Rewritten from an essay unpublished circa 1978.

Entailments, Beliefs and Brain-States
Revisiting the Gettier Infelicities

©2004 Edward G. Rozycki

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Is Brown in Barcelona? Are entailments beliefs?

Is it the case that if I believe X and X entails Y, I must believe Y? The following thought experiment appears to answer this question in the affirmative:

A. John believes that his brother, Sam, is in Venice. We point out to him that Sam's being in Venice entails that Sam not be in New York, since Venice is not New York. Thus, we press on, John must also believe that Sam is not in New York. Upon reflection, John might say that, even though he never entertained such a thought before, he does, in fact believe that Sam is not in New York.

Let us suppose that, if John entertains any proposition Z that we might put to him, he can upon reflection, ascertain whether or not he believes that Z. Does this mean that

B. For any Z, whether or not John entertains Z, John believes that Z or John doesn't believe that Z?

Or, more strongly,

C. For any Z, John believes that Z or John believes that not-Z?

Clearly, C is too strong, since John's not believing that X is not the same as his believing that not-X. ("Believe" is, as they say, an intensional verb, and the negation of it does not carry through to its object.)

Now, does John's believing that X require that he, to put it clumsily, be in a state of belief that X? I am not sure what being in a state of belief that X might mean but it need not entail entertains the thought that X. John can believe that X (by A., above) without entertaining the belief that X. But it is counterintuitive to think that people can be simultaneously in an infinite number of states of belief. The next set of considerations shows the beginnings of this difficulty:

Suppose John believes that 2.23 is greater than 1. Does John also believe that 2,345,167.111 is greater than 1? If so, then he "has" an infinite number of beliefs with respect to the number 2.23, since there is an Aleph-c infinity of the real line (or say, Euclidian plane) of numbers greater than 2. But this is true of any (real) number. So John has a set of beliefs about these numbers sized (Aleph-c x Aleph-c).

Now, if we suppose that to any belief there corresponds a brain-state, a problem arises. All the possible combination of human brain-states, supposing them to be defined most basically by positions -- or combinations of positions -- of atomic particles in the brain, is a very large, yet still finite set. Thus, whatever they may be, states of belief cannot correspond to brain states, given our original supposition, i.e. that if S believes X and X entails Y, that S necessarily believes Y.

One might object to the use of a mathematical example, since mathematical concepts have curious and often counterintuitive characteristics which might infect the arguments they are employed in. But consider

D. John tells us Sam is traveling. Assuming that John is indicating that he believes that Sam is traveling, we deduce that Sam is traveling or he is in New York City. Does if follow that John believes that Sam is traveling or in New York City? Again we ask him. Upon reflection -- since he knows propositional calculus -- he answers yes.

But can this be generalized? Consider the following

E. Sam is traveling entails

A.    Sam is traveling or Sam is in New York or Sam is not in York or Sam is in Tipp City or Sam has a Pekingese or...

B.    Sam is traveling or Sam is in Tipp City or Sam is not in York or Sam is not in Tipp City or Sam has two Pekingese or...

C.    Sam is traveling or...

The continuation dots are meant to indicate any of an indeterminately long set of an indeterminately long concatenation of propositions. Surely, this set is infinite if it includes the natural numbers in it. If we avoid the natural numbers, we still get a infinitely large set since there is no upper bound on the number of propositions that can be concatenated by or.

We are forced to conclude that it cannot be the case that for any X, a person P who believes that X can be said to believe that Y solely on the grounds that X entails Y.

We can understand the situation given in A above as a case of John's constructing a belief when presented with the opportunity. The logic gives him a reason for accepting as a belief something he hadn't necessarily believed before the situation arose. But there are, as we shall see below, other restrictions to be considered.

(The verb know is interesting in this way also. Everyone now living and competent to think reasonably, knows, when presented with the thought, that Julius Cesar's paternal grandmother was born before his or her own.)

The argument of this section applies very directly to Gettier's Guy in Barcelona argument ¹. Smith believes that p, Jones owns a Ford. Smith picks three random place names and constructs three propositions of the form p or q, where q asserts that Brown is in one of the randomly chosen places. Since p implies p or q, Smith accepts that he believes that p or q.

In fact, Gettier's argument fails. The "timeless quality" of logical deduction confuses the issue. What someone believes is not independent of what and when they consider or entertain certain thoughts. (We might, to save Gettier's argument, speak of a person's potential beliefs, but, then, this deflects from the original point of the argument to undermine the conception of knowledge as justified, true belief.)

If the arguments of this section are correct, then what it merely shows is that the axiom of the propositional calculus, p implies p or q, serves better the constructors of the propositional calculus (e.g. it brings truth-functional closure to a possible formula) than it serves our understanding of the relationship between knowledge and belief.

Gettier's Job Seekers -- the ambiguities of who

If John believes that p and John also believes that q, so that we might say John believes that p and q, should John believe that p entails q? Certainly, it is easy enough to show that

E. (p and q) entails (p entails q)

is necessarily true. I will argue that if John's beliefs contain definite references, then using E generates problems.

Gettier's original argument is this:²

Suppose that Smith and Jones have applied for a certain job. And suppose that Smith has strong evidence for the following conjunctive proposition:

(d) Jones is the man who will get the job, and Jones has ten coins in his pocket.

Smith's evidence for (d) might be that the president of the company assured him that Jones in the end be selected, and that he, Smith, had counted the coins in Jone's pocket ten minutes ago. Proposition entails:

(e) the man who will get the job has ten coins in his pocket.

Let us suppose that Smith sees the entailment from (d) to (e), and accepts (e) on the ground of (d), for which he has strong evidence. In this case, Smith is clearly justified in believing that (e) is true.

But should Smith accept (e) on the grounds of (d), if he recognizes that an ambiguity has been introduced into (e) by removal of references to Jones?

What Smith believes can be formulated as

1) x!, i.e., there is a unique x;

2) x = the person, Jones_s Smith is referring to;

3) (x will get the job)AND(x has ten coins in his pocket)

Put together this looks like

F. (x!){(x=Jones_s)AND((x will get the job)AND(x has ten coins in his pocket)}.

Gettier's reformulation plays on the ambiguity of "who" and can be formulated as Whosoever will get the job has ten coins in his pocket, i.e: in its weakest form, There exists an x (not necessarily someone referred to by Smith) such that, if x will get the job, x has ten coins in his pocket, i.e.

G. (the variable x ranges over all humankind) AND (∃x){(x will get the job) -> (x has ten coins in his pocket)}

Smith's beliefs include restrictions on the variable, i.e. its unique reference to Jones, that are not normally expressed in the orginal formulation, E, above. Smith does not believe nor should he accept (e). To insist otherwise is to insist that the structure of our beliefs be a necessarily impoverished particular form of the propositional calculus.

If the arguments of this section are correct, then Gettier's argument must fail to disestablish that justified, true belief is knowledge.

ENDNOTES

1. Edmund L. Gettier, "Is Justified True Belief Knowledge" Analysis 23 (1963): 121-123. I was newly piqued by this argument from its excellent presentation in Theodore Schick, Jr. & Lewis Vaugh Doing Philosophy. An introduction through thought experiments. (Mountain View, CA: Mayfield Publishing, 1999) page 430.

2. In Schick & Vaugh, pp. 429-430 from Gettier, op.cit. 121-122.

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