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However, the limit of the project of the paper is clear. It does not present a calculus for the logic of relations:. I regret that I am not in a situation to be able to perform this labor [a calculus, or art of drawing inferences based on the notation I am to describe]. Putnam While a good portion of that paper was devoted to classification among relations, the paper has important notions which became consistent themes throughout Peirce's numerous writings on logic: assigning logical interpretations to mathematical i.

Interestingly enough, Peirce again focuses on compositions of relations and predicts when complexity arises, which must have pushed him to the invention of a new notation. In , we see the appearance of these two quantifiers with bound variables:.

The Existential Graphs of Charles S. Peirce - Don D. Roberts - Google Livres

A relation is modeled by a pair of objects:. Even though the basic idea for ordered pairs existed in the paper, we can see more clearly how it is combined with quantifiers and bound variables in the Note CP A general relative may be conceived as a logical aggregate of a number of such individual relatives. In the same paragraph, another significant move was made toward the modern style of first-order logic, that is, to remove algebraic representation:.

It is worthwhile to note that multiple quantifiers, i. It did not escape Peirce's attention:. The logic of relatives is highly multiform; it is characterized by innumerable immediate inferences, and by various distinct conclusions from the same sets of premises. When the relative and non-relative operations occur together, the rules of the calculus become pretty complicated.

Obviously, Peirce noticed a fundamental difference between the logic of non-relatives and the logic of relatives, which corresponds to a difference between propositional and first-order logic. I suspect this is one of the reasons why Peirce did not share Frege's interest in developing an axiomatic system—at least for a while—until his EG was invented. For the rest of the Note, Peirce presents examples in the new notation involving multiple quantifiers CP He carries out his promise in section 3 of the paper by suggesting a list of methods of transformation, which is not complete.

The following are some of the rules involving quantifers a CP So far, we have argued that Peirce's insight on relations pushed him to extend the territory of logic from monadic, non-relational, propositional logic to relational quantification logic. This represented the beginning of modern logic as we know it. In this section, we take up a different angle of Peirce's adventure—to extend forms of representation from symbolic systems to diagrammatic systems. Believing that this aspect of Peirce's project is not yet fully appreciated, we present a story where his two different kinds of extension—one from non-relations to relations and the other from symbolic to diagrammatic—are connected with each other.

More recently, interdisciplinary research on multi-modal reasoning has drawn our attention to non-symbolic systems see, e. In that context, Shin focused on differences between symbolic versus diagrammatic systems and suggested a new way of understanding the EG system, though this was criticized in Pietarinen While Peirce mainly presented linear expressions in his official writings from to , [ 16 ] the notation adopted in Frege's Begriffsschrift is more iconic; it is at least not as linear as Peirce's in the above period.

However, it is Peirce, not Frege, who invented a full-blown non-symbolic system for first-order logic—Existential Graphs. As the EG system has been investigated more rigorously, philosophical questions involving Peirce's invention of the system have been raised as well.

The discovery of EG's power and novelty has naturally led us to other parts of Peirce's philosophy. Why and how did the invention of EG come about? What does EG reveal about Peirce's view of logic and representation? Many of us have pointed out Peirce's theory of signs, which classifies signs as being of three kinds—symbols, indices, and icons—as the foremost theoretical background for Peirce's EG.

We do think the connection is real, and the more we explore his view on signs the better we understand his EG Shin 22— But we do not think his sign theory could provide us with everything we need for the story of EG. There is a big gap between Peirce's talk about icons [ 18 ] and his invention of full-blown graphical systems; something else has to be brought into the picture to explain how Peirce arrived at EG from his talk about icons.

From a slightly different perspective, van Heijenoort's's distinction between Boole's calculus ratiocinator versus Frege's lingua characteristica could be related to the topic. Agreeing with both Hintikka's and Goldfarb's evaluation that Peirce belongs to Boole's tradition, Shin found a connection between the model-theoretic view of logic where Boole and Peirce are placed and EG's birth see Shin 14—16, and Pietarinen However, Peirce's awareness of the re-interpretation of language is necessary, but not sufficient, for his pursuit of a different form of representation.

While the acknowledgment of the possibility of different models of a given system was presupposed by Peirce's project for various kinds of systems, not every Boolean has presented multiple systems. Without challenging these existing explanations involving Peirce's EG, in this entry we bring in one overlooked but crucial aspect of Peirce's journey to EG so that our story may fill in part of the puzzle of Peirce's overall philosophy.

Peirce's mission for a new logic started with the question of how to represent relations, which led him to invent quantifiers and bound variables, as we discussed in the previous section. We claim that the same commitment—that is, to represent relations in a logical system—was a main motivation behind Peirce's search for a new kind of sign system: the iconic representation of relations.

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While improving Venn systems, Peirce realizes that the following defect cannot be eliminated:. It does not extend to the logic of relatives.


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Peirce CP Again, we do not think this is the crucial ingredient for the creation of EG, but one key element which works well with his theory of signs and his model-theoretic view of logic. After his own new notation came out in as seen above, why did Peirce revisit the logic of relations? The first paragraph of the paper provides a direct answer:.

Two things should be noted. That is, according to Peirce, algebra is not limited to symbolic systems. The other is that Peirce makes it clear that two different forms of algebra carry out the new logic of relations, not new logic s.

