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The distinction between the natural (inferential) and conventional signs refers to the nature of bonding which makes the interpretation possible: the bonding between the sign and its meaning, the object it signifies. For conventional signs, the basis on which the two are connected with each other is a convention, a preset agreement, or even a system of agreements between the users, a code. Natural signs are connected with their objects signified through the inference which, even though requires some prior knowledge or experience on the part of the interpreter, doesn't involve any preset conventions as to what should stand for this particular object. In other words, the connection is natural, and the interpreter only needs to learn about it, not to create it.
For Plato's linguistic theory, as it is presented in Cratylus, assuming that of the persons of the dialogue Socrates expresses the position closest to that of Plato, this distinction is crucial. In this dialogue, the language is basically discussed as a set of names serving for reference and denotation. The dispute focuses on the nature of connection between the names and the things they name. The opinion that we are obliged to take as Plato's mediates between a strict conventionalist view that "all is convention and habit of the users", that everyone can change names as long as there is an agreement between the users, that any name can be made correct, and a strict naturalist view that each thing has its true name, and the name is either a correct natural name, or no name at all, just a bunch of inarticulate sounds. The conjunction of these two extremes in a single theory is achieved through the introduction of two different levels where the naming occurs, at one of which the connection between the name and the named is natural, while at the other it is largely conventional.
First of all, the name is seen as an instrument for indication of the nature of the thing it names. As such, the name, if it is a proper, correct tool for naming of that thing (in other words, if it naturally belongs to it), should express its essence, thus actually naming its form-idea, the materialized abstraction of the actual thing. The persons of the dialogue are able to conceive of the only way in which this can be done: the name should be a vocal imitation of what it names, just as the signs of the sign language of the deaf are imitations of the things they represent.
Now that it is established that the connection between the ideal name and the named is natural and based on vocal imitation (sounds and syllables are taken as primary units of imitation whose "meaning", the ideas they imitate, motivates that of the primary names which in their turn motivate meanings of more complex, "secondary" names), Plato needs to account for what can be seen as the strongest evidence for conventionalism, namely the fact that a) different languages use different names for the same things, and b) there is insufficient motivation within a language, for example, "letters which are unlike [the things they represent] are significant equally with those which are like". Plato accomplishes it by introduction of the notion of the true natural name: just as actual things are realizations of their perfect forms-ideas, the names used by different languages to refer to the same thing are realizations of the perfect ideal name for that thing. Obviously, this perfect name should be a perfect imitation of the thing it names, and thus its connection with it is natural. At the same time actual realizations of the true natural name which are its attempted likenesses are established by legislators, and thus are conventional. It remains unclear, however, why the argument against iconic doubles is not seen as valid in this case. Why is there a need for a perfect representation of the idea of the thing? Why aren't the actual names used by the languages just seen as likenesses of the thing's idea-form itself?
The insufficient motivation within a language is dealt with in the same way: it is seen as a proof of arbitrariness in the assignment of the token-level expressions of the true natural name. So for them, the possibility of different degrees of "correctness" is acknowledged. At the same time, the true natural name itself is not affected by people's arbitrary "addition or subtraction of letters". So, by allowing both conventional and natural principles to serve as bases for correctness of a given name for a given thing, but assigning them to the separate levels of operation, Plato manages to integrate two basic theories of linguistic motivation.
In Aristotle's semiotic theory, the difference between conventional and inferential connection between the signifier and the signified is a basis on which he defines two distinct types of signs. Conventional signs are called "symbols" [symbola], and all linguistic (and cultural in general) signification is seen as symbolic. In other words, names are defined as spoken sounds which, being symbols of mental impressions, are significant "not as tools, but by convention".
The symbol-relation between the signifier and the signified is symmetrical. That is, if the distinction between encoding and decoding processes is disregarded, the names [spoken sounds] are symbols of mental impressions just as much as mental impressions are symbols of names. So in a sense, symbols are equivalents of what they stand for in a different domain. What appears to be an implication of this is that the compound thought structures [reflecting reality] can be replicated by complexes of symbols. So, a sentence would be an example of such complex (with parts significant) symbolic structure.
Natural signs are called "signs" [semeia]; they are "symptoms" of what they signify. Natural signs "cooccur" with the things whose present, past, or future existence they are evidence of (N.B. not causes or accidental coincidences). Unlike symbol-relation, sign-relation isn't symmetrical, but it is transitive. And so, as there are two possible directions in a non-symmetrical relation between two entities, there are two possible types of natural signs: necessary sign [tekmérion] and weak sign. If A is a necessary sign of B (i.e. "if A then necessarily B"), B is a weak sign of A (i.e. "if B then sometimes A").
