1

I sort of know that syntagmatic axis is how phonemes arrange in a language, is it true? What is it in general?

3

The basic idea is that the syntagmatic axis concern the way linguistic units (e.g. phonemes or words) are put together with other units. In phonology, for instance, you can study assimilation processes etc. Or, in syntax, you can say that the noun and the determiner agree in number features. All this are syntagmatic aspects.

In contrast, the paradigmatic axis concerns the relation between a linguistic unit and other units that could have been used in the same slot, so to speak (“the constitute a paradigm”). For instance, you can study the paradigm of modal auxiliaries in English.

That is, in (1):

           should
(1) Peter { can } bring out the trash.
            must

You can study the paradigm of {should, can, must, …} on the paradigmatic axis and you can study the relationship between, say, can and the verb it selects (that it cannot have tense, for instance).

2

The distinction is theory-specific, and makes sense only where there is a clear distinction between different "things". The strings [abcdef] and [bacdef] have a syntagmatic relationship made up of 6 distinct things in a particular order. The strings [abcdef] and [abcdex] have a paradigmatic relationship, where one unit is replaced by another in the pair. The problem is that in most theories of phonology, what look like nominally different "things" like [d] vs [t] are not completely unrelated, [d] is just [t] combined with something more (that is, they are syntagmatically related).

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