Liar Paradox, Russell's Paradox, and the Instability of the Subject

According to Aristotle, a true universal proposition must have the contrary proposition that is false. But there are universal propositions which have no contrary proposition. For example: X “All the propositions that contradict this proposition are true” does not have the contrary proposition (and does not have the contradictory proposition). In fact: FEX “All the propositions that contradict this proposition are not true” has a different subject: the subject of the proposition X is constituted by all the propositions that contradict the proposition X; by contrast, the subject of the proposition FEX is constituted by all the propositions that contradict the proposition FEX. According to Aristotle, a true singular proposition must have the negative proposition that is false. But there are singular propositions which have no negative proposition. For example: The proposition (or, rather, the pseudo-proposition) L “This same statement is not true” does not have the negative proposition because the proposition FNL “This same statement is true” has a different subject from the subject of the proposition L: the subject of the proposition L is the proposition L; by contrast, the subject of the proposition FNL is the proposition FNL. L and FNL cannot have the same subject. Aristotle in chapter 8 of “Peri Hermeneias” provides an example of something that has the form of a proposition but is actually a set of sentences. L has the form of a proposition, but is not a proposition because it lacks the corresponding negative proposition; L is actually a set of sentences containing an implicit sentence advising that, although L has the form of a proposition, L is not a proposition because it lacks the corresponding negative proposition. This is confirmed by the fact that L1 "The statement L1 is not true" and L2 “The statement L1 is not true” have different truth values (L1 has paradoxical truth values, L2 has non-paradoxical truth values) even though what is contained between the quotation marks is identical: in the case of L1 additional information is smuggled into the statement, the information that L1 is logically equivalent to L “This same statement is not true.” (which is not a proposition because it lacks the negative proposition). Thus, using the same procedure as above, the proposition G “This same sentence is written in English” would turn out to be a set of sentences (and would not turn out to be a proposition, because it is implicit the affirmation that G lacks the negative proposition, since FEG “This same sentence is not written in English” has a different subject), but it would be true that every word written in G (in the set of sentences G) is written in English. The second part of the paper is about Russell’s Paradox by introducing the method of "importing the subject into the predicate." Drawing a parallel with Hempel’s Paradox of the Ravens, the paper highlights hat the "set of all sets that do not contain themselves" is an impossible construct not because of an arbitrary axiomatic prohibition (as in ZF/ZFC), but because its predication results in a necessary falsehood. By importing the subject, the Russell Set reveals a self-contradictory predicate—analogous to a "black raven that is a non-raven"—which renders the existence of the set logically null. Following Aristotle’s Categories (13b), the author concludes that since the subject is impossible, affirmations regarding its existence are false by necessity, while their denials remain tautologically true.