Thoughts about Syntax and Semantics

Some stuff I've been thinking about:

Initial thoughts:

Take the proposition: 'A v B'.

This statement can be parsed by looking at the semantics of the proposition and the syntax of the proposition. 

The syntactic components of this proposition would be the signs: {A,B,v}

The semantics of this would be the meaning each symbol represents: 'A' would mean (for the sake of argument): the cat is on the mat, and 'B' would mean: the cat is outside. 

My initial thought is to say there is a semantics to the 'v' connective as well; the connective 'v' has meaning that can be expressed in a exclusive sense or an inclusive sense; a and b can both be true just as long as one is false (inclusive), and only one can be true (exclusive). 

The semantic content of the 'v' connective would then be the thought: 'at least one' or 'only one'.

So its clear that while the syntactic statement is 'A v B', the semantic content would be 'the cat is on the mat or the cat is outside' or 'either the cat is on the mat or the cat is outside'. (not sure if there is any obvious way to make this less ambiguous (the difference between inclusive and exclusive)

Development:

So let's say this occurs:

1. The person thinks the thought ''the cat is on the mat or the cat is outside' and writes the phrase out in logical form.

One could say there are many levels of syntax at work here: the syntax of universal grammar, the syntax of logic, and what I'll later call the syntax of thought.

a) U.G (universal grammar) : I'm assuming names, verbs, letters. (I'm not a pro in this sense of syntax)

b) Logic: the relation between the two instances of judgement being disjunctive, and the sense of the disjunctive itself.

c) Thought: the mental picture of what is being related, and how it is being related: the thought of A, B, and v. It seems to me like every thought stated here would have a structure. The only reasons for this I can give at the moment are two intuitions: everything that can be said to exist has a limiting structure, and something along the lines of: if thought didn't have structure, why aren't we thinking everything at the same time; thoughts as particulars. I guess a third reason I can provide, which is also intuitive, is that thoughts seem (from experience, which is fallible and probably false) to have a common form while differing in content. 

So here is what I think I can say tentatively:

1. 'or' has a sense, and that sense is contained in the syntax 'v'.

2. the sense of the syntactic connective 'v' is logical: it can be defined in some sense as, while I am not acquainted with the current state of the disjunction in the literature, as what I said above 'the connective 'v' has meaning that can be expressed in a exclusive sense or an inclusive sense; a and b can both be true just as long as one is false (inclusive), and only one can be true (exclusive)' seems to do the trick.

3. The sense of the syntactic object 'v' is the thought 'a and b can both be true just as long as one is false (inclusive)', and only 'one can be true (exclusive)'.

4. The thought I'm having right now is structured; the thought has syntax. 

5. What is the nature of the syntax of thought?

My question is: what is the structure/syntax of the thought that grounds logical syntax?