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«1. Peirce’s Method for Teaching Logic Peirce’s existential graphs (EGs) are the simplest, most elegant, and easiest-to-learn system of logic ever ...»

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These rules can evaluate Alpha graphs (propositional logic) in exactly the same way as Beta graphs. A model M for propositional logic would be a set of medads, such as {(p), (q), (t)}. A graph g with no negations is true of M if the set of medads in g is a subset of the medads in M. As an exercise, evaluate the EG for the Praeclarum Theorema in terms of some set of medads. Since it is a theorem, it should be true of every model, including the empty set.

To illustrate these rules for Beta graphs, consider the following relational graph as a model M of a state of affairs in which there is lightning at 1 pm, thunder and lightning at 2 pm, and thunder at 3pm.

As the first example, let the graph g be lightning—at—2pm. Since g has no negations, Graphist, the original proposer, wins because there is a mapping from g to the middle part of M. But if g happened to be lightning— at—3pm, then Grapheus would win, because this g cannot be mapped to M.

As an example that requires more than one pass through the procedure, consider Peirce’s graph in Figure 4, which represents the sentence “There is thunder, and not lightning.” That graph meets the conditions for Step 4 of the endoporeutic procedure: the subgraph g0 for the part outside the negation is —thunder. Graphist, the proposer, can choose any of two possible mappings of g0 to M: one for thunder at 2 pm and the other for thunder at 3 pm; this choice also determines the mapping of the line inside the negation. If Graphist chooses 3 pm, the subgraph g0 is erased, and the game continues. At Step 2, the negation is erased, causing the new g to become —lightning, but with the line forced to be mapped to the subgraph of M at 3 pm. Then the two players change sides, making Graphist the new skeptic. The game continues at Step 1, where g cannot be mapped to M because lightning occurred at 2 pm, not 3 pm. Therefore, the skeptic wins because this subgraph with this mapping is false. But the current skeptic is Graphist, who had been the original proposer for Figure 4, which is therefore true. Note that if Graphist had made a mistake in the original choice of mapping, then Graphist would have lost. The truth of a graph, however, is not determined by mistakes; it depends only on the existence of a winning strategy if both players make the best choice at each option.

As a final example, consider Figure 3, which adds one more negation around Figure 4. This graph meets the condition for Step 2 of the game, which removes the negation and causes the two players to change sides.

That makes Grapheus the proposer for a graph that is now identical to Figure 3, which has a winning strategy.

Therefore, Grapheus wins the game that started with Figure 4, which must therefore be false. Since Graphist and Grapheus may change sides during the game, a graph is true if and only if Graphist wins, independent of which role Graphist happens to be playing at the end.

To use endoporeutic to prove that Peirce’s rules of inference are sound, it is necessary to show that each rule preserves truth — i.e., if Graphist has a winning strategy with a given EG, no rule of inference can transform it to a graph that allows Grapheus to win. Therefore, the rules must monotonically increase the winning options for Graphist and monotonically decrease the winning options for Grapheus. Since the two players switch sides when a negation is removed (Step 2), Graphist is the proposer in every positive area and the skeptic in every negative area. The two steps of endoporeutic that depend on the number of options are Step 3, where the skeptic chooses one of the negations, and Step 4, where the proposer chooses one of the possible mappings of g0 to M.

1. Rule 1e erases an arbitrary subgraph in a positive area. Erasing a negation decreases the number of options for Grapheus, who is the skeptic in a positive area. Erasing all or part of a relational graph reduces the constraints on the mapping to the model M and thereby increases the options for Graphist.

Rule 1i, which inserts an arbitrary subgraph in a negative area, has the opposite effects: If negations are added, Graphist, who is the skeptic, has more options to choose. Adding a subgraph to a relational graph adds more constraints on the mapping to M and thereby reduces the options for Grapheus.

2. Rules 2e (deiteration) and 2i (iteration) have no effect on the truth value of any graph because any subgraph g that could be iterated or deiterated has its effect on the evaluation at its first occurrence.

