Opening Example
Consider the following two statements:
Statement 1
(1) For any two houses $H_{1}$ and $H_{2}$, if $H_{1}$ has more square footage than $H_{2}$ then house $H_{1}$ is more expensive than $H_{2}$.
Statement 2
(2) For any two houses $H_{1}$ and $H_{2}$, if $H_{1}$ has more square footage than $H_{2}$ then house $H_{1}$ is more expensive than $H_{2}$, all other things being the same (ceteris paribus).
Statement 1 (without the ceteris paribus condition) is false.
Statement 2 (with the ceteris paribus) condition is true.
Houses are used only to provide intuition; I really am seeking mathematical applications.
Given two vectors $v$ and $w$ taken from $\mathbb{R}^{3}$ if $v_{1} > w_{1}$ then $\sum_{k=1}^3 v_i > \sum_{k=1}^3 w_i$, all other things being the same (ceteris paribus)
End of Opening Example
MOTIVATION :
My question arose when I was trying to model the ceteris paribus principle in the standard mixture of logic and set theory in common use by contemporary mathematicians.
CONFUSING (BUT CONCISE) QUESTION :
What characterizes a a set of a objects, such that given a binary predicate on that set, if we eliminate all unary predicates contradicted by the binary predicate, are the leftover predicates mutually satisfiable?
Begin Additional Details
Let $\mathcal{HS}$ be the set of all houses.
The set of "Houses" is really a place-holder term.
I only used houses to provide some informal intuition for the problem.
You could have $\mathcal{HS}$ be the set of all real numbers $\mathbb{R}$.
A parameterized statement that one house has more square footage than the another house can be modeled as a predicate on two houses.
Let $Q$ be a subset of $(\mathcal{HS} \times \mathcal{HS}) \times \{\text{true}, \text{false}\}$ such that for all $H_{1}, H_{2} \in \mathcal{HS}$, $\begin{pmatrix} (H_{1}, H_{2}), \text{true} \end{pmatrix} \in Q$ if and only if $\begin{pmatrix} (H_{1}, H_{2}), \text{false} \end{pmatrix} \not\in Q$.
Predicate $Q$ accepts two houses as input, but there are predicates which accept only one house as input.
For example, $\forall H \in \mathcal{HS}$, we could have $P(H)$ be true if and only if house $H$ was built in the year $1987$.
Another example of a predicate which only takes one house as input is the predicate which is true if and only if both houses are painted blue.
If you assert $Q(H_{1}, H_{2})$ (the square footages of the two houses are different) but the houses are the same in all other respects, then for lots and lots of single-argument predicates $P$, we have $P(H_{1}) = P(H_{2})$.
- Both houses have the same number of bathrooms.
- Both houses have the same number of bedrooms.
- Both houses are located in the same city.
- etc...
Sometimes it is the case that for any houses $H_{1}, H_{2}$, $Q(H_{1}, H_{2})$ implies $P(H_{1}) \neq P(H_{2})$.
Other times, it is the case that $\exists H_{1}, H_{2}$ such that both of the following are true:
- $Q(H_{1}, H_{2})$
- $P(H_{1}) = P(H_{2})$.
For example, there exist two houses which have different square footages, but the paint-color of house \text{true} is the same as the paint color of house 2.
If you were to eliminate all single-argument predicates $P$ such that $\forall a, b, Q(a, b)$ implies $P(a) \neq P(b)$, would the remaining predicates be mutually satisfiable?
Suppose that $PS^{\prime}$ is the set of all predicates $P^{\prime}$ such that predicate $Q$ beging true does not imply that the one-argument predicates evaluate differently.
That is, $PS^{\prime}$ is the set of all predicates $P^{\prime}$ such that $\exists a, b$ such that $Q(a, b)$ and $P(a) = P(b)$.
Is $N = \begin{Bmatrix} k \in \mathbb{N}: \text{true} \leq k \leq \text{true}\text{false} \end{Bmatrix}$ a set of objects which have the desired property?
$N$ is the set of all whole numbers numbers $\text{true}$ through $\text{true}\text{false}$.