When symbols are combined to form complex structures it becomes possible for them to have a function other than that of single terms. Namely, while names can only have signification (i.e. to say what a thing is), complete sentences/propositions can perform referential function (i.e. to say that a thing is). So, in fact, if names can only function as symbols of mental impressions, propositions can refer to the actual things of which the impressions symbolized by single terms combined in a proposition are likenesses.
In other words, propositions are representations [symbols] of thoughts [in spoken sounds]. Thoughts are composed of mental images which are likenesses of actual things. So if a thought puts these mental images[Note 1] in a relation that corresponds to the composition of actual things that they are likenesses of in reality, there is truth in it (or falsity if it does not). Accordingly, truth or falsity about the actual things is contained in the thought's symbolic equivalent, the proposition expressing it.
The fact that propositions can contain truth or falsity about the actual things makes it possible to "translate" natural signs (which are the actual events in the world) into the true propositions of a certain type. So, necessary signs can be "translated" into universal affirmatives like "For all cases when there is A there is B". Weak signs can be translated into particular affirmatives like "For some cases when there is A there is B". Thus by representing the true knowledge about the world, such compositions of symbolic elements can be "hard-wired" to the world. They are "hard-wired" in the sense that for a given code relating linguistic symbols and thoughts, a true representation of actual things/events in spoken sounds can not be affected by convention any more[Note 2]. And so, in the event of reasoning, when propositions are combined in a syllogism like "All A are B, all C are A, therefore all C are B"(transitivity), what is inferred in reasoning becomes a sign of actual relationship in reality.
The fact that the difference between conventional and inferential signs is taken to reflect the distinction between the two different types of signification has an interesting implication: it becomes possible for the same thing to serve in both ways at the same time. So, if for Plato token-level names used by different languages conventionally "stand for" the true natural name (TNN) of the thing while the TNN naturally "stands for" the ideal form, for Aristotle the same entity, spoken sounds, can be both natural and conventional sign at once [though with different things signified]. So, for example, if you hear someone say, "All lions have large feet, all animals with large feet are courageous, therefore, lions are courageous", there are (can be) three things going on: a) symbolic interpersonal communication of a certain information about lions, b) syllogistic reasoning hard-wired to the world, large feet as a semeion of lions' being courageous, c) inference on the part of the listener that, say, the speaker is inclined to syllogistic reasoning. Apparently, Aristotle's account of this double-nature of the way spoken sounds function, laid the grounds for the debate over what should be studied by the study of language: its symbolon-function, i.e. the system of conventional signification that language sets (semantics of sign units), or its semeion-function, i.e. inferences in the process of communication (pragmatics of the discourse).
In the semiotic theory of the Stoics the focus is not so much on the difference between the conventional and inferential types of signification as on a slightly different dichotomy. What they regarded as essential was the difference between the mechanisms at work when indicating as opposed to representing (e.g linguistic) signs are involved.
In their view, the meanings of linguistic elements (of semainonta, the meaningful sounds [phonai semantikai]) are incorporeal, non-psychical, but also non-physical, and yet objective entities distinct both from the object referred to [tynchanon] and from the subjective psychical experience of the speaker uttering the meaningful sounds. These meanings [semainomena] arise from supra-individually fixed meanings of the components of the utterance. Thus they are objective and "given from without", but being incorporeal, they can only be realized in the subjective activity of thinking.
In Stoic doctrine, activities that corporeal physical bodies (things) engage in are also seen as incorporeal. In other words, when corporeal things interfere with each other, they are being causes of the (as incorporeal and non-material as time) processes of interference which are their effects and, being incorporeal, can never be causes themselves.
Sentence-meanings (the type of semainomena called lekta, "the things said") are considered to be of the same type as these incorporeal processes. In fact, they even are classified according to whether they are complete or incomplete processes. Propositions (axiomata, a subtype of complete lekta that, unlike questions or commands, can be true or false) are defined as complete processes. Predicates (incomplete lekta) are defined as incomplete processes. They are seen as propositional functions, i.e. they incorporate sentence structures but to become complete propositions they need subjects to fill in actantial positions, so they are meanings of sentences-to-be-completed.
In other words, lekta (meanings of both complete sentences and sentence-structures incorporated in predicates) are the incorporeal states-of-affairs [pragmata] that are the products of interferences between the corporeal bodies. They exist objectively, but can only be realized in corporeal activity of thinking. They can't be perceived, but are thought. They aren't corporeal, yet there is a correspondence between a certain type of lekta, propositions [axiomata] and reality: the truth of a proposition can be checked in reality.
