©2000 Edward G. Rozycki

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edited 8/22/14

Are humans naturally rational, or must they be trained to be so? Or, if humans are naturally somewhat rational, is that rationality necessarily enhanced by teaching them probability calculus and logic? There has been long discussion on whether "naive" or "natural" reasoning processes run the risk of error against rationality as determined by formalized reasoning procedures. This paper will demonstrate some shortcomings in probability theory and logic by examining reasonable practice in particular areas.Teachers, especially teachers of logic, statistics and mathematics tend to believe that teaching logic and probability necessarily helps students to deal with the world more successfully. But is it always the case that being more logical, or statistically subtle helps reasoning? |

"…to be rational is to reason in accordance with principles of reasoning that are based on rules of logic, probability theory and so forth. If the standard picture of reasoning is right, principles of reasoning that are based on such rules are normative principles of reasoning, namely, they are the principles we ought to reason in accordance with."

---- E. Stein, 1996^{1}

Are humans are naturally rational, or must they be trained to be so? Or if humans are naturally somewhat rational, is that rationality enhanced by teaching them probability calculus and logic? There has been long discussion on whether "naive" or "natural" reasoning processes run the risk of error against rationality as determined by formalized reasoning procedures.^{2} This paper will demonstrate some shortcomings in probability theory and logic by examining reasonable practice in particular areas.

**Acculturation to Formal Reasoning **

Teachers, especially teachers of logic, statistics and mathematics tend to believe that teaching logic and probability necessarily helps students to deal with the world more successfully. But is it always the case that being more logical, or statistically subtle helps reasoning?

Some years ago I presented a paper at a conference for mathematics teachers. My topic was "Mathematics education as acculturation." The point I was trying to make is that formal thinking rested on assumptions quite different from those one might pick up from experience with patterns of thought mediated by natural languages.

Perhaps, I suggested, the reason we find a general disparity between those students who do well in language arts and those who do well in mathematics is that the "culture" of mathematics is not "natural," at least not given its present day development, in the way the "cultures" of disciplines based on natural languages are. To improve mathematics learning one must focus on these special assumptions and teach them explicitly, rather than just expecting students to pick them up.

For example, students by first grade know about and play with ambiguity:

"When is a man not a man? When he turns into a store." "I bet I can make you talk like an Indian?" "How?"

What this means is that the famous axiom, a = a, needs to be explicitly stated, perhaps informally, as "In mathematics (formal thinking) words do not change their meaning. Unlike what you, your friends and most other people do, we don't make puns and we don't play tricks."^{3}

Another practice of mathematics educators is to use "obvious" examples and to infer from "obvious responses" a unanimity of internal algorithms:

"Continue the sequence: 1, 4, 9, 25, …"

Even if we all continue "…36, 49, 64, …" it does not follow we all are proceeding by square numbers. If a teacher wants students to learn a particular algorithm, he or she must check to see if the students inferred it from the example. As Wittgenstein comments, "There is no sharp boundary between a mistake made from not knowing a rule and a systematic mistake, i.e. one made by using a rule, but the wrong rule."^{4} *Mutatis mutandis*, one is not justified in assuming that because an answer is correct, it was arrived at by using a specific algorithm.

A third oddity is the use of universal statements, normally not justifiable in any individual's experience, and only shakily rationalized in endless tomes on "inductive logic." The famous syllogism:

- All men are mortal.
- Socrates is a man.
- Therefore, Socrates is mortal.

is strange, viewed "naturalistically." How do we know all men are mortal? Does this mean that if we found a man who seemed impervious to age and wounding, etc. we would not call him a man, rather than saying things like, "This man seems to be immortal?" or "As far as we know, this man is immortal."?

Clearly, the reason we accept universal generalizations -- among other things -- is so we can use formal thinking and get on with what we believe it helps us do. Formal thinking is justifiable on practical as well as aesthetic grounds. So we make assumptions that support the practice. So, teacher, be open about this! We don't need a theorist to tempt sophistry in trying to argue from "in all cases thus far observed, x is an A" to derive "All x's are A's".

