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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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In EGs, each argument or logical subject of a relation is linked to its line of identity by a peg. A medad, which has no arguments, represents a proposition. (Peirce’s term comes from the Greek μη for not.) In EGIF, a proposition represented by the symbol p is written as a relation with no bound labels: (p). A monad represents a monadic predicate, which is also called a property; a dyad represents a dyadic predicate or a binary relation; and a triad represents a triadic predicate or a ternary relation.

By treating a medad as a special case of a predicate or relation, Peirce avoided the need to distinguish

propositional logic from predicate logic. In other writings, he presented existential graphs in three parts:

Alpha is the theory of propositional logic, which avoids any lines of identity; Beta introduces relations and lines of identity; and Gamma introduces modal and higher-order logic. In MS 514, Peirce combined Alpha and Beta in a unified presentation. Pedagogically, this approach is highly effective because it enables beginners to represent complete sentences with their earliest examples.

Every indivisible graph instance must be wholly contained in a single area. The line of identity can be regarded as a graph composed of any number of dyads “—is—” or as a single dyad. But it must be wholly in one area. Yet it may abut upon another line of identity in another area.

To illustrate Peirce’s point that a line of identity may be composed of any number of dyads, consider the graph man—African, which may be read “There is an African man.” Replacing the dash with four copies of —is— would break the single line of identity into five separate segments:

Peirce’s method of reading such graphs would produce the sentence “There is a man that is something that is something that is something that is something African.” Each of the five segments corresponds to an existentially quantified variable, and each instance of the dyad —is— corresponds to an equal sign between

two variables. Following is the EGIF and formula for the above EG:

[*x] [*y] [*z] [*u] [*v] (man ?x) (is ?x ?y) (is ?y ?z) (is ?z ?u) (is ?z ?v) (African ?v) ∃x∃y∃z∃u∃v (man(x) ∧ x=y ∧ y=z ∧ z=u ∧ u=v ∧ African(v)) In EGs, equality is represented by joining lines of identity. In EGIF, a join of two lines is shown by a coreference node [?x ?y], which corresponds to x=y. A coreference node may enclose any number of bound

labels to show that all their lines of identity are joined:

[*x] [*y] [*z] [*u] [*v] (man ?x) [?x ?y ?z ?u ?v] (African ?v) The rules presented in Section 3 would simplify this EGIF to [*x] (man ?x) (African ?x).

Thus fig. 5 denies that there is a man that will not die, that is, it asserts that every man (if there be such an animal) will die. It contains two lines of identity.

Peirce considered the line of identity in the shaded area to be distinct from the line in the unshaded area.

To represent that interpretation, the following EGIF has a defining label *x for the part in the shaded area, another defining label *y for the part in the unshaded area, and a coreference node [?x ?y] for the connection

at the boundary:

~[[*x] (man ?x) ~[[*y] [?x ?y] (will_die ?y)]] Peirce sometimes included blanks in the names of relations, but those blanks, which are not permitted in EGIF, can be replaced by underscores. Since the pure graph notation has no labels, there is never a need to relabel lines of identity. But in linear notations, the coreference node [?x ?y] or the corresponding equality

would permit the identifier y to be replaced by x:

~[[*x] (man ?x) ~[[?x] [?x ?x] (will_die ?x)]]

The rules presented in Section 3 would then delete redundant copies of ?x and [?x]:

~[[*x] (man ?x) ~[(will_die ?x)]] ~∃x(man(x) ∧ ~will_die(x)).

Replacing ~∃ with ∀~ and converting the body of the formula to an implication produces the formula ∀x(man(x) ⊃ will_die(x)), which may be read “For every x, if x is a man, then x will die.” [Fig. 5] denies which fig. 6 asserts, “there is a man that is something that is something that is not anything that is anything unless it be something that will not die.” I state the meaning in this way, to show how the identity is continuous regardless of shading; and this is necessarily the case. In the nature of identity that is its entire meaning. For the shading denies the whole of what is in its area but not each part except disjunctively.

For an introduction to logic, this explanation is more confusing than enlightening. The simpler reading of Figure 6 is “There is a man who will not die.” For Peirce’s disjunctive reading, the graph in the shaded area must be viewed as a conjunction of two parts. To derive his reading, two copies of the dyad —is— or the

coreference [?x ?y] must be added outside the negation and two more copies inside the negation:

[*x] [*y] [*z] (man ?x) (is ?x ?y) (is ?y ?z) ~[[*u] [*v] (is ?z ?u) (is ?u ?v) (will_die ?v)]] Before converting this EGIF to a formula, change the last copy of ?v to the coreferent label ?z and use

parentheses to enclose the two equalities inside the negation:

∃x∃y∃z(man(x) ∧ x=y ∧ y=z ∧ ~(∃u∃v((z=u ∧ u=v) ∧ will_die(z)))).

Then use De Morgan’s law to convert the negated conjunction to a disjunction:

∃x∃y∃z(man(x) ∧ x=y ∧ y=z ∧ (~∃u∃v(z=u ∧ u=v) ∨ ~will_die(z))).

This formula may be read “There is a man x that is something y that is something z that is not anything u that is anything v, or z is something that will not die.” With practice, one can learn to “see” the disjunctive pattern without doing the algebra. Graphs enable the viewer to interpret a spatial configuration in different ways.

This concludes Peirce’s presentation of the EG syntax. As he showed, it has only two explicit operators:

a line to represent the existential quantifier and an oval to represent negation. Conjunction is an implicit operator, expressed by drawing any number of graphs in the same area. Equality is shown by joining lines.