Informally, suppose that $\mathcal{PS}$ is the set of all non-trivial predicates which accept a single whole number as input.
More formally, let $\mathcal{PS}$ be the set of all functions from $N$ to $\begin{Bmatrix} \text{false}, \text{true} \end{Bmatrix}$ which satisfy the following conditions:
- $\forall P \in \mathcal{PS}$, $\exists k \in N$ such that $P(k) = \text{false}$
- $\forall P \in \mathcal{PS}$, $\exists k \in N$ such that $P(k) = \text{true}$
Let $\mathcal{QS}$ be the set of all non-trivial predicates which accept a two numbers as input.
More formally, let $\mathcal{QS}$ be the set of all functions from $N \times N$ to $\begin{Bmatrix} \text{false}, \text{true} \end{Bmatrix}$ which satisfy the following conditions:
- $\forall Q \in \mathcal{QS}$, $\exists a, b \in N$ such that $Q(a, b) = \text{false}$
- $\forall Q \in \mathcal{QS}$, $\exists a, b \in N$ such that $Q(a, b) = \text{true}$
- $\forall Q \in \mathcal{QS}$, $\forall a, b \in N$ if $Q(a, b) = \text{true}$ then $a \neq b$
Let $Q$ be some arbitrary non-trivial predicate on two arguments.
Let $\mathcal{PS}^{\prime} \subseteq \mathcal{PS}$ such that $\forall P^{\prime} \in \mathcal{PS}^{\prime}$, $\exists a, b \in N$ such that $Q(a, b) = \text{true}$ and $P^{\prime}(a) = P^{\prime}(b)$.
Do there exist $a, b \in N$ such that $Q(a, b)$ and for all $\mathcal{P}^{\prime} \in \mathcal{PS}^{\prime}$, $\mathcal{P}^{\prime}(a) = \mathcal{P}^{\prime}(b)$
The negation of the above would be $\forall a, b \in N$ if $Q(a, b)$ then there exist two predicates ${P^{\prime}}_{1}$ and ${P^{\prime}}_{2}$ in $\mathcal{PS}^{\prime}$ such that at least one of the following is true:
- ${P^{\prime}}_{1}(a) \neq {P^{\prime}}_{1}(b)$.
- ${P^{\prime}}_{2}(a) \neq {P^{\prime}}_{2}(b)$.
If I understand your question correctly, take $Q$ to be the ordinary predicate $a < b$ on $\mathbb{N}$. Then $PS'$ is the set of unary predicates $P$ such that there exists $a < b$ such that $P'(a) = P'(b)$; this just says that $P$ takes the same value on two different positive integers at least once. Your question is whether there exists $a < b$ such that for all $P' \in PS'$ we have $P'(a) = P'(b)$, and the answer is clearly no. Given any $a < b$ we can consider the predicate $P'_b$ which is true for $b$ and false otherwise; this is in $PS'$ and $P'_b(a) \neq P'_b(b)$.
I'm not sure if this is a sensible formalization of the informal question you are trying to ask so let's go back to the house example. The problem with trying to reason informally here about what "all other things being equal" could possibly mean is that stuff like this does not actually make sense:
Where does this end? Both houses have the same number of rooms of every type. Is it also true that both houses have the same number of rooms of every type on the ground floor? Is it also true that both houses have the property that each room on the ground floor has the same square footage as the corresponding room in the other house? Doesn't that imply that both houses have the same square footage? At what point did we go from "other" respects to "not other" respects in this argument?
The term "other" in "all other things being equal" is ambiguous; in practice it refers to something like the following. We make a modeling assumption that
Then we can ask questions about the behavior of the function $C$ when we vary one of its inputs, keeping the other inputs fixed; this is a completely precise definition of "all other things being equal," and it depends on the choice of parameterization. We need to do this in mathematics, for example, to define partial derivatives. Although a partial derivative $\frac{\partial}{\partial x_i}$ is written as though it only involves one parameter $x_i$, actually defining a partial derivative requires a description of all parameters, because we have to know what we're keeping fixed.