My final examples dealt with the notions of a fair coin and fair dice. These are not items experienced, but mathematical constructs used to justify, circularly, a supposedly simpler concept of probability, often deliberately confounded with the natural language concept of "likelihood."^{5}

My audience, all mathematics teachers, objected. What could be more normal, more natural than mathematics? They themselves all could attest to the ease with which it could be acquired. I pointed out that they clearly had to consider their position to be biased, especially since it allowed them -- as carriers of knowledge highly valued in society -- to see themselves as superior intellectually to those who fumbled around with ill-defined concepts, thus disposed them, the math teachers, to rejecting evidence from anyone outside the field as hopelessly muddled.

At the time, I would probably have opted for the theory that probability theory and logic are normative for reason. Now, however, I believe otherwise.

Have you heard about Linda? Suppose you haven't. I will ask you, "Which statement is more probable?

a. Linda is a bank teller.

b. Linda is a bank teller and is active in the feminist movement.

This is a strange question. Why is it posed: "Which statement is more probable?" rather than "Which situation is more likely?" Suppose Linda is a cat. Both statements are equally unlikely, since no one makes cats bank tellers. If Linda is a real person and alive, then since we know nothing about Linda, we can't say which situation is more likely. Maybe where she lives there are no banks, etc.

You object. Obviously, -- "obviously": that famous word of math teachers -- this is meant be a problem in the analysis of probabilities. Statement a. postulates Linda to be a member of the class of bank tellers. Statement b. postulates Linda to be a member of the class B of bank tellers *and also* of the class F of people active in the feminist movement. Invoking a theorem of the probability calculus that tells us that

for any class X, we conclude easily that:

p(a) is greater than (or equal to) p(b).

* It is more probable that Linda is a bank teller than that
she is both a bank teller and active in the feminist movement.*

Lest we allow people to be overly impressed by anyone who can wave the probability calculus at us, we observe at this point that probability functions are defined on probabilizable spaces and it is a major -- **and likely false** -- assumption that arbitrary assemblages of English predicates define probability spaces.^{6} There is no justification here for the assumption that the probability calculus is applicable; and, certainly, no grounds for dismissing as lethally muddled earlier attempts at understanding the question about Linda.

To reject the applicability of the probability calculus is a radical move, but, I think, often justifiable. But we can reinterpret conditions a and b above in a way that saves application of the probability calculus and shows why the probability of **a** need not be higher that the probability of b. Indeed, it may be even less -- as when, in particular circumstances, bank tellers are a subset of all feminists.

The two conditions are:

a. Linda is a bank teller.

b. Linda is a bank teller and is active in the feminist movement.

But because b. implies a contrast not explicitly articulated by a, we can expand a. and get a better sense of comparison:

a. Linda is a bank teller and nothing more, i.e. not a feminist, nor a Presbyterian, nor a Phillies fan, nor a gourmand, nor a runner, nor a …b. Linda is a bank teller and is active in the feminist movement.

But the p(a) can now be interpreted as a more highly restricted

(where Z' is read "not-Z")

which is very much more likely to be less than

so that:

p(b) is greater than (or equal to) p(a).

*It is more likely (probable)
that Linda is both a bank teller and active in the feminist movement
than that she is a bank teller and nothing else.*

This -- though speculative -- explains two things: first, our sense that the original comparison was somehow strange; and second, why formally naïve persons believe b to be more probable than a.

I would like to reiterate an earlier point, because it bears on the question as to whether probability theory should be seen as normative for rationality. Probability functions are defined on probabilizable spaces and it is a major -- and likely false -- assumption that arbitrary assemblages of English predicates define probability spaces. The reason people might make "mistakes", is because they "realize" -- in some sense -- that the conditions for applying probability theory -- a discrete set of contrastive, partitionable classes -- do not obtain in the situation being considered. Let us go on. ^{7}

**Linda: Error of Representativeness**

Some of you may have recognized Linda. Here is some background information on her: .

Linda is thirty-one years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in antinuclear demonstrations.

Now, which of the following do you believe is more probable:

a. Linda is a bank teller; or

b. Linda is a bank teller and is active in the feminist movement.