All other operators of first-order logic are represented by combining these primitives. The following diagram

shows three common combinations:

Although these three operators are composite, their graphic patterns are just as readable as the algebraic formulas with the special symbols ⊃, ∨, and ∀. In fact, the explicit nesting of EG ovals emphasizes the effect on scope of quantifiers caused by the operators ⊃ and ∨. This effect is difficult to explain to students because the algebraic notation makes ∨ and ∧ look symmetrical. Following are the EGIF and algebraic representations. Note that the medads p and q are represented as relations with no bound labels: (p) and (q).

–  –  –

With experience, anyone who uses EGs begins to recognize many other common patterns, such as the

following combination of implication and disjunction:

The EG has no implicit ordering of subgraphs, but some ordering is imposed by any linear notation. The following EGIF is just one of 24 equivalent permutations for representing the above EG.

~[(p) ~[(q)] ~[(r)] ~[(s)]] Even when the four subgraphs are written in the same order, the choice of Boolean operators enables the EG to be read in many different ways as an English sentence or an algebraic formula. Following are three

examples of an English reading and the corresponding algebraic notation:

If p, then q or r or s — p ⊃ (q ∨ r ∨ s).

If p and not q, then r or s — (p ∧ ~q) ⊃ (r ∨ s).

If p and not q and not r, then s — (p ∧ ~q ∧ ~r) ⊃ s.

Each of these formulas has more permutations for each choice in mapping the EG to EGIF. For that reason, EGs are a good candidate for a canonical form that can reduce the multiple variations caused by different choices of Boolean operators or the order of writing them. This reduction of equivalent variations can be especially useful for reducing the amount of search in programs for theorem proving.

Reading a line as a concatenation of dyads of the form —is— can sometimes clarify the translation of an EG to a sentence or formula. As examples, the next three graphs show some ways of saying that there exist two things. In the graph on the left, the shaded area negates the connection between the lines of identity on either side. To emphasize what is being negated, the graph in the middle replaces part of the line with the dyad —is—. Therefore, that graph may be read “There is something x, which is not something y.” The graph on the right says that there exist two different things with the property P or simply “There are two Ps.” The following table shows the EGIF and formulas for these three graphs.

–  –  –

As these examples show, an oval with a line through it can be read as negated equality. It is equivalent to the symbol ≠ in the algebraic notation, but it is so readable that there is no need for a special symbol. With a nest of two ovals, the graph on the left below denies that there are two Ps. The graph on the right asserts that there is exactly one P.

For the graph on the right, the outer part says that there is a P, and the shaded part denies that there is another

P different from the first. Therefore, there must be exactly one P. Following is the EGIF:

[*x] (P ?x) ~[[*y] (P ?y) ~[[?x ?y]]] The direct translation of the EGIF to an English sentence or an algebraic formula would use two negations.

But the double negation could also be read as an implication. Following are both translations:

“There is a P, and there is no other P” — ∃x(P(x) ∧ ~∃y(P(y) ∧ x≠y)).

“There’s a P, and if there’s any P, it’s the same as the first” — ∃x(P(x) ∧ ∀y(P(y) ⊃ x=y)).

These examples can be extended with multiple lines of identity and negated equalities. The graph on the left below says there exist at least three things. The graph in the middle says there exist at most three things. The

graph on the right, which combines the previous two, says there exist exactly three things:

The graph on the left has three lines of identity, which consist of the three arcs of the circle that are not enclosed in any shaded area. Each of those three lines, which may be labeled *x, *y, and *z, is continued into the adjacent shaded areas, which show that it is not coreferent with its neighboring lines. The graph in the middle also has three lines of identity outside the shaded area. The point in the center of the shaded area, which may be labeled *w, is called a teridentity because it has three branches. The graph on the right is the conjunction of the other two. The following table shows the translations of the above graphs to English, EGIF, and formulas in predicate calculus.

English EGIF Formulas

–  –  –

Note that the graph in the middle could also be read as an implication whose conclusion is a disjunction:

“There is an x, a y, and a z, and if there is a w, then either w is x, w is y, or w is z.” To represent more things, EGs could be generalized to three dimensions with wires for lines of identity and bubbles for negation. As an exercise, generalize the above EGs to four things: represent each thing by a vertex of a tetrahedron; place a small shaded ball for ≠ on each of the six edges; place a large shaded ball inside the tetrahedron; place a dot to represent the nonexistent fifth thing at the center of the large ball; connect the center dot by four wires to each of the vertices; on each of those wires (but inside the large ball) place a small unshaded ball.

In the linear notations, the labels for variables or lines obscure the symmetry. The graphic form shows identity by joining two lines, but the linear versions require special notations, such as x=y or [?x ?y]. Those notations then require special axioms, such as reflexivity, symmetry, and transitivity. Pure graphs have no labels, no axioms for equality, no rules for substituting values for variables, and no rules for relabeling variables. Those axioms and rules are artifacts of the notation.

3. Rules of Inference for FOL All proofs in Peirce’s system are based on “permissions” or “formal rules... by which one graph may be transformed into another without danger of passing from truth to falsity and without referring to any interpretation of the graphs” (CP 4.423). Peirce presented the permissions as three pairs of rules, one of which states conditions for inserting a graph, and the other states the inverse conditions for erasing a graph.

In this commentary, the insertion rules are numbered 1i, 2i, 3i, and the inverse erasure rules are 1e, 2e, 3e.

There are three simple rules for modifying premises when they have once been scribed in order to get any sound necessary conclusion from them.... I will now state what modifications are permissible in any graph we may have scribed.

Peirce’s rules are a generalization and simplification of the rules for natural deduction, which Gentzen (1934) independently discovered many years later. For both Peirce and Gentzen, the rules are grouped in pairs, one of which inserts an operator, which the other erases. For both of them, the only axiom is a blank sheet of paper: anything that can be proved without any prior assumptions is a theorem. Section 6 presents a more detailed comparison with Gentzen’s method.

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