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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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1st Permission. Any graph-instance on an unshaded area may be erased; and on a shaded area that already exists, any graph-instance may be inserted. This includes the right to cut any line of identity on an unshaded area, and to prolong one or join two on a shaded area. (The shading itself must not be erased of course, because it is not a graph-instance.) The proof of soundness depends on the fact that erasing a graph reduces the number of options that might be false, and inserting a graph increases the number of options that might be false. Rule 1e, which permits erasures in an unshaded (positive) area, cannot make a true statement false; therefore, that area must be at least as true as it was before. Conversely, Rule 1i, which permits insertions in a shaded (negative), area cannot make a false statement true; therefore, the negation of that false area must be at least as true as it was before. A formal proof of soundness requires a version of model theory. Section 4 uses Peirce’s modeltheoretic semantics, which he called endoporeutic for “outside-in evaluation.” These rules apply equally well to propositional logic and predicate logic. Since EGs have no named variables, the algebraic rules for dealing with variables are replaced by rules for cutting or joining lines of identity (which correspond to erasing or inserting an equality or the graph —is—). Cutting a line in a positive area has the effect of existential generalization, because the newly separated ends of the line represent different existentially quantified variables. Joining two lines in a negative area has the effect of universal instantiation, because it replaces a universally quantified variable with an arbitrary term.

2nd Permission. Any graph-instance may be iterated (i.e. duplicated) in the same area or in any area enclosed within that, provided the new lines of identity so introduced have identically the same connexions they had before the iteration. And if any graph-instance is already duplicated in the same area or in two areas one of which is included (whether immediately or not) within the other, their connexions being identical, then the inner of the instances (or either of them if they are in the same area) may be erased. This is called the Rule of Iteration and Deiteration.

In other writings, Peirce presented more detail about applying these rules to lines of identity. Iteration (2i) extends a line from the outside inward: any line of identity may be extended in the same area or into any enclosed area. Deiteration (2e) retracts a line from the inside outward: any line of identity that is not attached to anything may be erased, starting from the innermost area in which it occurs. Peirce also said that no graph may be copied into any area within itself; it is permissible, however, to copy a graph and then make a copy of the new graph in some area of the original graph.

The proof of soundness of iteration (2i) and deiteration (2e) shows that they are equivalence relations: they can never change the truth value of a graph. First, note that a copy of a graph p in the same area is equivalent to the conjunction p∧p; inserting a copy of p by Rule 2i or erasing it by 2e cannot change the truth value. For a copy of a subgraph into a nested area, the proof in Section 4 shows that the subgraph makes its contribution to the truth value of the whole graph at its first occurrence. The presence or absence of a more deeply nested copy is irrelevant.

3rd Permission. Any ring-shaped area which is entirely vacant may be suppressed by extending the areas within and without it so that they form one. And a vacant ring shaped area may be created in any area by shading or by obliterating shading so as to separate two parts of any area by the new ring shaped area.

A vacant ring-shaped area corresponds to a double negation: two negations with nothing between them. The third permission says that a double negation may be erased (3e) or inserted (3i) around any graph on any area, shaded or unshaded. Note that Peirce considered the empty graph, represented by a blank sheet of paper, as a valid existential graph; therefore, a double negation may be drawn or erased around a blank. An important qualification, which Peirce discussed elsewhere, is that such a ring is considered vacant even if it contains lines of identity, provided that the lines begin outside the ring and continue to the area enclosed by the ring without having any connections to one another or to anything else in the area of the ring. Both rules 3e and 3i are equivalence rules, as the method of endoporeutic shows in Section 4.

It is evident that neither of these three principles will ever permit one to assert more than he has already asserted. I will give examples the consideration of which will suffice to convince you of this.

Fig. 7 asserts that some boy is industrious. By the 1st permission it can be changed to fig. 8, which asserts that there is a boy and that there is an industrious person. This was asserted as fig.

7, together with the identity of some case.

