The following appeared in Volume 96, Number 2 (Spring 1997) of the APA Newsletters.

A NOTE ON MR. DE MORGAN

Philip A. D. Schneider
Department of Philosophy and Religion
Coastal Carolina University

Most texts on symbolic logic cite what are variously called De Morgan’s laws, rules, or theorems. Named after their (relatively) modern proponent, Augustus De Morgan (1806-1871), these "laws" readily afford a method of simplifying expressions in the propositional calculus. The simplification arises by the reduction of expressions with a negation of a compound proposition to simpler expressions:

^ ( p . q ) = ^ p v ^ q

^ ( p v q ) = ^ p . ^ q

(I use here the usual notation of the tilde, dot and wedge as symbols representing the logical operations, respectively, of denial, conjunction, and disjunction. The equal sign represents logical equivalence-more on this later in Endnote 2.)

I have always thought it odd that Mr. De Morgan did not articulate a third "law" dealing with the denial of a compound implication. I suspect, but am not able to prove, that this omission resulted from the fact that the two De Morgan rules were predated by William of Ockham (1285-1349) in his Summa Totius Logicae¹ and that De Morgan merely updated Ockham’s two rules into De Morgan’s developing symbolic notation.

De Morgan’s "laws," "rules" or "theorems" are, of course, merely logical equivalences and should always be identified as such. Each equivalence (the dot and the wedge version, respectively) is readily demonstrated to be such by appeal to truth tables.

The De Morgan logical equivalence denying the dot, ^(p.q), is demonstrated by the identity of the truth tables in column 6 and column 9. The De Morgan logical equivalence denying the wedge, ^(p v q), is likewise demonstrated by the identity of the truth tables in columns 8 and 10.²

De Morgan’s dot and wedge rules are touted rightly as being useful "to get rid of a negated compound statement, which is usually hard to work with in a proof."³ It is important to note, however, that the same "usefulness" could exist for a negated material implication. For propositional calculi based on only four logical operators (tilde, dot, wedge, hook-material implication), students wonder why they can use De Morgan for dot and wedge, but can’t use a De Morgan-like simplifier for the hook. We can, and do of course, show them that a companion De Morgan-like equivalence can be derived from other equivalences. For example,

^(p>q) = ^(^p v q) = ^^p . ^q = p . ^q

where the first equivalence is supported by material implication, the second equivalence is supported by De Morgan’s Rule (!), and the third equivalence by multiple negation.

Why should there not be a companion third "De Morgan" rule that supports the simplification of a negated statement of material implication, i.e., ^(p > q)-the greater than symbol representing material implication? A number of texts seem to deny that there could be such a rule.⁴ But there is such an equivalence and it is readily demonstrated by a truth table.

^ ( p > q ) = p . ^ q

This third "De Morgan" equivalence is demonstrated by the identity of the truth tables in columns 5 and 6.

The usefulness of this third De Morgan-like equivalence is recognized only in a small minority of symbolic logic texts⁵ and, even there, may not be provided to students in the usual menu of "official" equivalences to use in simplifying compound propositions while testing for argument validity. Using truth table proofs for logical equivalence, the derivation of this third equivalence need not be based upon a chain of equivalences using other equivalences that, in some mysterious way, are thought to be more "fundamental."

How much more natural it would be to extend the traditional De Morgan equivalences by adding the third equivalence described here and demonstrate all three by appeal to the truth tables illustrated above. If, in the name of historical accuracy, we cannot bring ourselves to call this third rule "De Morgan’s,"⁶ then we can, at least, give it its rightful and useful place as an "extension" of De Morgan in the usual textual listings of bona fide equivalences available to students of the propositional calculus. I recommend this approach strongly to authors of future editions of existing texts and of any new text.

My proposal, then, comes down to the issue of what a minimal set of "existing and accepted laws" of logic should contain. In other words, why add a third De Morgan equivalence in preference to all the possible other equivalences that might be added? There does not seem to be agreement on what such a minimal set should be. Surely a pragmatic test for what to include in such a set would be simplicity and frequency of use in proofs of validity. For example, many texts include Constructive Dilemma as a "standard" argument form:

(p > q) . (r > s)

p v r

_____________

Therefore, q v s

There is, however, a "longer" way to derive the conclusion from the premises:

(1) (p > q) . (r > s)

