You can suppose any sentence you want in order to draw conclusions. To suppose a sentence does not mean that you are persuaded that it is true. And, of course, you can suppose conjunctions. After all, this is how reductio-proofs work. If, however, you insist to introduce a notion of supposition of "single thoughts" (suggested by your premise (4)), then you probably have to add one more premise to your argument with the strength of: A single thought is one which can be supposed (in the sense of the other premises) and of which every implication can be supposed. I.e. you would need to introduce a notion of supposition which is closed under implication. But then you would not need to suppose anything, because by supposing you would have to pre-suppose that you know all implications which your supposition has. So why bother to prove anything?

See All Liar, No Paradox for another analysis.

It's also worth noting that Kurt Goedel gave us, with his diagonal lemma, a way of constructing Liars which are purely statements of arithmetic.  There is a sentence phi involving nothing more than elementary arithmetical operations and quantifiers, such that Peano Arithmetic proves that phi is equivalent to "phi is not provable in Peano Arithmetic", the latter being an abbreviation for another purely arithmetical formula which merely encodes the intended statement.  (Peano Arithmetic can be replaced by even weaker theories, if you like.)  If the language is extended by a new unary predicate T (intended to be thought of as a truth predicate), the diagonal lemma even gives us a pure-arithmetic sentence phi such that PA proves phi is equivalent to ~T(_phi_) where _phi_ is a canonical code of phi.  Harvey Friedman and Michael Sheard wrote an excellent paper on this, http://philpapers.org/rec/FRIAAA, it's really a tragedy that it's barricaded behind an Elsevier paywall.

The unspoken premise: the statement can be trusted.
The statement claims to be untrustworthy. an obvious contradiction.

When a statement contradicts a premise you have a paradox. 
No need to suppose the liar... you've already started with/arrived at your paradox. so the premise is false. (the statement is not trustworthy) So any statements it makes cannot be useful even if it agrees with truth.

(if you assume the liar is telling the truth you actually never reach a paradox, so you learn nothing about the original premise.)

There is reasonable doubt concerning (4). Not only is the individuation of thoughts problematic, but it seems that to suppose X is NOT to entertain a single thought. The fact that (2) is true does not mean/ entail that I entertain a single thought. Nor can I assume that.

The idea of lying is not something that can have an epistemological disjunct, or truth value: there is no possible world in which the only epistemological content is the mere idea of lying. The act of lying is the act of misrepresenting something. But, the mere idea of that act is not that act. One can only conflate the two and end up with a sense of paradox.

But, the strength to which one unwittingly conflates them determines how much explication is required to dissolve the illusion. Even the impersonal self-descriptive statement, 'These six words are a lie', can be viewed from the frame of reference of the conflation, but one would hope that that frame of reference is entirely voluntary and recognizant.

The normal neurolinguistic habit is to assume that something genuinely serves as subject of misrepresentation in such statements. But, fortunately, the mere word 'misrepresentation' is not normally, if ever, used in such a way as to create a paradox on itself. What is it about which that word is misrepresenting? Is the very word 'lie' a lie??? LOL

Then, epistemologically adverse elements are not the only 'game in town'. One can as well (but perhaps not as readily) view the word 'True', self-referentially. Of course, it is not normally used to represent epistemological adversity, so it does not allow a paradox. The mere idea of 'something being true', such as a contingent condition, is not an actual, distinct member of epistemology, but merely a reflection of the language agent in the mirror of itself (metalinguistic).

A purely sentimentally adverse self-reference is not paradixical: 'I find it unpleasant to have to speak my mind.'

But, a combination of sentimentally and epistemologically adverse elements is possible, and thus allows a paradox by being interpreted as an instance of hypocracy: 'I feel that there is no point in saying anything.'

Further self-referential explications of the non-information of an incomplete statement are possible. For the liar statement, a rather lengthy such explication would be, ‘If you accuse me of lying about saying in this current run-on sentence that I am lying, and, indeed, I assert herein that I am lying, then you must prove purely therewith that I am lying, but, since you can’t prove it purely therewith without also proving that I’m telling the truth herein, then I think you misunderstand what this run-on sentence is actually about, namely your failure to see how I’ve spun you up in the mechanics of your own thinking.’

Re: All Liar, no paradox:

The statement, 'This statement is an instance of lying', is, from a certain frame of reference, correctly representing itself, in that the statement is not actually committing a lie, in which case its self-representation as an instance of lying is a misrepresentation of itself. But, other than that, there is no lie, and, since it is intended as self-referential, there is, in fact, a paradox.

The problem is that the statement does not, and cannot, stand on its own: it has no mind, no intention, in the sense that it is just a sequence of forms which do not constitute meaning. Only by way of a mind's accustomed interaction with that sequence is there a lie, a paradox, or a failure to inform about that sequence's normal functional content.

Taken self-referentially, the term 'lie', produces a paradox, because then the mind is in the act of contradicting itself as to whether the idea of lying constitutes an act of lying. Answering it in the affirmative is an instance of something being false: 'The idea of lying is an instance of lying' is false.

There are two necessary conditions for the Liar:
The Liar Sentence.and The Liar Definition. 

Example:
1 expression 1 is not true (Liar Sentence)
2 expression 1 = "expression 1 is not true" (Liar Definition)

IF there is no Liar Definition THEN the term "expression 1" 
in "expression 1 is not true" has no reference:
And the expression "expression 1 is not true" is NOT a statement!



Another way of explaining the Liar is to say that the two "sentences" are each others "contexts":They change each others internal meaning. Looked upon in isolation then "expression 1 is not true" can be about ANY sentence and will be only a statement function until the term "expression 1" is defined.

Thats where the Liar Definition comes to the rescue: 
Accepting it makes the liar sentence become a statement. BUT? 

What reason is there to accept the definition? 
Looking at it in isolation: x = "x is not true" we conclude that there is no such x!

Thats where the Liar Sentence comes to the rescue: 
If expression 1 IS a statement then the definition is correct, premiss 2 is true, premiss 1 is a supposition and the Liar becomes a Paradox.