Erasing a graph or subgraph usually simplifies a statement, but erasing part of a line of identity replaces one line of identity with two. To emphasize what is being erased, the EGIF or formula that corresponds to Figure 7 can be written as if it contained the dyad —is—, the coreference [?x ?y], or the equation x=y. The following table shows the EGIF or the formula for Figure 7, an intermediate form with the coreference or the equation, and the EGIF or formula for Figure 8.





–  –  –

Rule 1e, which allows any graph to be erased in a positive area, has the effect of erasing the equation from the intermediate form to derive the EGIF and formula for Figure 8. The result leaves open the question whether the boy and the industrious person are the same or different.

Fig. 9 asserts either there is nothing known for certain or else there is no communication with anybody. By the same permission this can be changed to fig. 10 which asserts that no communication with anybody deceased is known for certain. But this is fully included in the state of things asserted in fig. 9.

The simplest reading of Figure 9 is to append “It is false that” in front of the reading of the positive graph:

“It is false that some x is known for certain that is a communication with some y.” A more natural English reading would translate a negated line of identity as “nothing” or “no,” but then the second line of identity must be read as “any” in order to avoid a double negation. That would lead to the reading “No communication with anybody is known for certain.” The same procedure would lead to Peirce’s reading for Figure 10: “No communication with anybody deceased is known for certain.” Peirce’s reading for Figure 9 is another disjunctive statement, which is more confusing than enlightening in a tutorial for beginners. To derive that reading, start with the direct translation of Figure 9 to an algebraic

formula:

~∃x∃y(knownForCertain(x) ∧ communicationWith(x,y)) Then move the negation inward to convert the existential quantifiers to universals and apply De Morgan’s

laws to convert a negated conjunction to a disjunction of negated relations:

∀x∀y(~knownForCertain(x) ∨ ~communicationWith(x,y)) Literally, this formula says “For every x and y, either x is not known for certain, or x is not a communication with y.” Such a complex reading is distracting in an example intended to illustrate rules of inference.

In another passage of MS 514, Peirce said that the purpose of his rules is “to dissect the reasoning into the greatest possible number of distinct steps and so to force attention to every requisite of the reasoning.” To illustrate that point, he proved the syllogism named Barbara as a derived rule of inference. He starts with the premises “Any M is P” and “Any S is M” in Figure 11 and concludes “Any S is P” in Figure 16.

I will now, by way of an example of the way of working with this syntax, show how by successive steps of inference to pass from the premises of a simple syllogism to its conclusion.

The first step consists in passing to fig. 12 by the 2nd Permission [which allows the graph on the left of fig. 11 to be iterated (copied) into the graph on the right].

The second step is simply to erase “Any M is P” by the 1st Permission. The third step is to join

the two ligatures by the 1st Permission as shown in fig. 13:

It will be observed that in iterating the major premise, I had a right to put the new graph instance at any part of the area into which I put it; and I took care to have the ligature of the minor premise touch the shaded area of iterated graph instance. Now by the 1st Permission I have a right to insert what I please into a shaded area, and without making the new line of junction leave the shaded area, I make it touch the unshaded line of identity of the major premise.

Peirce’s explanations of lines that touch a boundary are sometimes confusing. If he had not made the boundary of the iterated graph instance touch the line of identity, the join would take two steps: an extension of the outer line into the shaded area by Rule 2i, and a join of the two lines by Rule 1i.

Before the two lines were joined, the inner copy of M could not be erased by deiteration because the two copies of M were attached to different ligatures. But after the join, both copies of M are attached to the same ligature, and the inner copy of M can be erased by Rule 2e.

This gives me a right in the fourth step to deiterate M so as to give fig. 14 by the second permission.

The fifth step is to delete the M on an unshaded field giving fig. 15 while the Sixth step authorized by permission the third consists in getting rid of the empty ring shaped shaded area round the P, giving fig. 16.

