I assume, based on the your posts elsewhere, that by 'sentence parts', you are referring to grammatical relations (GRs) like subject, object, etc. In the future, it would be clearer for you to call them that, as this is the standard terminology in English.

I think you are confused about argument structures and grammatical relations. Argument structures specific semantic roles like agent, patient, etc. These may be universal, but they do not have bijective maps with grammatical relations. Indeed, what motivates the use of GRs in linguistic analysis is the lack of one-to-one correpsondence between semantic roles and certain grammaticalised aspects of language (e.g. nominative case). Moreover, although argument structure can be easily expressed in terms of traditional predicate logic, grammar is not bound by the rules of any system of logic, so this is not a good argument for the universality of GRs.

With that said, I'd like to split the question into three different ones:

1. Do there exist necessary and sufficient conditions that allow us to conclusively identify GRs in any language?

The answer to this is definitely no. There are no objective criteria to help us pick out subjects, objects, indirect objects, oblique arguments, etc. in any language, in the same way that we can identify chemicals by spectroscopy. A set of criteria that works for English is not going to work for Chinese, and a set of criteria that works for Lakhota is not going to work for Swahili.

In this sense, GRs are not universal. We cannot give a formal characterisation of GRs that work for any language.

2. Is it possible to identify GRs in any language which are subject-like, object-like, etc.?

By this I mean: given a set of prototype characteristics of subjects and objects, can we identify subjects and objects in any language? The answer to this is uncertain. Certain languages, such as Archi and Acehnese, have been claimed by some linguists to lack GRs, and the role usually played by GRs can be replaced entirely be semantics and pragmatics.

However, in most languages it is possible to identify certain elements as subjects, objects, etc. based on a) overt coding (word order, case, etc.) and b) behaviour-and-control properties (pivots, control, raising, etc.) But it must be noted that none of these are necessary-and-sufficient conditions for GRs; GRs are a fuzzy concept here. A classic example is Keenan (1976), which gave a list of 30 characteristics of subjects. So, under this characterisation of universality, yes, all languages could have the same set of GRs; but this set is one of fuzzy categories.

3. Are there linguists working with the assumption that GRs are universal?

Yes; GRs are central to lexical-functional grammar (LFG) and relational grammar (RG); in these theories, GRs are assumed to be theoretical 'primitives'. In RG, grammatical relations are called 1, 2, 3; LFG has a fancier hierarchy of GRs: SUBJ > OBJ > OBJ(theta) > OBL(theta) > COMP > XCOMP > ADJ > XADJ.

Moreover, in typological work it is often necessary to assume the notion of grammatical relations, or it would impossible to make word order generalisations; however, this is purely methodological and does not involve 'reifying' GRs or assuming their universality.

I assume, based on the your posts elsewhere, that by 'sentence parts', you are referring to grammatical relations (GRs) like subject, object, etc. In the future, it would be clearer for you to call them that, as this is the standard terminology in English.

I think you are confused about argument structures and grammatical relations. Argument structures specific semantic roles like agent, patient, etc. These may be universal, but they do not have bijective maps with grammatical relations. Indeed, what motivates the use of GRs in linguistic analysis is the lack of one-to-one correpsondence between semantic roles and certain grammaticalised aspects of language (e.g. nominative case). Moreover, although argument structure can be easily expressed in terms of traditional predicate logic, grammar is not bound by the rules of any system of logic, so this is not a good argument for the universality of GRs.

With that said, I'd like to split the question into three different ones:

1. Do there exist necessary and sufficient conditions that allow us to conclusively identify GRs in any language?

The answer to this is definitely no. There are no objective criteria to help us pick out subjects, objects, indirect objects, oblique arguments, etc. in any language, in the same way that we can identify chemicals by spectroscopy. A set of criteria that works for English is not going to work for Chinese, and a set of criteria that works for Lakhota is not going to work for Swahili.

In this sense, GRs are not universal. We cannot give a formal characterisation of GRs that work for any language.

2. Is it possible to identify GRs in any language which are subject-like, object-like, etc.?

By this I mean: given a set of prototype characteristics of subjects and objects, can we identify subjects and objects in any language? The answer to this is uncertain. Certain languages, such as Archi and Acehnese, have been claimed by some linguists to lack GRs, and the role usually played by GRs can be replaced entirely be semantics and pragmatics.

However, in most languages it is possible to identify certain elements as subjects, objects, etc. based on a) overt coding (word order, case, etc.) and b) behaviour-and-control properties (pivots, control, raising, etc.) But it must be noted that none of these are necessary-and-sufficient conditions for GRs; GRs are a fuzzy concept here. A classic example is Keenan (1976), which gave a list of 30 characteristics of subjects. So, under this characterisation of universality, yes, all languages could have the same set of GRs; but this set is one of fuzzy categories.

3. Are there linguists working with the assumption that GRs are universal?

Yes; GRs are central to lexical-functional grammar (LFG) and relational grammar (RG); in these theories, GRs are assumed to be theoretical 'primitives'. In RG, grammatical relations are called 1, 2, 3; LFG has a fancier hierarchy of GRs: SUBJ > OBJ > OBJ(theta) > OBL(theta) > COMP > XCOMP > ADJ > XADJ.

Moreover, in typological work it is often necessary to assume the notion of grammatical relations, or it would impossible to make word order generalisations; however, this is purely methodological and does not involve 'reifying' GRs or assuming their universality.