There are two possibilities to consider: g is true, or g is false. If g is true, a copy of g inserted or erased from any area has no effect on the truth value of that area. If g is false, the original area in which g occurs is false; the truth value of any nested area is irrelevant.

3. The only effect of evaluating a double negation is to cause the proposer and skeptic to reverse their roles twice. Therefore, Rules 3e (erasing a double negation) and 3i (inserting a double negation) can have no effect on the truth value of any graph.

In summary, rules 1e and 1i monotonically increase the options for Graphist and monotonically decrease the options for Grapheus. Rules 2e, 2i, 3e, and 3i have no effect on the winning strategies for either side.

What distinguishes the game-theoretical method from Tarski’s approach is its procedural nature. One reason why Peirce had such difficulty in explaining it is that he and his readers lacked the vocabulary of the gameplaying algorithms of artificial intelligence. Peirce had written many pages about endoporeutic, but no one clearly deciphered them until Hilpinen noticed the similarity to game-theoretical semantics. The discussion in this section is a reconstruction and clarification in modern terminology of Peirce’s often cryptic notes.

Unlike Tarski’s definition, which maps all variables in a formula to individuals in a model M, endoporeutic is a “lazy” method that avoids mapping a line to M until the proposer chooses it in Step 4. Any subgraph that is not chosen is never evaluated; some very large or even infinite subgraphs can often be ignored. Lazy methods similar to endoporeutic are used in computational systems, such as evaluating an SQL query in terms of a relational database. Game-playing programs reduce the computation by a lazy method called the α-β algorithm. An application of α-β to endoporeutic could order the skeptic’s options in Step 3 with the smaller graphs first. If the best strategy for the skeptic is to choose one of the smaller graphs, the larger, possibly infinite graphs, might be ignored. Another optimization is to index the names of relations in M in order to speed up Step 4, where the proposer searches for a mapping from a subgraph to M. With these optimizations, the computational complexity of endoporeutic is the same as evaluating an SQL query in terms of a relational database.

5. How to Teach Logic Peirce’s approach in MS 514 is an excellent basis for teaching logic. The relational subset consisting of just conjunction and the existential quantifier is easy to understand, and it is widely used, even by people who have no idea that it is a subset of logic. Without any additional features, relational graphs can describe anything that exists. They are general enough to express the content of a relational database or anything expressed in the Resource Description Framework (RDF). Peirce (1882) tried to generalize relational graphs by adding other operators, but he couldn’t state everything expressible in his algebraic notation.

As an example, the sentence “A red ball is on a blue table” can be expressed by the relational graph on the left below. With small ovals that negate one relation at a time, the graph on the right says “Something red that is not a ball is on a table that is not blue.” Peirce had tried to add more features to relational graphs, but he couldn’t find a way to extend the scope of a negation or quantifier beyond such local modifications. In the 1960s, that same limitation plagued the development of semantic networks by the artificial intelligence community (Sowa 1987). Peirce’s breakthrough of 1897 was merely to use larger ovals, as in the next two graphs.

Since each of these graphs has four ovals, there are several different ways of reading them. They make good classroom exercises for soliciting different readings from the students. The graph on the left can be read as a simple if-then sentence, as an if sentence with a disjunctive conclusion, or as a universally quantified

sentence with a disjunctive body:

• “If there is a red thing that is not a ball, then it’s on a table that is not blue.” • “If there is something red, then either it’s a ball or it’s on a table that is not blue.” • “Every red thing is a ball or is on a table that is not blue.” Since the graph on the right has two lines of identity (quantifiers) inside the shaded oval, the English

pronouns must be supplemented with other words to distinguish them:

• “If a red thing that is not a ball is on something, then the latter is a table that is not blue.” • “If a red thing is on something, then either the former is a ball or the latter is a table that is not blue.” • “If a red thing x is on something y, then either x is a ball or y is a table that is not blue.” • “For every red thing x on something y, either x is a ball or y is a table that is not blue.” Letters or variables are used in the algebraic notation for logic or to supplement the pronouns of a natural language. Such supplements are necessary for a linear notation, but they are not needed in a graph notation that shows identity by direct connections.

