Semiosis, as a process, takes time. It cannot be well represented by a timeless logic, which generates self-contradictory propositions such as the ‘liar paradox’ by allowing a proposition to refer to itself as if it could exist as an individual (as the possible object of an index) before it is uttered, which in fact it cannot.
In his third Harvard Lecture of 1903, Peirce explained the relations between the subject of a proposition and the object of a sign in a way that shows why a genuine proposition cannot refer to itself. He begins by referring to the principle of excluded middle, which in classical logic says that a proposition must be either true or false – any option in between those two is excluded. He points out that this principle applies only when the subject of the proposition specifies an individual, and not when it is general.
That is, to say that the principle of excluded middle applies to S is no more than to say that S, the subject of the proposition, is an individual. But how can that be? We know very well that universal propositions have general subjects of which the principle of excluded middle is [not] true. That is, it is not true that “all men are either tall or not tall.” The logic of relatives furnishes the solution, by showing that propositions usually have several subjects, that one of these subjects is the so-called Universe of Discourse, that as a general rule a proposition refers to several Universes of Discourse, the chief of which are Singulars, and that all propositions whatsoever refer to one common universe,—the Universal Universe or aggregate of all Singulars, which in ordinary language we denominate the Truth. The analysis of the logic of relations shows that such is the fact, and by the aid of the categories we can easily see why it should be so. A proposition is a symbol which separately INDICATES its object, and the representation in the proposition of that object is called the subject of the proposition. Now to INDICATE is to represent in the manner in which an index represents. But an index is a representamen which is such by virtue of standing in a genuine reaction with its object; while a singular is nothing but a genuinely reacting object. It does not follow that the subject of a proposition must literally be an index, although it indicates the object of the representamen in a manner like the representation of an index. It may be a precept by following which a singular could be found. Take for example the proposition:
Some woman is adored by every Catholic.
This means that a well-disposed person with sufficient means could find an index whose object should be a woman such that allowing an ill-disposed person to select an index whose object would be a Catholic, that Catholic would adore that woman.Thus the subject of a proposition if not an index is a precept prescribing the conditions under which an object is to be had.
Consequently, though the subject need not be individual, the object to which the subject of a proposition applies must be the object of a possible index and as such it must be such as it is independently of any representamen or other Third. That is to say it must be real.
Consequently, it is impossible that a proposition should relate to itself as its object, since as long as it has not yet been enunciated it possesses characters which are not independent of how they may be represented to be.
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A semiotic process involving self-reference, when it takes the form of a proposition, takes its already-determined past (and not its living presence or semiotic functioning) as its object to determine its future as its interpretant. In the same way, a self-referential function in mathematics takes the result of its previous iteration to produce the result of the current iteration, which will in turn be used to produce the next result, and so on.