Implication in Russell’s Conception of Logic
In The Principles of Mathematics, Section 16, Russell argues that implication is indefinable. We will not be concerned with that argument6 but only with a consequence of the conclusion: knowledge of implication, and of indefinables in general, requires a kind of mental perception:
The discussion of indefinables ... is the endeavour to see clearly, and to make others see clearly, the entities concerned, in order that the mind may have that kind of acquaintance with them which it has with redness or the taste of a pineapple.
(1903,xv)
But how does “discussion” enable one to “see” such properties of implication as the “paradoxes” of material implication?
We can get at an answer through Kurt Godel’s well-known view that “ [d]espite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves on us as being true” (1964, 271 ).7 Godel’s ground for claiming that we have “something like a perception” of abstract objects is our acceptance of the axioms of set theory. One can extend this further: the perception of set-theoretic objects operates via doing set theory - thinking about the axioms and reasoning to the theorems. Russell has an analogous conception how “discussion” leads to seeing the properties of implication: we discover the properties of implication by examining “the ordinary procedure of deduction” (Whitehead and Russell 1910, 90). In particular, Russell grounds the paradoxical features of material implication in forms of inference that we accept as valid.
We see this in the unpublished lecture “Necessity and Possibility,” in which Russell argues for what I’ll call the “equivalence thesis”: the proposition (expressed by) “p implies q" is equivalent to the proposition (expressed by) “p is not true or q is true.” A part of this argument consists of arguing that the positive “paradox” is “unavoidable” given two forms of inference to whose validity we are committed. These are in fact axioms (8) and (5) of Principles, Section 18:s
(8) (pq z>r)=Up=>(q=> r))
Exportation
Simplification
(5) pq^p
The argument is no more than modus ponens applied to Simplification and the substitution instance of Exportation with p replacing r. The result shows that if a proposition q is true, then any proposition p, whether true or false, implies q. Thus, Russell argues for the positive “paradox” from forms of inference we accept to the consequences of the truth-values of propositions for the implications in which those propositions stand. In “Theory of Implication” Russell gives essentially the same argument and claims that unless we accept the positive “paradox,” we would be forced to reject one or the other of these two intuitively compelling principles of inference (1906,162). So, it’s at least not obvious that the positive “paradox” is a paradox, which is why I keep the scare quotes.
Russell’s view of implication changes between Principles and Prin-cipia Mathematica; in Principia implication is defined by the equivalence thesis:
The meaning to be given to implication in what follows may at first sight appear somewhat artificial; but although there are other legitimate meanings, the one here adopted is very much more convenient for our purposes than any of its rivals. The essential property that we require of implication is this: “What is implied by a true proposition is true”. It is in virtue of this property that implication yields proofs. But this property by no means determines whether anything, and if so what, is implied by a false proposition. What it does determine is that, if p implies q, then it cannot be the case that p is true and q is false, i.e. it must be the case that, either p is false or q is true. The most convenient interpretation of implication is to say, conversely, that if either p is false or q is true, then “p implies q" is to be true. Hence “p implies q" is to be defined to mean: “Either p is false or q is true.”
(1910, 19; emphases mine)
Why this change of mind over the definability of implication? The reason is that in Principia Russell has a different view of what counts as a definition. In order to see this, let’s begin by noting that Russell characterizes Georg Cantor’s definition of the continuum as giving “definiteness to an idea which had previously been more or less vague” (1910, 12). The same holds true of Russell’s definition of implication. It makes definite an idea that had previously been vague. The idea of implication is vague in that although there are propositions that we take definitely to stand in that relation and propositions that we take definitely not to stand in it, there are also a range of “borderline” cases. We make such an idea definite by keeping the definite cases fixed and putting each borderline case either in or out of the extension of the sharpened relation. There are, in general, many different ways of doing this, which is why according to Russell there are “other legitimate meanings of implication.” The definite cases for the vague idea of implication are given by what Russell calls “[t] he essential property that we require of implication,” namely true propositions do not imply false ones. This property is essential to implication because it is “in virtue of this property that implication yields proofs.” To understand this one has to bear in mind Russell’s conception of proof, which consists of establishing truths by inferring them from true premises9 by modus ponens-. if p implies q and p is true, then, provided that true propositions do not imply false ones, q is true.
If this property is the only constraint on a relation for it to “yield proofs,” then any non-empty relation R between propositions having the following feature:
whenever any propositions p and q satisfy the condition:
(*) p is true and q is false
p does not stand in R to q
yields proofs. There are many (extensionally) distinct relations that have this feature, differing on which of the propositions p and q that fail condition (*) count as standing in that relation. Russell in effect picks out from among these relations the one which holds of the most propositions: so long as propositions p and q fail to satisfy (*), p materially implies q.
But why is this definition more “convenient” for Russell’s purposes than others? Russell never says, so here we are in the realm of conjecture. One conjecture is that the definition is “convenient” because it does not require distinguishing among ordered pairs of propositions that fail (*). Another conjecture is that taking negation and disjunction instead of implication as primitive results in simpler definitions and more self-evident axioms for the propositional calculus.
In either case, definition in Principia is an ancestor of Rudolf Carnap’s notion of explication (see 1950,3). Russell’s definition is the precise delineation of an aspect of a pre-theoretical notion for theoretical purposes, where the result of incorporating the sharpened notion in theory is judged by pragmatic considerations.
All this, however, bring us back to the epistemology of implication. How are the logical axioms of Principia's propositional calculus selected? These axioms are supposed to describe which implications hold and so license deductions. But what are Russell’s grounds for thinking that the particular propositional axioms he picks do correctly describe the relation of implication? Since in Principia implication is explicated by material implication, one would expect that Russell’s reason for thinking that, for example, ^-1.3, q => q V p, is an axiom would be this: of any two propositions, q and q v p, either the first is false or the second is true. But if so, why does Russell not set forth these reasons when he presents the logical axioms? We will come back to this question in the following pages.