In the EGIF version of Peirce’s proof, the major differences result from using labels instead of lines of identity. The lines that cross borders between areas raise questions that Peirce answered in different ways in different writings. The labels in EGIF simplify the issues about lines crossing borders, but they add complications when lines of identity are joined. Following are the EGIF statements for the starting graphs in

Figure 11 and the six steps that produce Figure 16:

0. Starting graphs: ~[[*x] (M ?x) ~[(P ?x)]] ~[[*y] (S ?y) ~[(M ?y)]]

1. By rule 2i, iterate (copy) the graph on the left into the innermost area of the graph on the right:

~[[*x] (M ?x) ~[(P ?x)]] ~[[*y] (S ?y) ~[(M ?y) ~[[*x] (M ?x) ~[(P ?x)]] ]]

2. By rule 1e, erase the graph on the left:

~[[*y] (S ?y) ~[(M ?y) ~[[*x] (M ?x) ~[(P ?x)]] ]]

3. By rule 1i, join the line of identity with the label y to the line with the label x:

~[[*y] (S ?y) ~[(M ?y) ~[[?y] (M ?y) ~[(P ?y)]] ]] Note that the join causes the defining label *x and its bound labels ?x to become ?y. This operation is the equivalent of substituting one variable for another in the algebraic notation.

4. Since there are two copies of (M ?y), the more deeply nested copy may be deiterated (erased) by rule 2e. The node [?y], which is more deeply nested than the defining node [*y], can also be deiterated:

~[[*y] (S ?y) ~[(M ?y) ~[ ~[(P ?y)]] ]]

5. By rule 1e, erase the node (M ?y), which is in a positive area nested two levels deep:

~[[*y] (S ?y) ~[ ~[ ~[(P ?y)]] ]]

6. By rule 3e, erase one of the double negations. The result is the EGIF for Figure 16:

~[[*y] (S ?y) ~[(P ?y)] ] Peirce’s rules are fundamentally semantic: each one inserts or erases one or more meaningful units. For propositional logic (Alpha graphs), those units are relations (medads) and negations. First-order logic with equality (Beta graphs) adds lines of identity and connections between lines. In EGIF, those four kinds of meaningful units are expressed by nodes: relations, negations, defining nodes, and coreference nodes. Peirce treated functions as a special case of relations, and he used the same notation for both. The EGIF grammar presented in the appendix adds nodes called functions to support the full semantics of Common Logic. The

rules for propositional logic are a subset of the FOL rules because they do not mention lines of identity:

1. (i) In a negative (shaded) area, one or more nodes may be inserted.

(e) In a positive (unshaded) area, one or more nodes may be erased.

2. (i) One or more nodes in any area a may be iterated (copied) in the same area a or into any area nested in a.

(e) Any node that could have been derived by rule 2i may be erased. (Whether or not a node had previously been derived by 2i is irrelevant.) 3. (i) A double negation may be drawn around any collection of zero or more nodes in any area.

(e) Any double negation in any area may be erased.

Since EGIF does not use shading, a positive area is defined by an even number of negations, and a negative area by an odd number of negations. The conditions for first-order logic with equality include all the above conditions with further constraints and operations on the labels for lines of identity. Before stating those conditions, some definitions are necessary: the scope of a defining label; the EGIF equivalents for Peirce’s verbs prolong, join, and cut; and the verb simplify when applied to coreference nodes.

• If a defining node d occurs in an area a, the scope of d and its defining label is the area a and any area directly or indirectly enclosed by any negation in a. Any bound label in the scope of d that has the same identifier as d is said to be bound to d and to the defining label of d. In EGIF, the node d must precede (occur to the left of) all nodes that directly or indirectly contain labels bound to d.

• To prolong a line of identity with a defining node d into any area a in the scope of d is to insert a coreference node in a that has a single bound label that is bound to d.

• To join two lines of identity with defining nodes d and e in any area a that is in the scope of both d and e is to insert a coreference node in a that contains bound labels for d and e and no others.

• To cut a line of identity with defining node d in an area a in the scope of d is to insert a defining node e whose label is distinct from all other labels whose scope includes a and to replace some or all of the labels bound to d that are in the scope of e with labels bound to e.



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