1991. Peirce's semiotic version of the semantic tradition in formal logic

Peirce's semiotic version of the semantic tradition in formal logic

(New Inquiries into Meaning and Truth, ed. by Neil Cooper and Pascal

Engel, Harvester Wheatsheaf, St.Martin’s Press, 1991, p. 187-213)

Claudine Engel-Tiercelin As Van Heijenoort often used to point out, Charles Peirce obviously belongs to

what we might call the semantic tradition in logic (Boole, Schröder, Löwenheim) as opposed to the 'syntactic' trend (Russell, Frege) (Van Heijenoort, 1985). Peirce insists more upon such semantic notions as truth, validity and satisfiability of logical formulae than on syntactic notions such as demonstrability from a set of axioms or rules of inference. His treatment of the quantifiers,1 together with his correct definition of validity for the sentential calculus, 2 constitute major achievements. 3

Nevertheless, it is not to my present purpose to survey Peirce's many important formal contributions to the semantic view in formal logic. Rather it is to stress what seems to me characteristic of his position, compared, for example, with Boole's or Schröder's own semantic treatments of formal logic. In what follows, I shall argue that Peirce's semantic trend is part and parcel of his semiotic treatment of logic. His approach is distinctive because it places logic within the broader context of a general theory of meaning, understanding and interpretation, a theory of how signs function which enables him to classify different sorts of sign in a natural way. Thus, Peirce's joining the semantic trend is not merely a matter of chance or of following a certain tradition; rather, it is

* I would like to thank Neil Cooper and the referee for their very helpful comments on an earlier

version of this paper. I follow the usual method of citing from Hartshorne, C., Weiss, P., and Burks, A. (eds), The Collected Papers of Charles Sanders Peirce, Cambridge, Mass.: Harvard University Press, 1931-58 (by volume and paragraph number). NEM refers to Eisele, C. (ed.), The New Elements of Mathematics, The Hague: Mouton, 1976, 4 vols.

187

188 Claudine Engel-Tiercelin

because of his having philosophical reasons to rest his semantic approach in logic upon a semiotic perspective.

After a brief summary of Peirce's general understanding of logic as semiotic through its relation to mathematics and the general theory of signs, I shall examine Peirce's conception of inference, focusing on his distinction between the logical structure of the inference (the propositional content) and its assertive force. Finally, I shall present Peirce's analysis of deductive reasoning as observational or experimental, involving use of diagrams or icons, together with his claim that deduction is of two fundamental kinds: theorematic, which provides new surprising results, and corollarial, which largely draws obvious consequences.

1 LOGIC AS SEMIOTIC 1.1 Formal logic and mathematics It would be quite misleading to think that Peirce understood logic in a narrow

and in a wider sense, according to which formal logic, in the narrow sense, would be constituted by the deductive part of logic, whereas the wide sense would be covered by the theory of logic as semiotic: namely, the general theory of signs, or the study of anything whose function is to represent something.4 Such is not at all the case (4. 373). The basic distinction to be made is rather between logic and mathematics. In fact Peirce held, for example, the calculus of classes as a formal, deductive, symbolic system to be a piece of mathematics, not logic: 'Formal logic is nothing but mathematics applied to logic' (4. 263; cf. 3. 615).

Nevertheless, Peirce stresses the differences between them rather than the similarities: from the start, as a follower of Aristotle and Kant, he finds there is something more to mathematical logic than mathematics. That something more is precisely logic, which is always viewed from a philosophical and ontological perspective. 5

Hence an opposition between logic and mathematics; but it is not grounded on a distinction between two specific domains, since mathematics is not defined by its objects (space or quantity) but widely as the science of necessary reasoning. This is, of course, a wider use of 'mathematics' than we are used to; indeed, for Peirce, all a priori reasoning - both our everyday practice of 'necessary reasoning' and the more rigorous practice of professional mathematicians - counts as part of mathematics.6 Hence the real

Peirce's semiotic version of the semantic tradition 189

opposition between logic and mathematics lies between the theoretical or observational aspect of inference on the one hand, and its practical or operational part on the other. The mathematician practises deduction (2. 532; 4. 239; 4. 124; 4. 242), reasons deductively, whereas the logician studies deductive reasonings and arguments. According to a dictum of his father's, Peirce characterises mathematics as 'the science which draws necessary conclusions' (3. 558; 4. 229); logic, by contrast, is 'the science of drawing necessary conclusions'. Incidentally that makes mathematics a 'pre-logical science' which is in no need of logic, for a theory of the validity of its arguments: those are acritical and evident, 'more evident than any such (logical) theory could be' (2. 120).7

As a consequence, the respective aims and methods of logic and mathematics are very different: Peirce's apparent anti-logicism should rather be interpreted as a difference of attitude according to the position that is being adopted. From the mathematician's standpoint, the instrumental value of the calculus is decisive because he is only interested in finding the simplest and shortest way to get to the result (4. 239). Logical constructions are superfluous here (3. 222). But from the logician's point of view, it should be clear that his end 'is simply and solely the investigation of the theory of logic, and not at all the construction of a calculus to aid the drawing of inferences' (4. 373). Therefore, the mathematician's and the logician's purposes are incompatible:

The system devised for the investigation of logic should be as analytical as possible, breaking up inferences into the greatest possible number of steps, and exhibiting them under the most general categories possible; while a calculus would aim, on the contrary, to reduce the number of processes as much as possible, and to specialize the symbols so as to adapt them to special kinds of inference. (4. 373) Thus the calculus is an important tool of reasoning, but it is only a tool (3. 322;

3. 364; 4. 424; 4. 553), or a 'special system of symbols' for treating deductive logic. When Peirce criticises logicians like Boole and Schrdder for being excessively mathematical his objection is that they attempt to draw metaphysical conclusions from logical calculi whose merit consists in the ease with which calculations can be performed with them, rather than in their ability perspicuously to reveal the semantic structure of arguments and propositions. 8 But he is also convinced of the possible plurality of symbolic systems, applied to deduction itself, and above all of the superiority of his logical graphs compared to an algebra of logic (4. 617).

190 Claudine Engel-Tiercelin 1.2 Logic as the science of reasoning The logician is not interested in reaching conclusions, but in theories about their

relations to premisses (4. 239; 4. 370; 4. 481; 4. 533). Hence the natural purpose of logic is 'to analyze reasoning and see what it consists in' (2. 532). '

1. Since the business of logic is 'analysis and theory of reasoning' (4. 134; cf. 1. 417; 4. 242; 4. 373), its domain is widened so as to cover not only deductive reasonings but inductive and abductive seasonings as well; this is crucial for understanding Peirce's conception of logical and scientific inquiry.

2. It also means that the aim of analysis as opposed to that of a calculus will be guided not by simplicity, but on the contrary by complexity (as may be seen from the numerous steps involved in the graphic presentation), in order to reach the most basic and irreducible elements. It is precisely here that semiotic gets into the picture; indeed, the business of semiotic is to explain 'the gist' (2. 532) or the 'essence of reasoning', through the various functions exhibited by different signs, in order to discover the nature of arguments (1. 575; 4. 425).

