according to the Rule Governing Identites we may proceed (informally): let $S$ be $(a,b) = (a,b)$ and suppose that $a=c \wedge b = d$ thus applying the Rule Governing identities twice we infer $(a,b) = (c,d)$
In order for a formal derivation to go along these lines (as stated just above) to be correct we must be sure $(a,b)=(a,b)$ is in fact a open formula
Suppes gives the definition of an open formula as a formula with no quantifiers where the usual definition of a formula is given recursively, we know equality $a=b$ where a, b are terms is a formula. But the Main question of this post is:
Why is $(a,b)$ a term? Furthermore as the other example beneath why is the unit set {x} a term?
Here is Suppes definition:
a TERM is an expression which either names or describes some object, or results in a name or description of an object when the variables in the expression are replaced by names or descriptions.
My thinking was well $(a,b)$ is a term since if we replace $a$ with $1$ and $b$ with $2$ we get an expression which names the ordered pair $(1,2)$
Does this seem correct?
This question was inspired by the "truth of logic" converse for equalities as another example of this "truth of logic" converse $a=b \to ${a}$=${b} but once again in a formal derivation if we want to apply the Rule Governing Identities we must be sure that {a} is a term and if it is we may proceed by the above.
Best Answer
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.