https://www2.lawrence.edu/fast/ryckmant/Russell's%20Theory%20of%20Descriptions.htm
on three of Bertrand Russell's puzzles for his Theory of Descriptions.
Russell observed that a sentence like,
(1) George IV wished to know whether Scott is the author of 'Waverley'
might well be true, even though
(2) George IV wished to know whether Scott is Scott
was no doubt false while, in fact,
(3) Scott was the (initially anonymous) author of Waverley
was true.
But if (1) is true, then it would appear that George IV was uncertain about the truth of (3) and if (2) were true, it would appear that George IV was uncertain about the truth of
(4) Scott is Scott
As Kurt Goedel observed [1],
The problem is: what do the so-called descriptive phrases (i.e., phrases as, e.g., "the author of Waverley " or "the king of England") denote or signify and what is the meaning of sentences in which they occur? The apparently obvious answer that, e.g., "the author of Waverley" signifies Walter Scott, leads to unexpected difficulties. For, if we admit the further apparently obvious axiom, that the signification of a composite expression, containing constituents which have themselves a signification, depends only on the signification of these constituents (not on the manner in which signification is expressed), then it follows that the sentence "Scott is the author of Waverley signifies the same thing as "Scott is Scott."The ambiguities or apparent ambiguities of the English language in fact helped drive Russell's goal of formulating a more precise symbolical language that turned out to include the "exactly one" quantifier.
Russell pointed out [2] that his 1903 Theory of Descriptions is summarized by two definitions, written in logical symbolism, in Principia Mathematica (PM), which began publication in 1910. There the logician uses the existential quantifier and its "scope" (that part of the assertion to which the quantifier applies), along with the "exactly one" quantifier.
Let us put the problem thus:
George IV wished to know whether x = y.
But certainly the king did not wish to know whether y = y.
And yet x = y if and only if y = y.
So doesn't this suggest that His Highness actually did desire to know whether (x = y) <--> (y = y)?
I.e., George IV wished to know whether x = y, but not whether y = y -- even though (x = y) --> (y = y).
When Russell posed this question in 1903, he and Whitehead had yet to launch the revolution in logic wrought by Principia Mathematica. But by raising such questions in his early Theory of Descriptions and in PM, Russell sparked logicians to try to be more precise by what is intended by such words as identity, equality, equivalence and tautology. Various definitions and axioms have been used since.
Tarski [2a] gives Leibniz credit for this definition of equality or identity: x = y if and only if every property of x is held by y and every property of y is held by x.
Yet how does this differ from an equivalence, whereby x <--> y? In many cases, according to our position, equality and equivalence are indistinguishable.
For many writers, it is usual to apply the equivalence relation to constructs variously known as formulas, sentences, statements, propositions, predicates and sentential functions. These are the assertions in which a verb is implicit or explicit. For example, P may be the assertion 'x = y.' So we might write Ax Ey (x = y). So that assertion means "there is some y that is equal to any x."
Here, x and y are "bound" variables (i.e., they relate to the quantifiers).
Now though one ordinarily applies the equivalence relation to two assertions, P and Q, it is not customary to apply equivalence either to the variables of P(x,y) or of the terms of P(x,y), as in, say, P(x,t), where t is an instantiation of y (sometimes referred to as a "constant").
So then 'x = y' means that x and y are to be construed as interchangeable. That is, iff an instantiation of x is t, then an instantiation of y is t.
But, suppose x and y are variables representing assertions? In that case, we could properly write x <--> y.
Yet Russell's descriptions are verb-less "subjects" or "objects." How can descriptions then be construed as subject to equivalency? In the sense that if x is a description, then it is assumed that either 'x exists' or 'x doesn't exist.'
So the variables x and y of P(x,y) that are ordinarily said to be subject to the identity relation '=', are in our view first subject to the equivalence relation '<-->' -- where identity implies equivalence, but not the converse.
Note that all true assertions of an axiom-based theory are equivalent, in the context of the overall theory. For most, n/(n+1) <--> (n+1)/(n+2) is intuitively apparent. But '1 <--> pi' is not at all obvious. Another example: once we have established the set of natural numbers N -- perhaps by the von Neumann method -- we have that 1 <--> 2 and, further, 7 <--> 181.
So if we say description x <--> description y, we do not necessarily mean that they are identical.
Another example of non-identical equivalence: lim n-->inf. n/(n+1) = 1; lim n-->inf. (n+1)/n = 1. f(n) <--> f(n+1) but f(n) = f(n+1) only in the limit. Of course, we haven't bothered to offer a formal proof that f(n) <--> f(n+1), but if desired, the reader may do so, using some adequate number theory.
That is, we see that at any finite n, f(n) implies f(n+1) if and only if f(n+1) implies f(n) even though f(n) =/= f(n+1) except in the limit.
The Tarski/Leibniz identity definition fits nicely here: all properties held by f(n)'s limit point are held by f(n+1)'s limit point, and the converse. Yet, the non-identical curves approaching this point are equivalent in the sense that each intersects the other at infinity.
So, speaking loosely for a moment, King George had the "curve" x and the "curve" y and wished to know whether they intersected, thus establishing identity. He was uninterested in whether f(y) = y, as this fixed point provided no new information.