These diagrams illustrate some fundamental principles of logic, which are true of any notation, but which are

especially clear when expressed in EGs:

• Without negations, the operators of conjunction and existence are sufficient to describe anything that exists. That includes all the experimental data of any branch of science, since negations can only be inferred, never observed directly.

• Local negations, which negate an indivisible graph or atom, deny that a particular property or relation is true of the individual(s) designated by the pegs attached to the relation name.

• To deny an entire configuration of entities and relations, negations with a larger scope are necessary.

The scope can be demarcated by a larger oval in EGs, by a parenthesized expression in a formula, or by a phrase or sentence in natural languages.

• To state generalizations, implications, and disjunctions, a nest of two negations is required. In some notations, the nest of negations may be implicit in a single symbol, such as ∀, ⊃, and ∨.

Franz Brentano, who was born a few months before Peirce, also maintained the primacy of conjunction, negation, and the existential quantifier as the basic logical operators. Although Brentano’s notation was limited to expressing Aristotle’s syllogisms, it used the same three operators as EGs in logically equivalent ways (Simons 2004).

A good way to illustrate these principles is to start with a relational graph on any subject, overlay ovals of various sizes, and ask the students to translate them to their native language. The graphs can be drawn in any medium ranging from chalk on a blackboard to animated computer graphics. In classroom exercises, colored balls, blocks, and tables can be used to build structures. Then the students can compete to see who can be the first to describe a structure in a correct EG, translate an EG to English, or modify a structure when the EG is changed. Simple relational graphs are sufficient to describe any structure that can be built, but the teacher can ask the students to draw EGs that express generalizations or options that are true of several different structures. Another possibility is to draw a general EG and ask the students to find many different structures that make the EG true or false. These exercises serve two purposes: they teach logic, and they teach the students how to express themselves clearly and precisely in their native language.

Logicians sometimes object to teaching EGs because the algebraic notation is the one that students must use in order to get their papers published. But Don Roberts showed that EGs are a superior notation for learning logic, even when the students later use other notations. At the philosophy department of the University of Waterloo, Roberts and another professor were scheduled to teach two sections of an introductory course on logic. Roberts proposed a direct comparison: he would teach his section using EGs, and the other professor would use a traditional textbook based on the algebraic notation. At the end of the course, the students in both sections would take the same final exam, which would use only the algebraic notation. The other professor agreed. For almost the entire course, Roberts used EGs to introduce all the concepts of logic including Peirce’s method of proofs. In the last two weeks, he showed how to translate EGs to and from the algebraic notation, including some exercises for adapting Peirce’s proof methods to the algebraic notation. On the final exam, the section that began with EGs had a higher average score, and their scores on problems that involved proofs were much higher.

The comparison by Roberts and his colleague was not a controlled study, but it’s typical of the way students respond to existential graphs. For evidence of why they prefer EGs, look at the usual exercises in courses,

textbooks, and exams. Following is a problem from an exam that was posted on the Web:

Premises: r ∨ u. ~q ⊃ ~r. ~s. q ⊃ s.

Use the argument forms modus tollens, modus ponens, disjunctive syllogism, and conjunctive addition to deduce ~r ∧ u. Justify each step.

As an exercise, solve this problem in two ways: by the teacher’s method and by applying Peirce’s rules to the equivalent EGs. Note that the teacher had stated exactly which derived rules of inference to use and in which order to apply them (probably because too many students had failed previous exams). With EGs, the three permissions without any derived rules are simple and efficient: by 2e, derive ~q; by 2e, derive ~~~r; by 3e, derive ~r; by 2e, derive ~~u; by 3e, derive u; by 1e, erase everything but the desired conclusion, ~r ∧ u.

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