3. That logic should be concerned with reasoning makes it a normative science (1. 577), and even a branch of ethics (1. 611; 1. 573; 1. 575; 5. 35; 5. 130), for every reasoning is the product of a deliberate and self-controlled thought (1. 606; 5. 130) 'with a view to making it conform to a purpose or ideal' (1. 573). Logical criticism should apply to that type of reasoning alone; this is also a consequence of the principles of pragmatism, according to which all thinking is a kind of conduct (5. 534), so that reasoning is a kind of deliberate conduct, for which a man can be held responsible. Such a normative conception of reasoning is particularly decisive to understand the basic features of Peirce's theory of assertion as well as the principles governing inductive and abductive reasonings in the methods of scientific inquiry (1. 615).

4. Finally, such a definition of logic as a science of reasoning implies an appeal of some sort to psychology, but not in the sense that logic should be founded on it (as is the case with the German introspective tradition against which he fights).9 Peirce's antipsychologism never goes so far as to deny certain facts of psychology: namely, doubts, beliefs, etc. On the contrary, since logic is a positive science - contrary to mathematics, which is a science of pure hypotheses - it may, or even must, take into account certain facts or certain indubitable observations concerning mind: 'Formal logic must not be too purely formal; it must represent a fact


of psychology, or else it is in danger of degenerating into a mathematical recreation' (2. 710).

If such observations are indeed psychological, they must not be interpreted as observations, to be made only by empirical or experimental psychology. In fact 'they come within the range of every man's normal experience, and for the most part in every waking hour of his life' (1. 241), and are such that they constitute 'the universal data of experience that we cannot suppose a man not to know and yet to be making inquiries' (4. 116). Among these universal data is the fact that every reasoning is governed by an aim ('holds out some expectation' (2. 153)), proceeds by iconic constructions and assumes certain belief-habits which operate like leading principles or rules of inference, etc. Otherwise, logic would be confined to a grammar limited not only to abnormal but to non-scientific or irrational men (5. 438-63; 5. 502-37). Thus, when Peirce claims the possibility and the duty for logic to account for such psychological facts, it is because 'under an appeal to psychology is not meant every appeal to any fact relating to the mind' (2. 210).

1.3 Formal logic and semiotic One might still wonder whether Peirce's conception of formal logic both as a

theory of reasoning and as a theory of arguments may rightly be considered on the same footing as his general theory of signs. The current development of semiotics as an academic discipline distinct from logic and philosophy, which, moreover, tends to claim Peirce's inheritance,l0 may lead one to wonder whether these areas are similarly distinct in Peirce. We shall only understand Peirce's logic by seeing that they are not; just as Peirce's writings on signs are inseparable from the ontology on which they depend (hence from Peirce's semiotic realism: namely, his very particular position on the problem of universals)11, it seems illusory - often for the sake of saving Peirce as a logician at least! - to dissociate formal logic and semiotic.

One might object that, strictly speaking, Peirce's theory of arguments is only one branch of logic, what he calls his critic (1. 559, 2. 229), whereas semiotic, as a general theory of the interpretation of signs, is not limited to arguments but extends to concepts, terms and propositions as well (2. 229; 1. 559).

Yet, while Peirce holds to the triple division of logic into critic, speculative grammar and methodeutic, this indicates not so much a difference in nature between all three domains as a different


ordering of the various tasks to be achieved, whether signs are being considered according to their nature (speculative grammar) or to their classifications (critic) or to the methods to be followed to come to the truth (methodeutic) (1. 191). At all events, the critical part of logic is not limited to critic, for formal characters are present in the two other branches as well. This explains why 'logic, in its general sense, is . . . only another name for semiotic, the quasi-necessary, or formal, doctrine of signs' (2. 227).

What are the consequences of such an undertaking, and how can Peirce claim that its formal character is warranted?

This is so, in the first place, because of the task to be achieved by speculative grammar itself. First conceived in a general - both Kantian and Scotistic (2. 83) - way as a theory of knowledge and meaning (2. 206), speculative grammar is more and more restricted to the establishment of what 'must be true of the representamen used by every scientific intelligence in order that they may embody any meaning' (2. 229). Thus, grammar's first function is not to study every sign (symbols, icons and indices), but to 'treat of the formal conditions of symbols having meaning' (1.559, emphasis added). This is why semiotic as a whole will finally concentrate on the study of symbols alone (4.9), for propositional symbols are required for inference. Now, inference is the main concern of a scientific intelligence: that is, an intelligence which is incapable of intuition and has no other means than to follow the rules of inference (deductive but inductive and abductive as well) applied to experience, and which has accepted certain purposes and methods: namely, and among other things, that an assertion can have no meaning unless it deliberately aims at knowledge and truth. Therefore, 'the illative relation is the primary and paramount semiotic relation' (2. 444, n. l). And since a sign or symbol can have no meaning except as a part of the propositional or even assertive context in which it is involved (4. 583; 4. 56; 4. 551), a theory of meaning will necessarily be a propositional theory of meaning (cf. Frege), providing a formal analysis of the normal and rational act of assertion (3. 430) and capable of determining the conditions to be met by the symbols used by a scientific intelligence. But this does not imply that Peirce wishes to reduce meaning to truth, nor that a theory of meaning should simply amount to a theory of truth. This is why grammar must take into consideration not only a general theory of truth and of knowledge, but also an analysis of proposition, of assertion, of the conditions of communication, of the norms governing communication, and a theory of belief (cf. Brock, 1975, p. 125).

Peirce's semiotic version of the semantic tradition 193 But parallel to grammar growing more and more formal, critic itself undergoes

radical changes. Peirce's adoption of the semiotic perspective in logic has many consequences; it reveals, for example, that the common distinctions made by formal logic (and especially by syllogistics) between terms, propositions and arguments are secondary or mainly founded on grammatical misconceptions (3. 430). A logical analysis of their structure as signs will reveal, for example, that terms are implicit propositions (2. 341; 4. 48; 4. 56; 2. 356; 4. 583; 8. 183), that a proposition in turn is a rudimentary argument deprived of its force or power of assertion (2. 344; 2. 346). We are, then, entitled to say that the essence of logic is indeed critic: that is, the study of arguments (2. 203; 2. 710; 4. 9; 5. 159; 5.175), since propositions and concepts are merely degenerate forms of arguments. Critic assumes, moreover, that every asserted proposition is either true or false, and studies propositions as constituents of arguments or inferences (2. 205). Thus, to a certain extent, questions of truth and meaning are indeed reducible to questions of logical validity (5. 142; 2. 444, n; 3. 440). On the other hand, arguments are just one category of signs among others. Thus, to give a logical analysis of their structure we have not only to appeal to the wider conception of logic as semiotic but actually to introduce semiotic into logic. In that respect, any analysis of a logical argument will imply the study of 'the general conditions to which . . . signs of any kind must conform in order to assert anything' (2. 206), and particularly the investigation of the respective roles played by symbols, indices and icons in logical inference.

