Multiply
Suppose that now the machine must receive the instruction MULTIPLY 03 15 87, that is, multiply the content of the cell 03 by the content of the cell 15 and place the product in cell 87. Assuming that a three-addresses instruction has the format as represented in Figure 5.2, the instruction will be as represented in Figure 5.3 or, in binary digits, as in Figure 5-4·
digit 1 digit 2 digit 3 digit 4 digit 5 digit 6 digit 7 digit 8
operation code address 1 address 2 address 3
FIGURE 5.2
03 03 15 87
FIGURE5.3
000000 000011 000000 000011 000001 000101 01000 000111
FIGURE5.4
One realizes that both in the first and in the second position of the sequence there are two binary digits that, insofar as they are a numerical manifestation or a series of impulses, appear to be the same semiotic expression. In fact, it is not so. The expression /03/ in the first position must be referred to the operation code and must be read as MULTIPLY, whereas the expression /03/ in the second position must be referred to the address I – code and must be read as CELL 03. Likewise, it is the fact of being respectively in the second and the third position that ‘means’ that 03 and 15 are the cells whose content must be multiplied by each other, whereas being in the fourth position means that the corre-sponding cell is the one where the product must be placed. So it is the position which establishes of which sign-function a given numerical manifestation is the expression. In this sense, we are witnessing here three different levels of conventions: (a) a cipher a which correlates every decimal expression to a binary one, (b) a cloak β which correlates numeric expressions to operations to be performed, and (c) a cloak у which correlates a different address to each position in the sequence.
Such a language, composed of many simple correlational codes, is no more based upon mere equivalences. Its way of functioning is as follows: if—according to γ—the digit x is found in the position a, then the equivalence system is β1 , but if it is found in the position b, then the quivalence system is β2, and so on. Such a complex code implies contextual selections (see Eco 1976, 2.11). Let us disregard the objection that the machine does not make inferences; we are not interested in the Psychology’ of the machine but, rather, in the semiotics of the code —a code that, theoretically speaking, could also be ‘spoken’ by human be-ings and that undoubtedly complicated the equivalence model with the inferential one.
5.5.3. Codes and grammars
Let us consider now a code clearly conceivable for human beings but not structurally dissimilar to the previous one. Let us invent a way of labeling the books of a library in order to find them with a certain ease and to know in advance where they can be found.
Let us suppose that every book is designated by four numeric ex-pressions based on positional or vectorial rules (for vectors as modes of sign production, see Eco 1976, 3.6). From left to right, every position in the linear manifestation of the encoded message has a different mean-ing, according to the conventions as shown in Figure 5.5. Therefore, the expression /1.2.5.33/ will mean the thirty-third book on the fifth shelf of the second wall of the first room. According to Eco (1976, 3.4.9) this mode of sign production is ruled by ratio difficilis, and the form of the expression maps (or is determined by) the spatial organization of the content. Such a code has a lexicon (a semantics) as well as positional values (a syntax) and works, at a very primitive level, as a grammar.
Positions in the linear manifestation of the encoded message
Leftmost position
Center-left position
Center-right position
Rightmost position
System of architectural positions
room
wall
shelf
book
Reciprocal positions of each element of the system of architectural positions
1 = first room at left immediately after main door
2 = second room . . . and so on
1 = first wall at left when entering room
2 = second wall… and so on
1 = first shelf from floor, and so on, upward
1 = first leftmost book on shelf,
and so on, rightward
FIGURE 5.5
It is ‘a language’ because, with it, it is possible to generate infinite m e s s a g e s. One can c o n c e i v e of the e x p r e s s i o n /3,000.15,000.10,000.4,000/ which means: «the four thousandth book of the ten thousandth shelf of the fifteen thousandth wall of the three thousandth room». The only problem would be whether or not such a description has a referent in some possible world. There are no fictional difficulties in imagining a Borges-like library with thousands and thousands of enormous rooms, each structured as a bugeye megahedron with thousands and thousands of walls hosting billions of shelves, the whole construction free from gravitational laws.
Whether such a universe exists or not is a metaphysical problem; whether it can physically exist of not is a cosmological question; whether our imagination can conceive of it or not is an interesting psychological puzzle; what matters for the present purposes is that the structural logic of this code permits descriptions of this type. This code can only generate true or false sentences (such as there is a book so and so in a place so and so) and could hardly generate texts, except in a ‘me-Tarzan-you-Jane-jargon’. It provides a grammar for a very primitive holophrastic language, but it does not seem so handi-capped as compared with more respectable formal grammars. Our library-language not only is a system of correlation, it also involves infer-ential movements and provides sets of instruction.
It has been repeatedly said that a natural language is not a code be-cause it not only correlates expressions with contents but it also provides discursive rules. Cherry remarked that “we distinguish sharply between language, which is developed organically over a long period of time, and codes, which are invented for some specific purposes and follow explicit rules” (1957:7). If the difference is only in terms of historical growth, it is uninteresting for our present purposes. If the complexity of historical growth implies a greater organicity and flexibility, there is indeed a difference, let us say, between English and the library-code above.
But the difference lies in the complexity of inferential instructions displayed by the English language in comparison with the library-code, in the maze-like effect produced by this complexity, and in the fact that Eng-lish changes faster than the library-code (a difference that could be elim-inated if we decide to complicate the library-code day by day). However, from the point of view of their elementary logic, both codes display the same mechanism, where equivalences are complicated by instructions and where the principle of interpretability holds for both. Even the library-code can elicit many interpretations of its expression, except that, being rather stiff, it allows interpretations only through other semiotic systems and is not self-interpretable as a natural language. This is by no means a minor difference, but here we are not looking for differences (which are, besides, rather intuitive); we are looking for basic identities.
5.6. S-codes and signification
5.6.1. S-codes cannot lie
So far we have understood why the notion of code was used to designate many and variously complex systems of semiotic conventions. But, as we have seen, there was also a general tendency to consider s-codes and codes as fundamentally similar. What we must now do is to ascertain in what sense this apparent confusion took place and why. Codes as semiot-ic constructions can be used to produce propositions designating or mentioning states of the world. Therefore, with codes true or false, as-sertions can be generated.
This does not hold for syntactic systems or s-codes. With systems one cannot lie. Provided arithmetic is a system, one who says that 2 + 2 makes 5 does not lie; it is simply wrong, and one is wrong because one does not obey the tautological laws of the system. Obviously, there can be a teacher cheating his students and communicating to them false no-tions about basic arithmetical operations; but this is not a case of lying with arithmetic, it is an instance of lying about arithmetic by using a verbal or graphic language. This teacher in fact lies about the arithmetic we are used to recognizing as the true one in our ‘real’ world, but is building up an alternative system, no matter how internally coherent it may be.
In the same vein, one who says that fatherhood in the kinship system is expressed by the position G + I, f, L1 is wrong. If, on the contrary, one says that the English word /father/ corresponds to that kinship posi-tion, one lies about the English code.
One can lie by using the verbal or graphic names of the numbers (as when one says that there are three apples on the table when in fact there are four). But numbers as names are not numbers as elements of a math-ematical system: the former can be used to mention quantities that are not the case; the latter only allows for tautological assertions.
Nevertheless, s-codes, even though they do not permit acts of refer-ence or descriptions of possible situations but only tautological opera-tions, can produce strings of expressions that, by virtue of the internal logic of the system, make one expect a further course of systematic events. In other words, there is a sort of elementary signifying power in an s-code, since the sequence 5,10,15 makes one expect 20 as the fol-lowing event.
This leads us to the problem of the signifying power of monoplanar systems
