Angell - Note on a Less Restricted Type of Rule of Inference

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Note on a Less Restricted Type of Rule of InferenceAuthor(s): R. Bradshaw AngellSource: Mind, New Series, Vol. 69, No. 274 (Apr., 1960), pp. 253-255Published by: Oxford University Press on behalf of the Mind AssociationStable URL: http://www.jstor.org/stable/2252072 .Accessed: 27/01/2014 11:45

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NOTE ON A LESS RESTRICTED TYPE OF RULE OF INFERENCE

THE rule of inference which occurs most frequently in conventional axiomatizatioris of modern logic, that is,

R1. If FS and F(S : S'), then F S'. (Modus Ponens) (I.e. if S and (S n S') have both been set down as axioms or derived theorems, then S' may be set down as a theorem)

can be replaced in such systems by either of the two following alter,natives:

R2. If [S and (. ..(S S') ... ), then( ...S'...). (I.e. if S is a theorem, and there is a theorem, T, which contains (S n S') as a component, then a new theorem T' may be formed by substituting S' for (S n S') in T.) .

R3. If [S and (. . .(S n S'). . .) is a well-formed formula, then S' may be substituted for (S n S') in the well-formed formula.

To establish the substitutability of R2 or R3 for R1, it must be shown that (1) whenever an inference can be established by R1 it could be established as well by R2 or R3, and (2) that neither R2 nor R3, if substituted for R1 would result in additions or deletions from the class of theorems derivable by R1.

(1) That R2 and R3 will permit an inference whenever R1 will is established by the fact that R1 becomes a special case of each of the other two rules; it is the special case in which S is a theorem and S is also the main antecedent of another theorem in conditional form.

(2) That rules R2 and R3 will yield the same set of theorems as R, is established by reference to truth-tables. The set of propositional forms validated by truth-tables has been shown to be equivalent to the set derivable from conventional axiomatizations of the propositional calculus which use R1. Now the application of R3 to any formula- whatever will always yield another formula with a truth-table identical to the first. For when the first formula is of the form (. . .(S n S'). . .) and S is already a theorem (the conditions laid down in R3), the column under S will be all T's and thus the column under the component (S n S') will be identical with the column for S' alone. For if the antecedent of a conditional is true in a given case, then the whole conditional is true only if the consequent is true and false only if the consequent is false; since, in this case, by hypothesis, the antecedent is always true, it follows the whole truth-table for the component (S n S') is identical with the truth-table for S' alone. This being the case the truth-table for the whole expression (. . .(S S'). . .) will remain unchanged when the (S n S') is replaced by S'. That the same holds for R2 is clear, since R2 is merely a special case of R3 (in which (. . .(S n S'). .. has all T's as its over-all truth-table).

There are some other customary rules of inference which, on similar 253


254 R. B. ANGELL: grounds, could be replaced by similar alternatives. For example, the rule of inference,

R1'. If FS and (- S V S'), then IS', may be replaced by analogues, as could various other rules related to modus ponens through De Morgan equivalences. Not all rules of iniference can be treated in this way however. For example, Nicod's rule of inference,

-R1". If FIS and H(S/(R/Q)), then I-Q could be shown to be replacable by

R4. If FS and (S/(R/Q)), then if (... .(S/(R/Q))...) then p (... Q . )

but this is not an analogue of either R2 or R3. The analogues of either R2 or R3 would not apply in the case of Nicod's rule since the truth-table for (S/(R/Q)) is not always equivalent to that of Q if (S/(R/Q)) is not a theorem even though S is a theorem.

The advantages of R2 and R3, in systems to which they apply, are twofold. First, there is perhaps an element of elegance in bring- ing modus ponens into a form which is closer to that of other rules of inference. R2'and R3, for example, operate with the same freedom to make substitutions anywhere in a formula that is found in rules of definitional substitution. This introduces greater uniformity into the rules at the base of a system.

Secondly, and more important, the new rules permit greater economy of effort in derivations, especially in systems with a small number of axioms, by cutting down the number of steps required. For example, starting with the -ukasiewicz axiom set,

Al :((p nq) 0((q nr) n(p nr))) DI :(pVq) =df (- p :q) A2: ((- p n p) ap) A3: (p : (- p : q))

if one wishes to prove the theorem ((p n q) : (p n (q V r)) using R, (modus ponens) one must proceed through sixty-four steps (i.e., sixty-four applications of one or another rule of inference) to get ((q : r) : ((p n q) n (p n r))) from which in four more steps we get the desired theorem:

(65) ((q : (- q n r)) : ((p q) n (p : (-q : r)))) (by substitution for variables in (64)).

(66) q : (- q : r)) (by substitution for variables in A3). (67) ((p n q) : (p : (- q : r))) (By RI, (65), (66).) (68) ((p : q) : (p 0 (q V r))) (By DI, and rule of definitional

substitution). In contrast, if we use R2 (or R3) we can derive the same theorem

starting from the same axiom set in four steps: (1) ((p 3 q) : ((q 3 (- q : r)) : (p : (-q n r)))) (by substitu-

tion for variables in Al). (2) (q : (- q : r)) (Substitution for variables in A3). 1 Using the line of proof outlined in A. Church, Introduction to Mathe-

matical Logic, Vol. 1, 1956, in Sections 29.2 and 12.7.


RULE OF INFERENCE 255

(3) ((p : q) : (p n (- q 3 r))) (by R2 (or R3), (1) and (2)). (4) ((p : q) : (p n (q V r))) (by DI, and rule of definitional substitution).

The economy of steps in this case is due to the fact that the principle of the syllogism (Axiom 1) can be used with R2 without requiring that its major antecedent must be a theorem; if any antecedent of a component conditional is a theorem, a new theorem may be formed immediately without having to prove another principle of the syllogism with a different order of components. Note added in proof: It has come to my attention that R2 occurred as a derived rule in Quine's device of " bracketing out ". See Mathe- matical Logic, 1940, p. 101. R.B.A.

Ohio Wesleyan University R. BRADSHAW ANGELL


Article Contentsp. 253p. 254p. 255

Issue Table of ContentsMind, New Series, Vol. 69, No. 274 (Apr., 1960), pp. 145-288Front MatterPhilosophical Discoveries [pp. 145 - 162]On the Rationality of Persuading [pp. 163 - 174]Divine Necessity [pp. 175 - 186]Meta-Ethics and Normative Ethics [pp. 187 - 205]Some Uses of "Good" in Criticism [pp. 206 - 222]Particular Works of Art [pp. 223 - 233]DiscussionsThe Philosophical "Refutation" of Pareto [pp. 234 - 243]Mrs. P. Foot on Morality and Virtue [pp. 244 - 248]On Ceasing to Exist [pp. 249 - 250]A Note on the Redundant Axiom of Principia Mathematica [pp. 251 - 252]Note on a Less Restricted Type of Rule of Inference [pp. 253 - 255]Margolis on "The Identity of a Work of Art" [pp. 256 - 258]Kolenda on Science and Morality [pp. 259 - 262]

Critical Notice [pp. 263 - 266]New Books [pp. 267 - 287]Notes [pp. 287 - 288]Back Matter

Angell - Note on a Less Restricted Type of Rule of Inference

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