2 THE LOGIC OF REASONING 2.1 The content of inference: the logical or semiotic analysis of the proposition Any logical inference, be it deductive, inductive or abductive, may be defined as

'a passage from one belief to another’ (4. 53). But not every such passage is an inference: for example, if I observe that my ink is bluish and then gaze out of the window, my mind may dwell on the colour of flowers in the garden and I may particularly notice a poppy. But this movement of thought is not an inference, for in inference 'one belief not only follows after another, but follows from it' (4. 53).

This has an important bearing on our notion of what a logical reasoning should be: it cannot be described as a mere 'colligation of


judgements' (4. 55). Inference has two elements: the one is the suggestion of one idea by another according to the law of association, while the other is the carrying forward of the asserting element of judgement, the holding for true, from the first judgement to the second (4. 55). Thus, it is necessary to distinguish the content of the inference from its assertive force:

. . . No sign of a thing or kind of thing can arise except in a proposition; and no logical operation upon a proposition can result in anything but a proposition; so that non-propositional signs can only exist as constituents of propositions. (4. 583; cf. 4. 56, 551) Semiotic analysis must therefore start from the basic components of the

proposition. Peirce's logical examination of the structure of the proposition is achieved

through a complete reformulation of the traditional relation of predication, the respective functions of the subject and the predicate being completely sundered from any allusion to their grammatical functions. A proposition is a symbol, the subject of which is constructed as an index or a series of indices corresponding to our variables, and whose predicate or icon is obtained by erasing the logical subject or subjects of the proposition (2. 95). Thus in a proposition such as 'Socrates is a man', the predicate is ' is a man', a form called a 'rheme' or 'rhema' (cf. 4. 438; 2. 272; 4. 560), which is very near the Fregean notion of unsaturated predicate (incidentally, that it should be obtained by erasing the logical subject or subjects indicates that for Peirce the correct way of understanding a term is as a derived form of proposition instead of a kind of block out of which the proposition would have been built):

In order properly to exhibit the relation between premisses and conclusion

of mathematical reasonings, it is necessary to recognize that inmost cases, the subject-index is compound and consists of a set of indices.Thus, in the proposition, 'A sells B to C for the price D', A, B, C, D form a set of four indices. The symbol 'sellstofor the price of refers to a mental icon, or idea of the act of sale, and declares that this image represents the set A, B, C, D, considered as attached to that icon, A as seller, C as buyer, B as object sold, and D as price. If we call A, B, C, D four subjects of the proposition and 'sellstofor the price of a predicate, we represent the logical relation, well enough, but we abandon the Aryan syntax. (2. 439)

You will, of course, have noticed that the copula has been integrated as a part of

the predicate; this means that the logical analysis of the proposition exhibits the signification of the proposition quite independent of its being used or asserted. Indeed, Peirce even goes so far as to define a proposition as any sign which indicates its object separately or independently (NEM, IV, p. 242);


thus a proposition can be the object of many different linguistic acts and mental attitudes, and since it can be expressed in any language (NEM, IV, p. 248), we cannot identify a proposition with a particular linguistic form or sentence type.

However, if the logical analysis clarifies the structure of the proposition, it does not provide its meaning, and, as we saw, that meaning consists in its 'assertive' element. In other words, we should not be too quick in Platonising Peirce's account12

of the propositional content of the inference; it is one thing to have the 'character of the sign', quite another to show what it is for the proposition to 'act as a sign' (2. 247).

2.2 The assertive force of inference If it be admitted that the normal use of a proposition is to assert it (1. 485), first,

what is its most natural mode of expression, and second, what are the respective roles played by the index, and the icon, in the assertion?

1. The logical analysis has shown that the indicative form is only conventionally linked to the act of assertion, so that the proposition in the sentence 'Socrates est sapiens', strictly expressed, is 'Socratem sapientem esse' (NEM, IV, p. 248). But we might suppose that the indicative mode is the most natural mode of the assertion. Here again, Peirce's position can sound paradoxical; according to his identification of categorical and hypothetical propositions, principally based on his Scotistic treatment of a proposition as a consequentia, he claims that there is no logical difference between the indicative and the hypothetical mood, and that it is even more natural to express any proposition under its conditional form (see Hookway, 1985, pp. 135 ff.). Thus, we can always paraphrase a sentence such as 'All humans are mortal' as a sentence involving a conditional open sentence within the scope of a universal quantifier: 'Anything is such that, if it is human, then it is mortal': it means that ordinary conditionals involve an implicit quantification over'possible cases'. In each case, we find a 'Boolean' (a conditional open sentence) within the scope of a quantifier. That quantifier works as an index signifying what universe of discourse has been selected. It also limits our universe to the actual case, according to the Scotistic interpretation of the conditional as a consequentia de inesse.l3 In fact, the conditional is very near material implication (3. 374-5). But is also explains why any proposition is at heart an inference which might be expressed in the following way: (x) (Ax < Bx) (the expression < representing the Peircean con-


nective of inclusion and later of implication). 14 Expressed in semiotic terms, all this means that, for Peirce, 'the subject of a categorical proposition is a sign of the predicate, and the antecedent of a conditional a sign of the consequent'.

2. Let us now turn to index and icons. In his grammatical theory of judgement and inference, Peirce gives the following description of what takes place:

A judgement is an act of consciousness in which we recognize a belief,

and a belief is an intelligent habit upon which we shall act when occasion presents itself. Of what nature is that recognition? It may come very near action. The muscles may twitch and we may restrain ourselves only by considering that the proper occasion has not arisen. But in general, we virtually resolve upon a certain occasion to act as if certain imagined circumstances were perceived. This act which amounts to such a resolve, is a peculiar act of the will whereby we cause an image, or icon, to be associated, in a peculiarly strenuous way, with an object represented to us by an index. This act itself is represented in the proposition by a symbol, and the consciousness of it fulfills the function of a symbol in the judgement. Suppose for example, I detect a person with whom I have to deal in an act of dishonesty. I have in mind something like a 'composite photograph' of all the persons that I have known and read of that have had that character, and at the instant I make the discovery concerning that person, who is distinguished from others for me by certain indications, upon that index at that moment down goes the stamp of RASCAL, to remain indefinitely. (2. 435) Inference seems to involve three basic elements: an icon, namely some

'composite photograph' of a certain experienced generality (for example, in such a proposition as'it rains', all the rainy days the thinker has experienced); an index, functioning as an indicator of reality distinguishing 'that day' as it is placed in his experience, whereby it stamps that day as rainy (2. 438); and a symbol, asserting the existence of a link between the two.

So the index has a function of designation but not of signification: the icon and the symbol, on the contrary, both function at the level of description and signification; the symbol or token ensures the functioning of generality which is essential to any reasoning (3. 363), it gives the general terms in which description may be carried; the icon, an image, although abstract and formal, allows the representation of description itself, it provides the observational element without which no reasoning could take place (3. 364).