Kahnemann and Tversky in "Judgments of and by representativeness" report that people are inclined, in what Kahnemann and Tversky claim to be violation of theorems of the probability calculus, to pick b, given the background in the paragraph about Linda.^{8}

But as we have seen above, the selection of b as more probable than a. can be seen in either of two lights:

a. not irrational since the probability calculus does not clearly apply to this situation; or

b. in accord with the probability calculus if we make explicit some of the background assumptions which permit the problem to be interpreted.

In general then, apparent violations of the probability calculus via representativeness are only apparent. Far from being normative of rationality, application of the probability calculus requires rational justification.

The universal principles of logic, arithmetic and probability calculus … tell us what we should … think, not what we in fact think. … If our intuition does in fact lead us to results incompatible with logic, we conclude that our intuition is at fault.

-- Piattelli-Palmerini^{9}

It is a not uncommon inclination among logicians to imagine that idealized languages are superior to natural ones. For example Fagin, *et al.*^{10} in their text, *Reasoning About Knowledge*, state quite cavalierly that English is not as good a language to discuss knowledge in as modal logic is.

Let us consider an example where entailments lead to inconsistencies in practical judgment. Suppose p entails q. Is the following principle true?

Gettier's Principle of Justified Entailed Belief:^{11}

If I am justified in believing that p, and if I am aware that p entails q, then I am justified in believing that q?

More generally, are the following statements true, given that p entails q?:

I know that p, therefore I know that q.

I believe that p, therefore I believe that q

It will be argued below that the answer is "sometimes not." The method will be to show that to assume that entailments always work to enable further claims of knowledge or belief leads to severe inconsistencies.

**ALT vs OPT: Excrescence vs. Viability **

Consider there are two senses of the word "or":

- the first I will designate with ALT: it is the truth-functional conjunction defined in the propositional calculus by a truth table that yields T, only if either p or q (or both) in (pALT q) is T.
- the second sense of "or" I will designate with OPT. It is used in informal discourse in the following sentence to identify viable options. (This has logical properties quite different from ALT):

Let us consider the following example:

a. The rightmost digit of any (or a given) whole number is either 0 OPT 1 OPT 2 OPT 3 OPT 4 OPT 5 OPT 6 OPT 7 OPT 8 OPT 9.

For both ALT and OPT the following is the case: from the fact that I know (or justifiably believe -- too weak here) that **a**. is the case, it does not follow that I know (or justifiably believe that)

b. The rightmost digit of any (or the given) whole number is 2.

However, for OPT but not for ALT, we can make the stronger claim that if I know or justifiably believe pOPTq. then **I do not know** (or justifiably believe)

Suppose I know that John is home in his two room efficiency apartment consisting of a bedroom and an general purpose room, then I am justified in believing that he is in the bedroom or (OPT) the general purpose room, but not justified in believing he is in the bedroom nor justified in believing he is in the general purpose room

In general,

"know (believe) pOPTq" entails "not know (believe) p" and "not know (believe) q"

Using The Principle of Entailed Belief, and **conflating** ALT with OPT we can argue as follows:

c. John justifiably believes that p and is aware that p entails q.

Let us make q = p or s, since, by the propositional calculus, p entails (pALTs). Then

d. John justifiably believes (p or s).

Now interpret "or" as OPT and it follows that

e. John does not justifiably believe that p.

Then, by transitivity of entailments,

f. If John justifiably believes that p, then John does not justifiably believe that p.

Thus, the Principle of Entailed Belief leads to a contradiction, so long as the distinction between ALT and OPT is not observed. But pOPTq is not entailed by p. So we forsake the entailment from p via ALT, or we lie. ^{12}

It is easy to construct parallel arguments for knowing that (or just believing that) p, and p entails q.

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. ^{13}

** ALT, OPT and Epistemological Status**

In everyday speech, "or" is used most likely as OPT, rather than ALT. We are normally not only concerned with the content of our assertions, but also with their epistemological status, e.g. are the knowledge claims, belief claims, surmises, assumptions, etc. Using ALT and its associated entailments, wreaks havoc on these distinctions. For example, if you ask me where John is and I, knowing he is at school, answer, "John is either at school or at the supermarket", I am misleading you. Actually, I am lying. (Unless the Principle of Entailed Belief is correct.) Even though

p = John is in school

entails

pALTq, where q = John is at the supermarket.,

our normal interpretation of "or" as OPT, does not permit us the equivocation. Bald assertions are normally interpreted as knowledge claims, and knowledge claims or weaker assertions that begin with "I believe that...," "I assume that..." permit normally only OPT as the interpretation of "or."