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In thinking about the scope of the logic of relations, the question arises: Why did Peirce feel the need for another form of representation different from the notation? See the entry on Peirce's Theory of Signs. Finally, Peirce's hallmark of the pragmatic maxim leads us to the third grade of clarity:. It appears, then, the rule for attaining the third grade of clearness of apprehension is as follows: Consider what effects, which might conceivably have practical bearings, we conceive the object of our conception to have.

Then the whole of our conception of those effects is the whole of our conception of the object. In order to understand what a relation is, we need to know what follows from it.


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Then, the question is how we know what its consequences are. The third grade of clearness consists in such a representation of the idea that fruitful reasoning can be made to turn upon it, and that it can be applied to the resolution of difficult practical problems. Therefore, how a relation is represented is crucial in figuring out what follows from a relational state of affairs.

It is obvious that in the paper Peirce intends to search for more desirable representations.

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Influenced by A. Kempe's graphic representation, [ 19 ] Peirce finds an analogy between relations and chemical compounds:. A chemical molecule consists of chemical atoms, and the manner in which atoms are connected with one another is based on the number of loose ends of each atom.

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For example, chemical atom H has one loose end and chemical atom O has two. So, the following combination is possible, and it is a representation of the water molecule, H 2 O:. An analogy to the logic of relations goes like this, putting Peirce's ideas into more modern terms: A proposition consists of names proper names or indices and predicates, and each predicate has a fixed arity.

Peirce created a novel and productive analogy in representation between chemistry and the logic of relation by adopting the doctrine of valency as the key element for the analogy, as shown in the above two diagrams. Believing that this graphic style of representation would help us conceive the consequences or effects of a given relation in a more efficient way, Peirce presents Entitative Graphs, which is a predecessor of EG.

EG keeps the representation of a relation developed in Peirce's paper, and it remains as his final and the most cherished notation for the logic of relations CP: 3. EG consists of three parts—Alpha, Beta, and Gamma—which respectively correspond roughly to propositional, first-order, and modal logic. After presenting the Alpha system in a formal way, we discuss the Beta system of EG focusing on Peirce's novel ideas in expanding a propositional graphic system to a quantificational graphic system. Below we introduce Alpha Graphs as a formal system in a standard way, that is, to present its syntax inductively and its semantics recursively.

In order to place Peirce's graphic systems in the traditional well-developed discourse of logic, we introduce an intermediate stage, that is, to read off Peirce's graphs into symbolic language. This will make Peirce's graphs more accessible, and at the same time support our claim that Peirce extended forms of representations under the same scope of logic. As a result, the reader might notice some differences between Peirce's terminology and ours since some of Peirce's terms are modified versions of familiar phrases. Inevitably we will end up simplifying many of Peirce's ideas behind EG.

We also would like to emphasize that this is not the only way to approach Peirce's EG.


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For example, some claim that game-theoretic semantics were foreshadowed by Peirce, and thus argue for a more dynamic understanding of EG from the game-theoretic point of view Burch ; Hilpinen ; Hintikka ; Pietarinen Here we present two equivalent reading methods for the system.

The Endoporeutic reading algorithm, formalized based on Peirce's suggestion, is a traditional way to understand EG. Each of these two readings has its own strength. The Endopreutic reading assures us that the Alpha system is truth-functionally complete, since it has power to express conjunction and negation.

Two questions may be raised: i Is there a redundancy in the Multiple readings method? For example, is 4 b above dispensable in terms of 3 and 4 a? The Endoporeutic reading allows us to get the first reading only, but we may obtain different sentences by the Multiple Readings. Of course, all of these sentences are logically equivalent. Here is an interesting point: In the case of symbolic systems, we need to prove the equivalence among the above sentences by using inference rules.

But, derivation processes are dispensable in the case of the Alpha system when the Multiple readings are adopted. Since we have the semantics for propositional logic and our reading methods translate Alpha diagrams into a propositional language, we can live without the direct semantics.

However, if one insists on the direct semantics:. Emphasizing the symmetry both in erasure versus insertion and in even versus odd number of cuts, Shin rewrote the rules , 84—85 :. Peirce did not aim to present a new logic by inventing a graphic system, but rather to present another new notation for the logic carried out by quantifiers and bound variables. He almost took it for granted that a graphic representation of relations helps us observe their consequences in a more efficient way.

Visual Logic: Peirce’s Existential Graphs

Hence, the Beta system may be considered to be the final stop of Peirce's long journey, which started in at the latest. We will not go into the formal details of the Beta system in this entry but will instead refer to Chapter 5 of Shin, where three slightly different approaches to Beta graphs—Zeman's, Roberts', and Shin's—are discussed at a full length. While Zeman's reading is comprehensive and formal, Roberts' method seems to appeal to a more intuitive understanding of the system.

Taking advantage of the merits of these two existing works, Shin developed a new reading method of Beta graphs and reformulated the transformation rules of the system. In the remaining part of the entry, we would like to examine how the essence of the logic of relations is graphically represented in the Beta system so that the reader may place EG in the larger context of Peirce's enterprise. The introduction of quantifiers and bound variables is believed to be one of the key steps of first-order logic in symbolic systems. If this is the case, then how does Peirce represent quantifiers and bound variables in Beta graphs?

Interestingly enough, when Peirce considered a graphic system his first concern was representation of relations, not representation of quantifiers. Hence, the arity of a predicate is represented by the number of lines radiating from the predicate term. Next, Peirce extends the use of a line to connect predicates:. In many reasonings it becomes necessary to write a copulative proposition in which two members relate to the same individual so as to distinguish these members.