Why is that so? Precisely because Peirce's definition of inference as the 'conscious and deliberate adoption of a belief as a consequence of another cognition' (2.442) implies that the link between premisses and conclusion should not be operated in a


mechanical way. This is what distinguishes any kind of belief-habit from the belief that takes place in an inference:

What particularly distinguishes a general belief or opinion, such as is an inferential conclusion, from other habits, is that it is active in imagination. If I have the habit of putting my left leg into my trousers before the right, when I imagine that I put on my trousers, I shall probably not definitely think of putting the left leg on first. But if I believe that fire is dangerous and I imagine a fire bursting out close beside me, I shall also imagine that I jump back . . . Thus, when you say that you have faith in reasoning, what you mean is that the belief-habit formed in the imagination will determine your actions in the real case. This is looking upon the matter from the psychological point of view. Under a logical aspect your opinion in question is that general conditions of potentialities 'in futuro', if duly constructed, will under imaginary conditions determine schemata or imaginary skeleton diagrams with which percepts will accord when the real conditions accord with those imaginary conditions; or stating the essence of the matter in a nutshell, you opine that percepts follow certain general laws. (2. 148)

We shall see that such imaginary, though formal, icons play a decisive role in

deductive inference. 3 DEDUCTIVE INFERENCE AND ICONIC REASONING The central idea developed in Peirce's account of necessary deductive reasoning

is that it proceeds by constructions of diagrams, which are a species of icons. This is as true for logical reasoning as it is for mathematical reasoning, which is in fact the paradigm of deduction. Such a conception has important bearings not only for a conception of iconic logic, but for certain peculiarities that are attached to mathematical deduction as well.

3.1 The main characters of the icon An icon is a sign which 'refers to the Object that it denotes merely by virtue of

characters of its own, and which it possesses just the same, whether any such Object actually exists or not' (2. 247). Although Peirce is sometimes misleading in that he says that the icon is 'similar' to its object (1. 558; 2. 247; 2. 255; 2. 276; 2. 314), we must be careful not to limit the icon to a material resemblance with its object. As a matter of fact, an icon may be 'like' its object in several ways: (1) as a resemblance in respect of simple qualities (the icon is then an 'image', e.g. a photograph); (2) where the relations of the icon's parts are matched by analogous relations of the object


parts (we then have a 'diagram'); (3) when there is, rather than a 'matching', a more general 'parallelism' of relations or characters (the icon is then a 'metaphor').

So, as examples of icons, we find such various things as pictures, photographs, geometrical diagrams, logical graphs, algebraic formulae and even ideographic signs pertaining to certain natural languages. We may even say that the'material resemblance' is most often overlooked by Peirce. If, for example, we represent the relations of the various kinds of sign by the following brace:

Icons

Signs { Indices Symbols

we have used an icon which only 'resembles' what it signifies, in that it shows the relations to be'as they really are' (2.282). The basic feature of the icon is that it can represent the formal aspects of things: 'No pure Forms are represented by anything but icons' (4. 544). Thus the icon has less a function of similarity to its object (it is the only sign that can function even when it has no object) than a function of exhibition or exemplification of its object (cf. 2. 282; 3. 556; 4. 448; 4. 531) (cf. Burks, 1949, pp. 673-5). In that respect, we must not overlook the fact that, for the icon to function as a sign, it has to be interpreted (in Peirce's terms, it must have an 'interpretant' that recognises that it is such): when one reads the sentence 'This sentence is in italics', written in italics, one learns what italics are by making use of the fact that they are exhibited by the sentence.

That the icons should be formal rather than pure empirical images implies that such 'skeletons' should imply certain efforts of abstraction, in order to be represented (3. 434). Such a mechanism which Peirce defines as 'hypostatic abstraction' 15 helps us to feel the difference between the thing and its copy.

3.2 The use of icons in deduction Such features of the icons may explain why they should be of particular help in

deductive reasoning (cf. 4. 479; 4. 410--11; 3. 363; 3. 418; 3. 429; 7. 619). Thus Peirce claims the superiority of his graphic presentation of logic to the purely algebraic one. But his point is stronger than this. His thesis is that icons not only help us in reasoning but are essential constituents of all necessary deductive logical and mathematical - reasonings. This strong thesis is a consequence, first, of the fundamental pragmatist principle that all


thought is not only in signs but made of signs and, second, of Peirce's conviction that a symbol alone is incapable of conveying any information unless it be accompanied by indices and icons:

Remember it is by icons only that we really reason, and abstract statements

are valueless in reasoning except so far as they aid us to construct diagrams. The sectaries of the opinion I am combating seem, on the contrary, to suppose that reasoning is performed with abstract 'judgments', and that an icon is of use only as enabling me to frame abstract statements as premisses. (4. 127; cf. 2. 278).

We may understand how geometrical diagrams may be considered as such icons,

but to insist on the iconic character of algebraic formulae seems less obvious. Is it not more natural to consider them as conventional compounded signs (2. 279)? On the contrary, though it be true that such icons have been rendered such through the rules of commutation, association and attribution of symbols, and that their likeness is aided by conventional rules, the symbolic character is in them secondary compared with the iconic one (3. 363). This is mainly because of one remarkable property of the icon: namely, that 'by the direct observation of it, other truths concerning its object can be discovered than those which suffice to determine its construction (2. 279; cf. 3. 363).

Let us now try to specify how such ideal experimentation and observation works in Peirce's paradigm of deductive reasoning, namely, in mathematics, and why he also thinks that formal logic is essentially iconic.

3.3 Iconic reasoning: Theorematic and corllariat deduction

The first things I found out were that all mathematical reasoning is diagrammatic and that all necessary reasoning is mathematical reasoning, no matter how simple it may be. By diagrammatic reasoning, I mean reasoning which constructs a diagram according to a precept expressed in general terms, performs experiments upon this diagram, notes their results, and expresses them in general terms. This was a discovery of no little importance, showing as it does, that all knowledge comes from observation. (NEM, IV, pp. 47-8) How does the procedure work? First, the mathematician has:

to state his hypothesis in general terms; second, to construct a diagram, whether an array of letters and symbols with which conventional 'rules’, or permissions to transform, are associated, or a geometrical figure, which not only secures him against any confusion of alt and some, but puts before him an icon by the observation of which he detects relations between the parts of the diagram other than those which were used in its construction. This observation is the third step. The fourth step is to

Peirce's semiotic version of the semantic tradition 201 geometry) is not, contrary to what has been sustained, a matter of

incompleteness of axiomatic formalisation (p. 306). Moreover, by stressing the 'logical step' occurring in theorematic reasoning, Peirce would have anticipated Hintikkâ s claim for modern quantification theory, that a valid deductive step is possible if it 'increases the number of layers of quantifiers in the proposition' (p. 307). Indeed, Peirce links the capacity of proofs to provide surprising results to the fact that, in the logic of relations, intermediate stages in the proof may consider complexes of individuals with more constituents than are referred to in either premisses or conclusions. Thus we imagine something which 'goes beyond what was given in a full articulation of the problem' (NEM, IV, pp. 38, 288). Unfortunately, while such comparisons are relevant to point up some of Peirce's insights in that area, it is very difficult to think that they do not misrepresent his thought.

