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Here are some Examples of the Phrase

All other things Being the Same

  • All other things being the same, the juice of a Granny Smith Apple is more acidic than a the juice of a Red Delicious Apple.

  • All other things being the same, a house with a large number of bedrooms has a higher asking price than a house with a smaller number of bedrooms.

  • All other things being the same, a student who does their homework will get a better grade on the final exam than a student who does not do their homework.

Without the phrase, "all other things being the same" or "ceteris paribus", the statements are all false.

Statement Counter Example
For every red delicious apple r and Every granny smith apple g, g has more acidic juice than apple r Inject the red delicious apple with muriatic acid. Do not inject the red delicious apple with muriatic acid. Now the red delicious apple is more acidic (sour) than the granny smith green apple
For every pair of houses H1 and H2, if H1 has more bedrooms than H2 then H1 has a higher asking price than house H2 There exist three bedroom houses which sold in the city of La Junta, Colorado in the year 2020 for a smaller dollar amount than two bedroom houses in the city of Denver, Colorado in the same year. Note that the cities are different. Also maybe the square footages of the inside of the houses totaled across all rooms
For every pair of students s1 and s2, if student s1 does their homework and student s2 does not do any homework, then student s1 will get a better grade on the final e(x)am than student s2 There exist students who ace the final e(x)am without doing any homework

How would you define "all other things being the same" using formal logic?


Below is my failed attempt

First, let us have some examples of predicates.

Below, we have predicates on students learning logic:

P₁(x) is True if and only if student x already knows what the symbol means in formal logic.

P₂(x) is True if and only if student x has a computer programming background knows that an int is a positive whole numbers, such as 591.

P₃(x) is True if and only if student x has a philosophy background and also remembers that Aristotle wrote about Greek islands floating on top of the ocean.

P₄(x) is True if and only if student x is 20 years old.

Suppose that PS is the set of all predicates on students.

Perhaps when we say that something is true for any two students x and y ceteris paribus, then we mean that:

  • there exists PS ℙ𝕊 such that P ∈ PS**, we have P(x) = P(y).

Additionally, for any predicate P′ ∈ ℙ𝕊 if P′ ∉ PS then for every pair of students x′ and y′ we have ∃ P ∈ PS ⋃ {P′} such that P(x) ≠ P(y)

For lots of properties, the two students have the same property. For example, maybe both students were born in France.

One way to look at it is that everything which is true about both students has already been said in family of predicates PS and there does not exist one more predicate which is equal for the two students.


The Mistake

For any two students. you can find a set of predicates PS such that PS is the set of all predicates such that P(x) = P(y).

No additional predicates can be added to the set without there existing a predicate P such that P(x) ≠ P(y)

The notation P(x) ≠ P(y) means that P is true for x and P is false for y or P is false for x and P is true for y

We wanted to say that the two students are maximally similar.

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    See default logic, "reasoning often involves facts that are true in the majority of cases but not always. A classical example is: “birds typically fly”."
    – Conifold
    Jun 7, 2023 at 0:56
  • "Inject the apple with muriatic acid" is a bit overkill. "The two apples are at different stages of maturity" would already be enough for a counterexample.
    – Stef
    Jun 7, 2023 at 10:51
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    I suggest thinking of your functions as depending on more than one parameters. For instance, call A the acidity of apples. The acidity depends on a lot of factors: the species x_1 of the apple, the maturity/ripeness x_2 of the apple, the quantity of rain x_3 there has been this year, the pollution x_4 in the area, etc. Now, what "all other things being equal" means that for any values x_2, x_3, x_4 we have: A(granny smith, x_2, x_3, x_4) > A(delicious red, x_2, x_3).
    – Stef
    Jun 7, 2023 at 10:58
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    If you remove the condition "all other things being equal", you end up with a much stronger, and blatantly false, statement: for all values x2, x3, x4, y2, y3, y4..., we have A(granny smith, x2, x3, x4) > A(delicious red, y2, y3, y4)
    – Stef
    Jun 7, 2023 at 10:59
  • @Stef The injection of acid or the maturity stages would not be counter examples, yet the other two comments explain the idea quite well.
    – haxor789
    Jun 7, 2023 at 12:06

2 Answers 2

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∃y∃z∀x (x = y OR x = z OR x ∈ A) SUCH THAT (y AND A) IMPLIES z and (NOT y AND A) implies NOT Z.

x, y, and z are propositions about the object of discussion, and the universe of discourse is all propositions about the object of discussion.

There exist y and z such that if a proposition is not y and not z, it is part of the set A which is every proposition about the object of discussion except y and z. When y is true, so is z, and if y changes, z does too, with all other propositions (A) remaining unchanged.

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This may or may not be helpful...

It's easy, if you are talking about the logic used in digital computation. This is because in that realm, there exists something called a don't care condition where the processor carrying out a logical operation can be instructed to ignore a subset of the logical input signals and carry out its program without them. This yields a condition in which "nothing else matters" (all other things can assume any values at all).

To get the "nothing else is allowed to change" condition, you carry out an operation in which a subset of the logical inputs are assigned fixed values regardless of the state of the other inputs, so they never change.

The challenge then is to invent a logical operator/function that asserts independence (A does not depend on B or C) and another which asserts assignment (when evaluating A, B is always X and C is always Y).

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