To appreciate how important this is, imagine you had seen John at the scene of a crime, in school, at a particular time. When asked under oath by a court officer whether you knew where John was at that particular time, you replied that you knew that John was either in school or at the supermarket. Unless the court officers themselves were aficionados of the Principle of Entailed Belief, you might find yourself faced with a perjury charge, should you, under cross-examination admit you had seen him at that time in school.

We currently have no clear idea about how to translate some of the most fundamental statistical considerations into guides for reasoning in everyday life.

--- Nisbett,et. al.^{14}

By the arguments given previously it would seem to follow that probability theory and logic, even if useful in sophisticating certain considerations, cannot be seen as normative of rationality. Probability theory fails because the conditions of its application are far from clear, often violated, and must be justified. Logic fails because its need for set-theoretic closure for the truth-functional calculus requires it to define (*p* implies *p or q*) as true. This produces the anomalies with ALT versus OPT shown above.

To reiterate a consideration introduced early above: perhaps the teaching of probability calculus and logic serves not so much the enhancement of rationality as it serves the practice of the probability calculus and logic, and socially and academically more importantly, the status of their practitioners.

**REFERENCES**

**1**. E. Stein, *Without Good Reason* (Cambridge, MA: MIT Press, 1996) p. 4

**2**. An interesting review of this dispute is given in Richard Samuels, Stephen Stich and Michael Bishop, "Ending the Rationality Wars: How to Make Disputes About Human Rationality Disappear" Available on line.

**3**. With occasional exception as when one demonstrates that normal people have eleven fingers by counting down from finger ten to finger six and then adding the six to the remaining five: 6 + 5 = 11.

**4**. My translation from paragraph 143 *Philosophische Untersuchungen* (Augsburg: Surhrkamp, 1967) p.77.

**5**. This, in turn, generates inconclusive argument over probability as a frequency measure versus a measure of potential; also, over whether unique events can have a probability, etc.

**6**. See Edward G. Rozycki, "The Functional Analysis of Behavior: theoretical and ethical limits." Online.

**7**. See Richard E. Nisbett, David H. Krantz, Christopher Jepson and Geoffrey T. Fong, "Improving Inductive Inference" in Daniel Kahnemann, Paul Slovic and. Amos Tversky, *Judgment under uncertainty: Heuristics and biases* (New York: Cambridge, 1982) pp. 445 - 459.

**8**. Amos Tversky and Daniel Kahnemann, "Judgments of and by representativeness" p. 92 in Daniel Kahneman, Paul Slovic, Amos Tversky (eds) Judgment under uncertainty: Heuristics and biases. (Cambridge: Cambridge U. Press, 1982)

For more on this topic, see Edward G. Rozycki, "Is the Notion of Conjunctive Fallacy Based on Fallacy? " (Online)

**9**. M Piatelli-Palmerini, *Inevitable Illusions: How Mistakes of Reason Rule Our Minds.* (New York: Wiley & Sons, 1994) p., 158.)

**10**. Ronald Fagin, Joseph Y. Halpern, Yoram Moses, Moshe Y. Vardi, *Reasoning About Knowledge* (Cambridge, MA: MIT Press, 1995)

**11**. Cf. Gettier, Edmund L., "Is Justified True Belief Knowledge?" *Analysis 23* (1963): 121 - 123

**12**. Fagin, et al define possibility as the duality of knowledge "A takes x to be possible" <=> "A does not know that not-p". Conflating OPT and ALT for OR generates the interesting result with this definition that {(possibly p OR possibly q) is equivalent to ((not-knowing that possibly p) OR (not-knowing that possibly q))}.

**13**. The argument of this section applies very directly to Gettier's Guy in Barcelona (2) argument. 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. Smith ought not accept it.

For more on this, see Edward G. Rozycki, "Entailments, Beliefs & Brain-States" (Online)

**14**. Nisbett, Krantz, Jepson and Fong, p.446.