Before going on any further, I shall give illustrations of the various kinds of deduction.

1. From the beginning, Peirce analyses the Aristotelian syllogism in such a way as to render it plain that

All deductive reasoning, even simple syllogism, involves an element of observation . . . For instance, take the syllogistic formula All MisP S is M So, S is P This is really a diagram of the relations of S, M and P. The fact that the middle term occurs in the two premisses is actually exhibited, and this must be done or the notation will be of no value. (3. 363) This will become obvious when the notation for the logic of relations is

introduced, in which predicates function as icons. Let us start again with a simple syllogism in Barbara.

All humans are animals All animals are mortal All humans are mortal Here, we have a corollarial deduction, in Peirce's sense, in so far as one only has

to understand the premisses to bring them together in order to see that the conclusion follows from them. If we now adapt Peirce's notation, such an inference could be diagrammatised thus:

(dx) (Hx -< Ax) (box) (Ax ~ Mx) (`dx) (Hx -~ Mx)

202 Claudine Engel-Tiercelin Or: HestA AestM HestM Or, in the notation of the beta system of existential graphs: humans animals humans animals mortal mortal 1, All humans are animals All animals are mortal All humans are mortal (Premiss 1) (Premiss 2) (Conclusion 3) Peirce's idea is that all these descriptions provide diagrammatical experiments

thanks to which we observe and imagine the permissible transformations and conclusions to be made. Thus we can see that the elements of experiment on a diagram are present in the simplest forms of corollarial reasonings such as a syllogism (cf. Hookway, 1985, p. 195).

2. A syllogism is only a form of corollarial deduction. In such an observation we only have to inspect the diagram of the premisses to see that the diagram of the conclusion is directly presented therein. In theorematic reasoning, the procedure is somewhat different, but only in degree. This is because the logic of relatives is but a generalisation of the procedures of transformation and substitution. But we can use theorematic reasoning - that is, experimenting on a created diagram because of our incapacity to detect the conclusion through a mere direct observation of the premiss diagram - thanks to two basic devices introduced by the logic of relatives, first, because of the iconic dimension of the predicate and, second, because of the common logical structure of hypothetical and categorical propositions.

As concerns the first point, it means that predicates are unsaturated expressions with a definite valency. The predicate expression (such as '() red' or '() loves ()') is an iconic sign because its unsaturated bonds correspond to the valency of the relation that it expresses. The notation 'diagrams' a feature of the relation ex

Peirce's semiotic version of the semantic tradition 203 pressed. For example, if I assert that a dyadic relation holds between two objects

by using a representation in which a dyadic relation holds between indices for the two objects, by saturating the two bonds of the relational expression, it also means that, when I use a predicate to ascribe a property to an object, the expression I use will occur in many other statements that I accept. For example, I shall accept that 'Whatever is red is coloured' or 'Whatever is red has such and such wavelength'. It means that my understanding of the predicate expression involves a grasp of the relation to other predicates and relational expressions articulated in other statements, which form a set illustrated by a diagram of the property and its relations to other properties. Such an iconic character of the predicate aids reasoning and imaginative experiments (see Hookway, 1985, pp. 133-4).

But, second, the common logical structure of categorical and hypothetical proposition, allied with the Peircean interpretation of the fundamental logical relation'-<' as a sign-relation (so that both modus ponens and Barbara reflect the transitivity of the signrelation) allow all kinds of substitutions on the basis of their being a representation of another. Any deductive inference may thus be interpreted in the following way:

If A is a sign of B, and B is a sign of C, then A is a sign of C The benefit of such an analysis is that, when the subject-predicate distinction is

generalised to allow for multiple subjects in one sentence, some complex patterns in the logic of relations are seen to continue to instantiate Barbara in that general sense (NEM, IV, p. 176). But at the same time, we have increased our information, since, so understood, Barbara covers a wide range of inferences and may be presented as a chain of inferences, the premisses of each inference being either the conclusion of an earlier inference in the chain, a premiss of the overall argument, or a statement which spells out our understanding of a logical constant. For example, if we reason

Socrates is a man So. Socrates is mortal we only have to formulate the 'leading principle' we use and add it to the

premisses: namely If Socrates is a man, then Socrates is mortal So there is no limit to the process of adding to the list of premisses

204 Claudine Engel-Tiercelin by articulating the leading principle relied upon in the inference, and it is always

possible to add premisses to an inference in order to make it rely on modus ponens. In that sense we have not abandoned analyticity and yet have captured a wide range of inferences (cf. Hookway, 1985, pp. 196-9).

All this, I hope, will help to explain why I have some doubts about Hintikka's interpretation.

1. First, though it is true that Peirce uses the structure of Euclid's proofs, as the description of the mathematical procedure above testifies (cf. NEM, IV, p. 238), he does not reach his account of theorematic reasoning by generalising from Euclid. The Euclidean framework only provides a convenient and elementary example of the figurative as opposed to the algebraic kind of diagrammatical reasoning (cf. Ketner, 1985, p. 414). The concept of construction that Hintikka wants to impose on Peirce is not appropriate, since, as should be clear from what I said before, Peirce talks of constructing diagrams in a much wider sense, including figures as well as other signs, algebras, maps or language. We could also say that Peirce's constructionism is not the same as that associated with the school of Brouwer, although it has some similarities: for example, one shortcoming of the intuitionist school that Peirce would not buy is the intuitionist rejection of reasoning involving infinity. Peirce claims that we can reason effectively about infinity (Fisch, 1982, pp. 37-43) . If we wish to find a historical parallel, we should rather look for it in the Kantian schematism of which Peirce gives, to a certain extent only, an empiricised and semiotic version..

2. Hintikka seems to hold that Peirce's distinction captures well enough the analytic-synthetic distinction. Thus, 'corollarial' is equivalent to 'analytic', while 'theorematic' would coincide with his 'Synthetic Ill', 'the best rational reconstruction of the Peircean concept of theorematicity' (1980, p. 307). But here again the Peircean reformulation of the analytic-synthetic distinction is such that Peirce's basic thesis is precisely that analyticity involves both corollarial and theorematic deductions (which means that mathematical judgements are not, as Kant maintains, synthetic, but analytic), analyticity being defined as logical compossibility: for Peirce, to be analytic is either to be a definition, or to be logically deducible from a definition (cf. 6. 595). An analytic consequence of a hypothesis may then be said to be 'involved' in it, in the sense that it can be 'evolved' from it. In that way, we should not be surprised to find unobvious consequences evolved from it. The contrast Peirce draws between analytic and synthetic statements refers not to the distinction between concepts and constructions of concepts,

Peirce's semiotic version of the semantic tradition 205 since, as we saw, 'Deduction is really a matter of perception and

experimentation, just as induction and hypothesis are', but rather to the fact that, in the first case, 'the perception and experimentation are concerned with imaginary objects, instead of with real ones' (6. 595, emphasis added). In other words, 'analytical reasoning depends upon associations of similarity, synthetical reasoning upon associations of contiguity' (cf. 1. 383).

