Writing is an example of a language isomorph in that it has a close part-to-part correspondence with natural language and scientific formulae are extensions of language in that they begin with language and then go on to constructions which are language-like, but not actually used in natural language. Symbols are still wider generalizations of language than either isomorphs or extensions. In the widest sense a symbol is anything, linguistic or non-linguistic, which stands for or "symbolizes", something else. The symbol "a" stands for the sound [a], the visual symbol "|", whatever you call it, stands for the number 'one' in more than one system of writing. A repeated low-pitch horn may stand for a warning that there is a heavy fog in the harbour. A system however has to be something which cann be conveniently produced, presented, and perceived without necessary perceiving the object it stands for.
we shall follow the therminology of Charles W. Morris (...) Moris deals with is not symbols, but, as the title of his work implies, signs, of which symbols form a special case. For example, lowering clouds are a sign of rain, a shiny wet road is a sign that the road is slippery, but the road sign which says "Slippery When Wet" is not only a
sign, but also a symbol. In general, as we have noted above, a symbol is something which can be conveniently produced and has a conventionalized, usually arbitrary, relation to what is symbolized.
what is one symbol?
In analysing a complexity thing, such as symbolic systems, there is always the twofold problem of (1) identification or differentiation on the one hand, and (2) idividualization or segmentation on the other. We have already met with similar problems in the case of phonemes and words. More generally, we can ask: What similar of different things can be classed together as instances of the same symbol? This is a question of kind. Or we can ask: How much a chunk of a thing extending in space or time or both in space and time (such as gestures) shall be considered one piece of a symbol? This is a problem of size. To revert to our linguistic interest in things grammatic, the former is a paradigmatic problem, while the latter is a syntagmatic problem. This is in fact not too far from Morris's terminology, since he calls the study of the structure of signs (including symbols) themselves syntactics, which of course has a much wider application that syntax in the grammatical sense.
(1) To take the problem of identity of symbols first, it will be more convenient to regard a symbol, not as one event or one thing, but as a collection of events or things considered as members of a class, in other words, a symbol is usually taken as a type rather than a token. On the other hand one instance of a symbol, or token, such as an utterance made on one occasion, is often termed a 'signal'. In common usage, one speaks of signals usually in connection with special forms of visual and other forms of communication other that linguistic forms, but there is no reason why a signal in the sense of one instance of the use of a symbol should not include language.
(2) As to the problem of segmentation, there are two sides to consider: (a) What is one symbol and what is a complex of symbols? (b) Where does a particular symbol begin and end? These are obviously generalizations of corresponding linguistic problems of subunits of language, with which we are already familiar. As for the complexity of symbols, no upper limit can be set. As Rudolph Carnap has noted, to any sentence which is reputed to be the longest sentence possible, one can always add the co-ordinate clause 'and the moon is round', which makes it a longer sentence. (...) The lower limit to the size of a symbol is not the smallest physical element which is perceivable, but a symbol which, even if perceivable when subdivided, whould no longer be a symbol (or a set of symbols) in the system of which it is a part.
Symbol and object
(1) Symbols and icons. The normal relation between a symbol and its object, or denotatum in Morris's terminology, is conventional, arbitary, and fortuitous. There is usually no similarity or casual relation between the two. There is for example nothing intrinsically long about the English word 'long' or intrinsically short about the word 'short'. In fact the word 'short' is longer not only graphically but also phonetically and foreigners often tend to pronounce it 'shot' in order so make it sound more symbolic -- symbolic in the popular sense we noted above: 'fitting, expressive, consonant, appropriate', which is precisely the opposite of 'conventional, arbitrary', etc. In this popular sense red is symbolic of danger, stop, etc., because it is physiologically more impressive.
4. Ambiguity, vagueness, and generality. Symbol and object may correspond in the relation of one to one, one to many, many to one, or many to many, understanding of course that one symbol may consist of a class of various members whose differences do not matter.