The Stoics' conceptions of the processes that the things engage in and the meanings that linguistic expressions reveal as incorporeal states-of-affairs realized in thought and checked for truth in reality predetermined their views on the distinction between the indicating (inferential) and the representing (linguistic) signs. They regarded the indication [semeion] as a thing that cannot be perceived, but is thought. The inferential connection is not between the actual happenings in the world, but between two type-level states-of-affairs [pragmata]. That is to say, the connection is between two incorporeal propositions, such that they are both true, and the existence of what one of them [semeioton] states can be inferred from what the other one [semeion] states [Note 3]. And this inferential connection is realized in thought of the interpreter.
Apparently, the situation is quite different with linguistic signs. An utterance realizing a sentence[Note 4] leads to apprehending of the type of state-of-affairs in a way that does not involve the inference of the corresponding actual state-of-affairs from the acoustic event of utterance. So rather than being an indication of these incorporeal states-of-affairs, uttering (and thinking) is a way of their realization. The language is seen as a form of realization of logical syntax in which rational presentations (that can be sent forth by speech) are formulated. The incorporeal states-of-affairs [pragmata] that are seen as the variables of inference exist in the conformity with these rational presentations. Thus by realizing a true inferential connection made at the incorporeal level of type-states-of-affairs, the event of reasoning is made into a hard-wired to the world event. So, representing (linguistic) signs in fact are seen as tools for reasoning, which is hard-wired to the world by means of true inferential connections.
Though with the Stoics the units of reasoning could only be realized in the mind of the thinker, no theory has actually considered the process of interpreting the sign to be involved in defining the kind of signification. Peirce introduced a new element into the system. His notion of the interpretant sign formed by the interpreter changed a singular event of signification into a progressive activity of interpretation. He saw this activity as occurring in the series of signs-representations each of which is closer to the true representation of the original object than the one that it as an interpreting sign represents and by which it is "determined". Of course, with this importance attributed to the process of interpretation (as opposed to "objective" signification, independent of the interpretation given to it, which was the primary concern of the previous theories), no natural hard-wired to the world inference is possible. Instead, there is an infinite approximation to truth in a sequence of representational symbols.
In view of this conception of inference, the distinction between inferential and representing signs looses its traditional significance. Instead, Peirce distinguishes between the three other types of "grounds" on which a sign can be related to the object it represents. The first two types do not depend on interpretation as the sole reason for the connection between the signifier and the signified. These two types, iconic, resemblance-based, and indexical, based on actual physical connection grounds, are seen as more primitive, basic. The third type, symbolic, is entirely defined by the interpretation. These three types are seen as hierarchically organized. That is, a more internally complex, higher-order type of connection necessarily involves the grounds of the lower order which, however, are not seen as primary. Thus an index in order to function has to involve to a certain degree the resemblance peculiar to icons, just as any symbol needs to involve both iconic and indexical types of connection. This trichotomy, along with two others, produces ten distinct sign types defined by a featural matrix from which the distinction between inferential and conventional types of signification is eliminated, and a dynamic hierarchical system of finer distinctions takes its place.
Thus each of the four theories utilizes its own approach to integrate two distinct types of signification into a single system. Each of these two types plays a role determined by the views taken by the theory on signification in general and on the relationship between signification and the revelation of truth (i.e. the ways in which the truth is/can be through signs).
Footnotes
[Note 1] Of course, this is true only with regard to the thoughts where there is a "composition" or "division", which correspond to affirmation and negation in the medium of spoken sounds.
[Note 2] Thus propositions are more motivated than words with respect to both traditional and Saussurean axes of motivation. Unlike words, they can be hard-wired to the world because their structure permits them to perform referential function. Also because of their more complicated structure they have more "calculable" meanings than words.
[Note 3] Stoics also distinguished between the type of inference that should be realized to indicate the existence of different types of objects. Thus occasionally non-evident ones required associative inference, while naturally non-evident ones required indicative inference. There is normally a causal relationship between the sign [semeion] and the inferred [semeioton] whose direction is usually opposite to that of signification. But not always: for example, in the syllogism "if a man looses as much blood, he will die" the actual state-of-affairs denoted by the sign-proposition causes the one denoted by its consequent.
[Note 4] The type-level incorporeal sound patterns were absent from the Stoic doctrine of language.
Bibliography
Aristotle
De Poetica.
De Interpretatione.
Plato
Dialogues with Socrates. Cratylus.
Umberto Eco
Semiotics and the Philosophy of Language
Sextus Empiricus (excerpts)
Against the Logicians.
Outlines of Pyrrhonism.
Peirce, Charles Sanders
Collected Papers of C.S.Peirce
Parmentier, Richard
Peirce Divested for Non-Intimates (in Signs in Society, 1994).