3. Finally, it is not at all certain that Peirce wants to make a specific logical point by the stress he puts on theorematic reasoning; rather it seems to me to be an epistemic point, first, because he once conjectures that the need for theorematic reasoning reflects the current state of mathematical ignorance - which means that it is only a question of time before all theorematic reasonings are reduced to corollarial reasonings (NEM, IV, p. 81 or p. 289). 'Perhaps, when any branch of mathematics is worked up into its most perfect forms, all its theorems will be converted into corollaries.' He also expresses his puzzlement whether this distinction is 'inherently impossible in some cases' (IV, p. 290), or the reflection of some psychological convenience rather than of some logical truth. Thus he writes:

There is still some question how far the observation of imaginary, or artificial constructions, with experimentation upon them is logically essential to the procedure of mathematics, as to some extent it certainly is, even in the strict Weierstrassian method, and how far it is a merely psychological convenience. (NEM, IV, p. 158)

So it seems to me that we could summarise Peirce's position in the following way. Theorematic reasoning is important as an epistemic procedure; this is owing to the fact that the main 'business of the mathematician is to discover new theorems', while 'leaving the grinding of them down into corollaries to the logician (NEM, IV, p. 289). So Peirce can hold both that, even in such cases where there are resisting cases to it, his distinction still remains valid (p. 290) and also that, to a certain extent, it is logically dispensable:

[Theorematic reasoning] is an operation of necessary, or rather, of compulsive reasoning . . . which formal logic cannot possibly take into account, since from the point of view of formal logic, there is no essential distinction between it and the most obvious corrolarial transformation. Yet, the key to mathematical methodeutic lies hidden here. It will consist in the transformation of the problemor of its statement- due to viewing it from another point of view. (NEM, III, p. 491)

But if the distinction tends to evaporate from a logical point of view, this is precisely because grinding down theorems to corollaries

206 Claudine Engel-Tiercelin does not imply at all that the observational, experimental or iconic part is then left out of account. When Peirce at times presents corollarial deductions as trivial or somewhat sterile deductions, it is to oppose them either to the fertility of mathematical discoveries or to the traditional way of interpreting corollarial deduction, as signifying that'the operations of demonstrative reasoning are nothing but applications of plain rules to plain cases', thus trivially verbal. Such thinkers assume that Barbara in all its simplicity represents all there is to necessary reasoning (see 4. 427). Peirce's point is that, even in the simplest syllogistic form, something like an iconic representation is required, so that the development introduced by the logic of relations is no departure from that view but only a generalisation. So rather than a strict distinction between three kinds of deduction (syllogism, corollarial deduction, theorematic deduction), we only have a hierarchy of degrees in iconicity. 4 CONCLUSIONS I would like to make three kinds of remark as a conclusion to my presentation of Peirce's semiotic-semantic conception of logic. 1. First, I would like to draw attention to his distinction, concerning the logical structure of inference, between what we might call the propositional content and the assertive force. Apart from the intrinsic interest we may find in his structural analysis through its logical form instead of its grammatical form, in the modern interpretation he makes of subject and predicate, it seems to me that his insistence on the superfluous character of the copula,together with the claim that the indicative mood is no part of the logical structure of the proposition, may help, as Risto Hilpinen (1986, p. 194) rightly pointed out, to refute certain criticisms addressed to those wishing to develop a deontic logic or a logic of imperatives because of the supposed original prevalence of the categorical mode. On the other hand, by underlining the difference between the logical form and the assertion proper, and by adding that the meaning of the proposition finally lay in the act itself, Peirce has stressed the need of a close association of any theory of meaning with a correct theory of speech-acts (Brock, 1981a, 1981b; Chauviré,1979). Again, by developing the normative character of the act of assertion, involving the responsibility of the speaker and the interpreter both engaged in a game submitted to certain rules of

Peirce's semiotic version of the semantic tradition 207 interpretation, to certain norms and ideals, Peirce may, this time rightly, be

considered as an anticipation of Hintikka's gametheoretic semantics (Hilpinen, 1982; Brock, 1980).1 Still there would be a lot to say here, on the relationships between the semantic interpretation Hintikka makes of the quantifiers and the pragmatic interpretation Peirce makes of both subjects and predicates, particularly with regard to the vagueness of the predicate. But this would lead us too far into what is undoubtedly one of Peirce's most interesting views about logic: that is, his project for a logic of vagueness.

2. My second remark concerns the emphasis Peirce puts on the need for an iconic logic. As Hintikka said, it is surely in that respect that Peirce may be considered as the direct opponent of the mainstream of modern logic during the crucial period from Frege to Herbrand, just as it also might explain why this served to alienate most of Peirce's contemporaries among logicians, with their predilection for the purely symbolic aspects of modern 'symbolic' logic, from his insights (Hintikka, 1980, p. 313). The connection between Peirce's distinction of two kinds of deduction and modern discussions on decidability are also obvious. Just as modern logicians stress that it is the presence of relational predicates that renders elementary logic undecidable, so Peirce insists that it is only with the development of the logic of relations that the character of theorematic reasoning becomes readily discernible (NEM, IV, p. 58). Given Peirce's interest in whether the future work of necessary reasoning could be left to machines, it is tempting, as Hookway suggests, to construe Peirce's doctrine as 'an early dim anticipation of the undecidability of the logic of relations' (1985, p. 199). But it is also striking, as far as his reflection on logical machines is concerned, to see the changes of his thought on this matter, his beginning by attributing to machines capacities of reasoning and then becoming more and more sceptical about it, the more he is convinced of the importance of the part of self-control and imaginative experimentation in inference (cf. Engel-Tiercelin, 1984).

3. It seems to me also that it would be interesting to follow his steps in what he says concerning the role of the belief-habit principle of inference, if we accept another dimension of his conception of inference. For Peirce the understanding of any inference involves the grasping of all such kinds of substitution that are warranted. For instance, to accept a universal proposition means to have a habit of using it as a rule in deriving conclusions by Barbara. If that be the case, a semantic account of what takes place

208 Claudine Engel-Tiercelin might be in order, not necessarily under the form of a psychological account, but

under the form of a correct theory of interpretation. Again, Peirce is not ready to subscribe to any kind of Platonic version of logical necessity; he wishes to justify it, thanks to the power of our using formal icons in imaginative associations (cf. 3. 390). It is true that, to a certain extent, our belief-habits are irreducibly vague and therefore in no need of any kind of justification. 18 But in so far as we should look for some explanation of what is going on, it might be not only permissible but logically safer to turn to empirical psychology. Would that commit us to the psychologism otherwise denounced by Peirce? It seems to me to be quite a different story - and I am sure that Peirce, who was at the same time so opposed to any account of perception in terms of images and so enthusiastic about an iconic treatment of logic, which looks, in many respects, very near Johnson-Laird's mental models, would have agreed to it (Johnson-Laird, 1983).19

It remains that Peirce's main challenge in proposing such a semiotic version of formal logic lies perhaps not so much in his attempt to follow Kant's steps by holding a new version of schematism, or to reintroduce psychology without psychologism in logic. For him, the meaning of his original iconic treatment of deduction very likely consisted mainly in his trying to think through the possible relations between formal logic and metaphysics. We may have some doubts about the well-foundedness of such a position, not only because it supposes the relevance of Peirce's realistic solution to the problem of universals, but also because it rests upon the claim that formal logic should have something to do with metaphysics.