(2) p v r

(3) r > s 1, Simplification

(4) ^p > r 2, Material Implication

(5) ^p > s 3,4, Hypothetical Syllogism

(6) ^s > p 5, Transposition, Double Negation

(7) p > q 1, Simplification

(8) ^s > q 6,7, Hypothetical Syllogism

(9) s v q 8, Material Implication and Double Negation

(10) q v s 9, Commutation

Thus, we clearly could do proofs of validity without using the argument form for Constructive Dilemma. However, I suspect only a masochist would choose eight steps over one. My proposal for a third "De Morgan" equivalence is motivated by the same search for simplicity and ease of use. Quickness of proof, using simple yet powerful logical laws, is a laudable goal-and one which will prevent adopting every derived equivalence as part of the "standard" repertoire. I believe my proposal passes this test.⁷

Endnotes

1. William Kneale & Martha Kneale, The Development of Logic, (Oxford: Clarendon Press, 1988), 294-5.

2. From a truth functional perspective, where the calculus defines the sole and total meaning of logical operators by means of their truth tables, logical equivalence means, pure and simple, that the equivalent propositions have identical truth tables. Material equivalence means merely that, when the material equivalence is true, the propositions have the same truth values. Unlike the relation of logical equivalence, the material equivalence (ME) operator is defined by a truth table:

W. V. Quine, Mathematical Logic, (Cambridge: Harvard University Press, 1983), 30 is particularly clear about the relationship between material equivalence and logical equivalence: "... statements are logically equivalent when the biconditional formed from them is logically true," i.e., the statements have identical truth tables. Robert J. Rafalko, Logic for an Overcast Tuesday, (Belmont, California: Wadsworth Publishing Company, 1990), 339 correctly notes that logical equivalences can be rewritten as tautologous material equivalences. In a nutshell: p ME q merely expresses the specific truth table cited earlier in this note. Only if p ME q is tautologous (where "... a tautologous statement is describable as one which proves true by the truth-table method under every assignment of truth values to its ultimate truth-functional components," Quine, op. cit., 52), are p and q logically equivalent. The following example illustrates this point:

Here the two logically equivalent propositions (p>q and ^pvq), indeed, have their material equivalence as a tautology. The definition of a logical equivalence as a tautologous material equivalence is an alternative to defining two propositions to be logically equivalent when they have identical truth tables. This close relationship between material and logical equivalence, however, does not justify using the three-bar (ME) operator as a surrogate symbol for logical equivalence.

A number of logic texts employ symbology that confuses logical equivalence with material equivalence. For example, William J. Kilgore, An Introductory Logic, (New York: Holt, Rinehart and Winston, 1968), properly notes (101) that "... propositions are equivalent if they have identical truth values." But Kilgore persists in using the three-bar operator (expressing material, not logical, equivalence) to express the logical equivalences, such as De Morgan’s equivalences,(181). Materially equivalent propositions in general do not, of course, have identical truth values. Even Quine’s symbology, e.g., "^(p.q) ME ^p v ^q," (op. cit., 57), misleadingly uses the three bar material equivalence operator to define logical equivalences. Patrick J. Hurley, A Concise Introduction to Logic, (Belmont, California: Wadsworth Publishing Company, 1994) corrected such usage in his fourth edition; until then, he used the three-bar operator (ME) to express logical equivalence throughout the text and on the back cover summary of the rules of inference and axioms of replacement in the propositional calculus. In the fourth edition, Hurley adopted a four-dot symbol to represent logical equivalence. I, myself, prefer to use the equal sign.

3. David Kelley, The Art of Reasoning, (New York: W. W. Norton, 1994), 355.

4. Hurley, op. cit., 371: "When applying De Morgan’s rule, one should keep in mind that it holds only for conjunction and disjunction (not for implication ...)." Also, Paul Herrick, The Many Worlds of Logic, (Fort Worth, Harcourt Brace, 1994), 167: "Note that De Morgan’s rule applies only to conjunctions and disjunctions-it does not apply to horseshoes ... ." It might be that these authors are looking primarily to the distributive property of the tilde in the De Morgan equivalences for dot and wedge-the tilde "moves" across the parenthesis and attaches to each of the component propositions, whereas it does not so move for hook. Well enough, but this focus on distribution seems a bit fanciful since, along with the "moving tilde," the operators are also changing-the dot to a wedge, and the wedge to a dot. It seems more natural to treat the De Morgan equivalences de novo rather than as specially contorted distributional rules. This way, logic texts can, in good conscience, add a "De Morgan" hook equivalence rather than seeming to deny its availability.

5. W. V. Quine, op. cit., 57, calls the De Morgan-like equivalence for hook a "minor variant" of De Morgan’s wedge equivalence. Also, Robert Paul Churchill, Logic: An Introduction, (New York: St. Martin’s Press, 1990), 280-1.

6. This seems overly fastidious because the "De Morgan" rules would, more accurately, be called Ockham’s Rules (see Endnote 1) anyway.

7. I am indebted to the reviewers of an earlier version of this paper who provided insightful and incisive challenges to my original exposition.

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