logic symbol for 'unlike, differing from'

Question

I'd like to express 'unlike, differing from', in the most 'academic/professional' fashion using logic symbol(s) or mathematical operators. The descriptions of the corresponding Unicode block seem to show that there's not one symbol for it.

Maybe it's more of a formal semantic/linguistic issue, if so please let me know.

Example: Unlike set B, set A contains X (or X belongs to A).

Answer 1

How you interpret 'unlike' can really change depending on the context. For example, if I say that 'set A is unlike set B', then I would probably take this as a simple $A \not = B$. But with your example, we mean something more specific, namely that set $B$ is not like set $A$ in this particular respect. So, I think at the very least your desired operator $*$ would have to take this 'in this particular respect' into account. So we'd be looking for something like:

$*(x,y,\phi(z))$ if and only if $\phi(x) \land \neg \phi(y)$

(that is, one of $x$ and $y$ has property $\phi(z)$ while the other does not ... I think it would be handy to make this symmetric as far as the first two arguments go)

which I suppose would make it a second-order logic operator.

So, for your example, we could use this operator and say that $*(B,A,X \in z)$

Well, I have not seen anything like this, but it could be useful, sure! (though of course, why not just say $X \in B \land \neg X \in A$ ?)

Answer 2

There is nothing unprofessional about using words like "unlike" in your mathematical narrative. If you were working on a modal logic with a modality for "likeness" and "unlikeness" you might be justified in adopting a new symbol for it, but otherwise it is best to stick with well-known notation.

Answer 3

Try the following,

$$\forall x \forall y ( x \neq y \leftrightarrow \left. \left( \neg x \neq x \wedge \left( \begin{array}{c} ( y \neq x \rightarrow x \neq y ) \\ \wedge \\ ( x \neq y \rightarrow y \neq x ) \\ \end{array} \right) \right) \right)$$

$$\forall x \forall y ( x = y \leftrightarrow \exists z( x = z \wedge z = y ) ) $$

$$\forall x \forall y ( x \neq y \rightarrow ( x = x \wedge y = y ) )$$

$$\forall x ( x = x \rightarrow \neg x \neq x )$$

$$\forall x \forall y ( \neg x = y \rightarrow x \neq y ) $$

In the classical logic of identity, there is an objection to what is called the priniciple of identity of indiscernibles. A singular term is said to convey "self-identity". So, "unlikeness" is neither a primitive or a simple negation (if it were, identity would follow from the inability to show that two singular terms differ). The second sentence above had been developed by Tarski. Taken as a definition for a non-eliminable identity, the definiens provides a semantic warrant for uses of the sign of equality. Because a diversity relation cannot convey "self-identity", a true diversity relation must imply the self-identity of its terms. If an individual term is self-identical, it ought not be diverse from itself. And, if an identity statement is false, a corrresponding diversity relation ought to be true. Note, however, that a conditional is used rather than a biconditional in order to respect the objection to the identity of indiscernibles.

Now, consider the eliminable defined equivalence relation obtained from the set-theoretic primitive,

$$\forall x \forall y ( x \approx y \leftrightarrow ( \forall z ( x \in z \leftrightarrow y \in z ) \wedge \forall z ( z \in x \leftrightarrow z \in y ) )$$

If the sentences,

$$\forall x \forall y ( \forall z( x \in z \leftrightarrow y \in z ) \rightarrow \neg x \neq y )$$

$$\forall x \forall y( \exists z \neg( z \in x \leftrightarrow z \in y ) \rightarrow \neg x = y )$$

$$\forall x \forall y ( \forall z( z \in x \leftrightarrow z \in y ) \rightarrow \forall z ( x \in z \leftrightarrow y \in z ))$$

are taken as axioms, one can prove

$$\forall x \forall y ( x \approx y \rightarrow x = y )$$

$$\forall x \forall y( \neg x \approx y \rightarrow x \neq y )$$

on the basis of the usual notion of extensionality in set theory as expressed through the third axiom.

However, to finish what has been done here, the sentence

$$\forall x \forall y ( \exists z ( x = z \wedge z = y ) \rightarrow x = y )$$

would have to be introduced into the logical schema with adjustments to the quantifier rules.

Philosophers would object to this on the basis that such logic rules would suggest what they call "contingent identity". This is fundamentally a metaphysical objection. So it is important to remember that the situation with set theory is such that the standard axiom of extension is not an eliminiable, defined equivalence. That axiom is structured specifically to defer questions about identity to the first-order predicate logic. That set theory is formulated in this way is typically attributed to the influence of Skolem.