Paradoxically enough, it should be noted that Peirce is one of those logicians who have most contributed to the development of contemporary logic, and who have also framed a logical method of conceptual clarification (pragmatism), the aim of which was to put an end to the 'melancholy disintegration of philosophic thought due to loose reasoning' (2. 18) and to the 'moonshine' demonstrations of metaphysicians. And it is the same man who has insisted so much on the necessity to remember Aristotle's or Kant's lesson, in order for logic to avoid triviality or being 'an art far worse than useless' (3. 404). In that respect, it should be inaccurate to understate Peirce's obvious -and classical - message: if formal logic may have any meaning at all, it is mainly because it may help to demonstrate that 'the commonest and most indispensable conceptions are nothing but objectifications of logical forms'. 20

Peirce's semiotic version of the semantic tradition 209 NOTES 1. As early as 1870, Peirce begins to represent binary relations by matrices,

finite or infinite, whose elements are ordered pairs of individuals. In 1882 he takes a further step: a relative is represented as a Boolean sum of ordered pairs (taken in a certain domain), each pair being affected by a coefficient, which is 1 or 0 depending on whether the relation obtains between the elements of the pair or not. The mathematical signs E and II thus come to play the role of the existential and universal quantifiers. But contrary to what is often too much emphasised, Peirce's treatment of the quantifier is very remote from the way Frege handles his quantifiers: the big difference is that there are no axioms about them. Their interpretation and handling are tied up with a specific domain, changeable at will. Moreover, they are generally used only in initial position. It is also well known that another feature of the 'syntactic' stream in logic is that Frege and Russell are engaged in a total logical reconstruction of our world; logic is universal. Such a conception of the universality of logic has an important consequence: wishing to build a universal system and starting from primitive notions, they make their quantifiers range over all objects. The range of these quantifiers is a fixed unique universe: The Universe (Van Heijenoort, 1985, p. 45), Frege's ontological furniture of the world is divided into objects and functions. On the contrary, just as Boole had his universe class, Peirce follows De Morgan and his universe of discourse, denoted by 1. The universe has no ontological import. It only comprehends what we agree to consider at a certain time, in a certain context. For Frege, it cannot be a question of changing universes. As Van Heijenoort says, one could not even say that he restricts himself to 'one' universe. His universe is 'the universe', which consists of all that there is, and it is fixed (p. 13). In Russell, we have a stratified universe, but each stratum is a fixed domain (p. 45). For a good account of Peirce's approach to quantification, see Martin (1980) and Thibaud (1975, pp. 84 ff.).

2. With a fixed universe (as Fregé s or Russell's universe) the semantic notions of validity and satisfiability lose their applicability, while with a variable universe like that of Peirce they come to the fore. As early as 1885, Peirce gives the correct definition of validity for the sentential calculus: 'To find whether a formula is necessarily true, substitute f and v for the letters and see whether it can be supposed false by any such assignment of values' (3. 387). Peirce even goes so far as to suggest a test for validity which is very near the refutation tree method. For a detailed study of the matrices used, cf. Thibaud (1975, pp. 41ff.).

3. Another important consequence of such a conception of logic as a calculus, as opposed to the conception of logic as lingua characteristica is certainly the project entertained by Peirce of the possibility of a

10 Claudine Engel-Tiercelin non-classical or triadic logic, linked with a wider programme for a logic of

vagueness (cf. MS 339 'Logic Notebook', published in Fisch and Turquette, 1966). On the other hand, it is also obvious that Peirce, just like Schrdder and Ldwenheim later on, is much more aware than Frege or Russell, who do not hesitate to go beyond first-order logic, of the difference in complexity between first-order logic and a higher-order logic. One can easily measure the influence of such a separate consideration of first-order logic upon the development of modeltheory. On Peirce's logic of vagueness, see Brock (1979), Nadin (1980), Engel-Tiercelin (1986, 1989).

4. Cf. 1. 192; 1. 227; 1. 444; 1. 529; 2. 93; 2. 227; 4. 9; 6. 129; 5. 488. On Peirce's insistence on the 'formal' character of semiotics, see NEM, IV, pp. 20-1, 54.

5. I have developed this theme in Engel-Tiercelin (1985). See also Burks (1943, p. 188).

6. One consequence of such a conception is the fact that any proposition can be looked upon as a mathematical theory and used for mathematical reasoning. Therefore philosophy may be rendered mathematical (MS 438). See Hookway (1985, pp. 182 ff.).

7. For a good account of the acritical character of mathematics, see Hookway (1985, pp. 183 ff.).

8. Peirce was not himself completely immune from the Platonism which he denounced, especially in his philosophy of arithmetics. There would be a lot to say here about the way Peirce uses his Scotistic realism as a better candidate than Platonism for a convincing realistic approach to mathematics.

9. And especially all the psychology of introspection, based on such criteria as self-evidence or the favourite motives of the German tradition holding that the ultimate test of valid inference is an immediate, instinctive feeling of rationality. Sigwart is associated with such a'Gefühl-criterion' (5. 85; 2. 232; 2. 210; 5. 329), as is Schrdder (one of Sigwart's followers, according to Peirce (5. 85)), who defines 'logical consequence as a compulsion of thought' (3. 432). Thus Peirce refutes the view, made popular by Stuart Mill, that logic-theoretic grounds are wholly borrowed from psychology: 'Logic is not the science of how we do think; but in such sense as it can be said to deal with thinking at all, it only determines how we ought to think; nor how we ought to think in conformity with usage, but how we ought to think in order to think what is true' (2. 52). So, by contrast with the 'German theory of logic', supposing a non-cognitive faculty as final authority, Peirce claims his linkage with the 'English' or 'objective conception of logic' (2. 185), which makes the criterion of logicality state the condition of its own testing (2. 153). But again, this anti-psychologism does not imply that no appeal to certain facts of psychology should be taken account of, nor that Peirce would have been that hostile to a kind of project such as Dewey's, of a 'natural history of thought' (cf. 'Review of Studies in Logical Theory', The Nation, 1904, vol. 79, p. 220).

