Antirealists deny the real existence of some kind of thing. One can be antirealist about numbers, the past, (other) minds, propositions, substances -- almost any remotely controversial ontological category. But this ontological story is not the sum total of antirealism.

Truly generic modern antirealism takes the form of a denial that an objective reality underlies bivalence of truth-values. The motivating principle is this idea: realists unwarrantedly assume that all the objects of discourse are ultimately accessible for a determinate truth-evaluation. (For an example of this critique applied to realism in mathematics, we have Dummett's intuitionism.) So antirealism is reconceived as a conservative judgment of the what is meant by the availability of truth-conditions -- as a reactionary analysis of what truth is, and its importance. The modern view is 'linguistic' because the conservative view of truth implies a separation between on the one hand meaning or (human) conceptualization, and on the other truth. Put one way:

Realism: The availability of truth/falsity for our utterances underlies but is not necessarily limited to the availability of meaning.
Antirealism: The availability of truth/falsity for our utterances is strictly limited to the availability of meaning, but does not underlie it. Instead, some other conditions for utterance are what individuate meanings.
(Crispin Wright is, as far as I know, the foremost such antirealist critic currently at work. Refer to him if my explication seems garbled.)

However, it is contentious whether the dichotomy of realism and antirealism presents genuine alternatives, or rather constitutes illegitimate, unsound or vacuous theorizing. For instance, it might be supposed that no arguments for antirealism have any force to convince realists, and vice versa, that the realist thesis is no 'stronger' than its premises either. That is: that any substantive theory in this arena begs the question against the alternative. (Wittgenstein is one modern thinker who would see the contemporary debate here as improper, but for programmatic reasons of his own.)

Kant is a realist about phenomenal things, i.e. about everyday empirical statements; but holds a transcendental idealist, quasi-antirealist view of knowledge and meaning as a whole.

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