Well, how is the king to know whether "curves" x and y intersect? How is he to know whether x = y? Suppose he knows that Scott cannot be ruled out as the author of Waverley. In that case, Waverley implies a set of which Scott is a member and Scott is a member of a set that implies Waverley. So the asserted existence of the novel W strictly implies the asserted existence of the set S. Even so, the set S does not equal the novel W.
But the king in this case in fact wishes to know whether the author of Waverley and Scott pose an identity, which is here indistinguishable from an equivalence. That is, the king is curious as to whether it is so that the anonymous writer has all Scott's properties if and only if Scott has all the anonymous writer's properties.
This simple point brings us forthwith to the notion of deduction, as used by mathematicians and logicians. Rigorous deduction ordinarily requires a set of axioms (some systems include "axiom schema" which are tailored to questions of infinity). That is, we are to use a chain of modus ponens, or implication arrows, or nand connectives (they're all equivalent) to establish the truth value of a claim. This process is cited in the Deduction Theorem, which the reader can find online or in a number of logic textbooks.
The Deduction Theorem is required by our limited minds. If one had a large enough mind, one would not need to prove a claim holds by running through a chain of modus ponens. One could see the chain of reasoning and result all at once. An analogy can be drawn from the case of those people who can read a sentence in one glance as opposed to the majority who read word by word. And we also face the caveat of Goedel's proof that PM and related systems contain assertions that show that the Deduction Theorem can be put into a non-linear loop.
Well, we have discussed identity and equivalence. What about tautology? (Russell's protege, Wittgenstein, was among the first to use truth tables, as well as adopting the word tautology for logical redundancy, though it had long been used to imply rhetorical redundancy.z1)
A tautology describes the situation in which there are no false possibilities for a proposition. All truth table possibilities hold.
For example, the formula P --> (Q --> P) is always true. We simply rewrite this as
~P v (~Q v P), which is,
~P v ~Q v P = (P v ~P) v ~Q
Without resort to a truth table, we can see easily that, no matter what value P or Q has, the formula above always holds.
And so King George's mind was not so large that he could at a glance determine whether x = y is a tautology. Yet it certainly was large enough to determine that y = y is a tautology -- although we can in fact prove that y = y is tautological. That is, "=" implies "<-->" (but not the converse), and so we have y = y implies y <--> y. So y --> y & y --> y. Hence, (~y v y) & (~y v y), which holds for all truth table possibilities. (We have not reduced this last to ~y v y because that would introduce circularity into the proof.)
Of course, our proof relies on the rules of transformation of logical connectives. But a similar proof could have been supplied using either modus ponens arrows only or nand connectives only. This would seem like folderol to His Majesty, but it helps us see more clearly what is at issue in Russell's Waverley problem.
We have belabored the obvious, you say. Yes, but in doing so, we have made more precise our notions of identity, equality, equivalence and tautology. In addition, we have argued for the use of the equivalence relation for the variables of assertions, though then of course we wander into the issue of exactly what constitutes "first order predicate calculus" versus "second order predicate calculus." This last is left for further philosophical efforts.
Another take on this issue is the use of ordered pairs. Consider the set of people who may be writers, as in x e X. As each set is defined by its elements, each element differs from the next and so must have some specific property. If X were singleton, the king wouldn't be curious -- whatever the specific property. If not, then the king is aware that there is some element x that has the property "Scott" and another element y e Y with the property "sole author of Waverly."
So we have a set of ordered pairs, as in X X Y, with elements < x, k >, where k is the constant denoted as the "author of Waverly."
Thus the king's puzzlement is represented by the fact that the set X is non-singleton. He is asking which instantiation of x pairs with k, or which element of X X Y can be expressed as a true assertion, as in f(x,k).
Also see Morris Weitz [3] for a useful discussion of negations and the clarification of the Waverley puzzle and G.E. Moore [4] for a diligent, but largely outdated, overview.
1. "Reply to Criticisms" by K. Goedel in The Philosophy of Bertrand Russell (The Library of Living Philosophers 1946), Paul Arthur Schilpp ed.
2. "Russell's Mathematical Logic" by B. Russell in The Philosophy of Bertrand Russell (The Library of Living Philosophers 1946), Paul Arthur Schilpp ed.
2a. Introduction to Logic and to the Methodology of the Deductive Sciences by Alfred Tarski (Dover 1995 reprint of Oxford second edition 1946).
3. "Analysis and the Unity of Russell's Philosophy" by M. Weitz in The Philosophy of Bertrand Russell (The Library of Living Philosophers 1946), Paul Arthur Schilpp ed.
4. "Russell's 'Theory of Descriptions' " by G.E. Moore in The Philosophy of Bertrand Russell (The Library of Living Philosophers 1946), Paul Arthur Schilpp ed.
z1. A footnote in Elements of Symbolic Logic by Hans Reichenbach (Macmillan 1947) tells us:
Truth tables were used by L. Wittgenstein, Tractatus Logico-Philosophicus Harcour, Brace, New York, 1922, p. 93, and by E.L. Post, Amer. Journal of Math., XLIII, 1921, p. 163. Materially, the definition of propositional operations in terms of truth or falsehood was used earlier, for instance, in B. Russell and A.N. Whitehead, Principia Mathematica, Vol. I, 1910, p. 6-8. Furthermore, C.S. Pierce employed this definition...
No comments:
Post a Comment