+Peirce's semiotic version of the semantic tradition 211 10. In fact, the list of the so-called followers would be endless, from C. Morris

to T. Sebeok, Julia Kristeva, U. Eco, etc. 11. And particularly the Scotistic realism which he espouses. For a development

of Peirce's position on the problem of universals, see Engel-Tiercelin (1986). 12. E.g., Quiné s comment on Peirce's account of proposition in his introduction

to vol. II of Collected Papers. 13. For a fuller development, see Engel-Tiercelin (1985, esp. pp. 244-5). 14. Hookway (1985, p. 136) and Thibaud (1975, pp. 15 ff.). Such a notation

only forbids the representations of singular propositions. 15. This is one of the most important forms of abstraction (beside precisive

abstraction). It consists in introducing abstract entities, which we only identify by virtue of their relation with something with which we are already familiar (1. 559). For example, if we conceive of 'heat' as of something explaining such phenomena as the feelings of heat, expansion, contraction, etc., or what goes from what we call heat to what we call cold, then such a concept is formed by hypostatic abstraction. Such is the case of the 'dormitive virtue' of opium, introduced to explain the tendency of certain people to sleep after taking opium; what such a 'hypostatic abstraction means', according to Peirce, 'is simply that there must be some particularity in opium which must be the occasion of sleep' (5. 534); but a concept so introduced cannot have any possible application unless a certain corresponding assumption should be true. Such a procedure is the essential vehicle of deductive reasoning, as is obvious in mathematics, where mathematicians always speak of 'lines', 'numbers', 'surfaces', 'points', etc.

16. Such is the interpretation developed by Chauviré (1987). 17. For a general account of Hintikkâ s game-theoretical semantics, see

Hintikka (1979) and other papers by Hintikka in the same volume. 18. A lot more should be said here about Peirce's remarks on the constraints of

logical necessity due to our 'hereditary metaphysics': namely, the way we use and practise deduction, and of their 'Wittgensteinian' outlook (see, for example, 2. 173; NEM, IV, p. 59; NEM, IV, p. xiv).

19. Like Peirce, Johnson-Laird believes that the rules governing reasonings are semantic rather than syntactical rules, and that it is through those rules of inference that people build mental models of the premisses of a reasoning and look for mental models of the conclusion. But contrary to Peirce, who extends his analysis to all kinds of reasoning in the logic of relations, Johnson-Laird essentially builds his theory upon an analysis of the syllogistic reasoning, and particularly upon the 'figural' effect: namely, the fact that people tend to achieve better performances according to the figures of the syllogisms. But we may, of course, trace this back to what Aristotle does in Prior Analytics. About the semantic procedures used by Aristotle, see Granger (1976, pp. 127 ff.).

212 Claudine Engel-Tiercelin 20. Second draft of 'A new list of categories', cited by Murphey (1961, p. 412). REFERENCES Brock, J. (1975) 'Peirce's conception of Semiotic', Semiotica, vol. 14, no. 2, pp.

124-41. Brock, J. (1979) 'Principal themes in Peirce's logic of vagueness', Peirce Studies,

Institute for Studies in Pragmaticism, Lubbock, Tex., pp. 41-9. Brock, J. (1980) 'Peirce's anticipation of game-theoretic logic and semantics', in

Herzfeld, M. and Lenhart, M. D. (eds), Semiotics 1980, New York: Plenum Press. Brock, J. (1981a) 'An introduction to Peirce's theory of speech-acts',

Transactions of the C. S. Peirce Society, vol. 17, no. 3, pp. 319-26. Brock, J. (1981b) 'Peirce and Searle on assertion', in Ketner, K. L. et al. (eds),

Proceedings of the C. S. Peirce Bicentennial International Congress, Lubbock, Tex. Burks, A. (1943) 'Peirce's conception of logic as a normative science',

Philosophical Review, vol. 52, pp. 187-93. Burks, A. (1949) 'Icon, index and symbol', Philosophy and Phenomenological

Research, pp. 673-89. Chauviré, C. (1979) 'Peirce: le langage et l'action', Les Etudes Philosophiques,

no. 1, pp. 3-17. Chauviré, C. (1987) 'Schématisme et analyticité chez C. S. Peirce', Archives de

Philosophie, vol. 50, no. 3, pp. 413-38. Engel-Tiercelin, C. (1984) 'Peirce on machines, self-control and intentionality',

in Torrance, S. (ed.), The Mind and the Machine: Philosophical aspects of artificial intelligence, Chichester: Ellis Horwood.

Engel-Tiercelin, C. (1985) 'Logique, psychologie et métaphysique: les fondements du pragmatisme selon C. S. Peirce', Zeitschrift für allgemeine Wissenschaftstheorie, vol. 16, no. 2, pp. 229-50.

Engel-Tiercelin, C. (1986) 'Le vague est-il réel? Sur le réalisme de Peirce', Philosophie, special issue: 'La métaphysique de Peirce', no. 10, pp. 69-86.

Engel-Tiercelin, C. (1989) 'Le projet peircien d'une logique du vague', Archives de Philosophie, pp. 553-79.

Fisch, M. (ed.) (1982) Writings of C. S. Peirce, Bloomington: Indiana University Press.

Fisch, M. and Turquette, A. R. (1966) 'Peirce's triadic logic', Transactions of the C. S. Peirce Society, vol. 2, no. 2, pp. 71-85.

Granger, G. (1976) La Théorie aristotelicienne de la science, Paris. Heijenoort, J. Van (1985) Selected Essays, Naples: Bibliopolis. Hilpinen, R. (1982) 'On C. S. Peirce's theory of the proposition: Peirce as a

precursor of game-theoretical semantics', The Monist, vol. 65, pp.182-8. Hilpinen, R. (1986) 'The logic of imperatives and deontic logic', in Vuillemin, J.

(ed.), Mérites et limites des méthodes logiques en philosophie, Paris: Vrin.

Peirce's semiotic version of the semantic tradition 213 Hintikka, J. (1979) 'Quantifiers in logic and quantifiers in natural language', in

Saarinen, E. (ed.), Game-Theoretical Semantics, Dordrecht: Reidel. Hintikka, J. (1980) 'Peirce's first real discovery', The Monist, July, pp. 304-15. Hookway, C. (1985) Peirce, London: Routledge and Kegan Paul. Johnson-Laird, P. (1983) Mental Models, Cambridge: Cambridge University

Press. Ketner, K. L. (1985) 'How Hintikka misunderstood Peirce's account of

theorematic reasoning', Transactions of the C. S. Peirce Society, vol. 21, no. 3, pp. 407-18.

Martin, R. (1980) 'Individuality and quantification', in Peirce's Logic of Relations and Other Studies, Dordrecht: Reidel.

Murphey, M. (1961) The Development of Peirce's Philosophy, Cambridge, Mass.: Harvard University Press.

Nadin, M. (1980) 'The logic of vagueness and the category of synechisxri , The Monist, vol. 63, pp. 351-63.

Thibaud, P. (1975) La Logique de C. S. Peirce, de l'algèbre aux graphes, University of Provence.

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