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The question should be trivial, but I still can't get my head around it:

Question. Suppose we have an object $y$ belonging to the type $T$ (e.g $y$ may be an integer, matrix, set etc). Now suppose we define $x := y$. Is it true for arbitrary property $P$, $P(x)$ is true if and only if $P(y)$ is true?

In an attempt to further clarify my question, let me give more concrete example (definitions below are taken from the Terence Tao's Real Analysis book):

Definition 1 (Integer). We define integer to be an expression of the form $a $$−−$$b$ , where $a,b$ are natural numbers. Furthermore, integers $a$$−−$$b$ and $c$$−−$$d$ are equal, if and only if $a+d = c + b$.

Now suppose that without verifying whether the substitution axiom holds for the aforementioned definition of equality, we proceed by defining a sum of integers:

Definition 2 (Sum of integers). For arbitrary integers $a$$−−$$b$ and $c$$−−$$d$, we define the sum of an integers as $(a$$−−$$b)+(c$$−−$$d) := (a+c) $$−−$$ (b+d)$

Based on the definitions above, can I safely assume that:

  1. $(a$$−−$$b)+(c$$−−$$d)$ is an integer

  2. $P\bigr((a$$−−$$b)+(c$$−−$$d)\bigl) = P\bigr((a+c) $$−−$$(b+d)\bigl) $ for all functions and operations $P$

?

In other words, for arbitrary mathematical statement or expression, can I substitute $(a+c) $$−−$$ (b+d)$ for $(a$$−−$$b)+(c$$−−$$d)$ (and vice versa) without either changing the truth value or the meaning?

Stokolos Ilya
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1 Answers1

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Yes, this is the case. However, the rules governing equality are "one level lower" than the more "mathematical" axioms like commutativity, associativity, etc. - they're logical rules, at the same level as modus ponens, "From '$A$ and $B$' we can deduce $A$ and we can deduce $B$," and so forth. This level of logic is often not made explicit in texts, which can lead to some confusion; any good book on logic however will present them explicitly.

Noah Schweber
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