Speranza

There is a venerable philosophical tradition which describes human beings
as rational creatures.

Famously, Aristotle said:

i. Man is a rational animal.

It may be added that it is so unless we are very tired or angry or drunk,
but generally we agree with that idea.

However, during the last 40 years, various psychologists have suggested the existence of much deeper problems
than the above-mentioned exceptions.

They pointed out that, given certain
circumstances, people systematically fail to apply simple rules of classical logical inference.

At this point, we could legitimately ask ourselves a very
basic question.

Let’s call it The Question: “Are people irrational?”.
2

We shall present where the psychologists’
worries about human rationality originated from, namely the Wason
Selection Task or ‘The Task’, as we shall call it.

(Wason went to Oxford, but left -- Grice never did -- "Oxford made me").

Actually, we will see that,
according to “traditional”cognitive psychology, Wason's Selection Task provides sufficient
grounds to give a positive answer to The Question.

But we will take a broader perspective and will show that, in answering The
Question, the interpretation of a certain type of conditional sentence, that
is indicative conditionals, is crucially relevant.

Firstly, we must obviously take a brief
look at a pragmatic defense of rationality through Wason’s notion of
confirmation bias and Grice’s theory of conversational implicature.

Grice had coined 'implicature' in Oxford BEFORE Wason experimented in London. Grice's theory (and his attack of Strawson on 'if' came from a few years later, when he was invited to deliver the bi-annual William James lectures at Harvard. The fourth lecture is dedicated to "Indicative Conditionals").

We should introduce Adams’ probabilistic view of indicative
conditionals and will give reasons for preferring his account to those
aforementioned.

It is then we can evaluate The Question in the light of
the new standpoint acquired.

In particular, all interpretations that we will
be mentioning favour a negative answer to The Question, but Adams’, in
virtue of its higher generality, seems to grasp the very essence of the
problem.

Conditional sentences are traditionally classified in two main groups: indicative and
subjunctive conditionals or counterfactuals.

J. L. Mackie ("Oxford made me") denies this, and Grice seems to agree with Mackie ("Do not multiply the senses of 'if' beyond necessity").

An example of the first kind is

ii. If Brutus didn’t
kill Caesar, then someone else did.

The corresponding counterfactual would be

iii. If Brutus
hadn’t kill Caesar, then someone else would have.

In general, indicative conditionals are of
the form did/did, while counterfactuals has the had/would have form.

Most importantly,
we would tend to accept the former example, but not the latter.

In fact, we would say
that

iii. If Brutus hadn’t kill Caesar, then someone else would have.

-- is false, because, contrary
to what the counterfactual seems to express, there is no necessity in the fact that Caesar
was murdered.

Once a friend of Grice's came to him and said “I want to challenge your
reasoning skills”and produced four cards.

Each card had a number on one
side and a letter on the other.

Two cards were showing their number-side
and the other two their letter-side.

The cards were showing something like
the following string: A, K, 2, 7.

The Task was to determine which card/s
should one turn in order to verify the following conditional sentence or
‘rule’:

iv. If there is vowel on one side of the card, there is an even
number on the other.

The Task, which was originally developed by the Peter Cathcart Wason (sometime Oxford) was meant to show that the majority of people do not reason
consistently with the principles of classical logic, therefore the
psychologists’ worries.

But let us be more precise, starting with spelling out
how The Task actually works.

As a start, we can formalise the rule in the
following straightforward way:

v. p⊃q

where p stands for the 'if' utterance's antecedent:

vi. There is a vowel on
one side.

and q stands for the 'if utteranc's consequent:

vii. There is an even number on the other.

‘⊃’
represents the material implication of classical logic -- vide Grice's famous second William James Lecture. He had lectured on other operators earlier on, notably negation. It was all provoked by his tuttee's "Introduction to Logical Theory" that gave him the idea to deal with all the 'formal devices' and their vernacular counterparts and explain the alleged divergence identified by Strawson as 'implicatural' in nature and thus totally cancellable.

In this way, the string
p, ¬p, q, ¬q (where ¬ means “not”) would represent the previous
arrangement of the cards.

Let’s now take a look at a typical distribution of
results regarding The Task.

It’s interesting to note
that, after many years and innumerable iterations of the experiment, the
distributions is still found to be the same.

Now, anyone who has some acquaintance with propositional logic -- or Myro's System Ghp ("conceived in gratitude to H. P. Grice) should be
quite startled by the data.

In fact, the results show that only a small
percentage of the participants choose the right cards, namely p and ¬q.

At
this point, it should be pointed out that, in the last few decades the study
of human reasoning had been dominated by roughly two tendencies.

On the
one hand, some researchers (such as Wason himself) considered classical
logic as the ultimate criterion for rationality, possibly out of ignorance of
111
Card/s selected # Persons (%)
p, q 46
p 33
p, q, not-q 7
p, not-q 4

Other combinations 10

A typical outcome regarding the Wason selection task (Wason and Johnson-Laird, 1972)
the alternatives.

On the other, some others (such as Herbert Simon and
Gerd Gigerenzer) were too quick in abandoning logic altogether.

Having
said this, it seems clear why cognitive psychology tended to interpret the
previous data as undermining the idea that humans are rational.

Nonetheless, the psychologists’ reaction is justified only in case "if" is interpreted correctly as the classical Material Implication.

Furthermore, classical logic cannot capture the notion of uncertainty, which
is an essential feature of reasoning.

Two questions may then arise.

Does
classical logic yield the best criterion of validity of inferential reasoning?

What if we interpret "if" differently?

We should deal
with the latter question.

In particular, we should explore the idea that
different interpretations may be able to give a better account of The Task
outcome and, consequently, cast some light on the notion of rationality.

However, let us conclude our analysis of The Task first.

We said that the correct answer to The Task was choosing p and ¬q.

Why?

The key is given by the classical rule of inference called Contraposition,
that is

p⊃q ∴ ¬q⊃¬p

and the corresponding axiom

(p⊃q) ≡ (¬q⊃¬p).

Contraposition tells us that any argument with

p⊃q

as premise and

¬q⊃¬p

as conclusion is valid.

Now, one way to make clear how this is relevant for
solving The Task is saying that Material Implication puts a ‘condition’ on
its antecedent.

This means that, when we consider

p⊃q

we should pick the
card

p

since that’s the relevant one.

On the other hand, if we consider the
logically equivalent

¬q⊃¬p

the antecedent is

¬q

Therefore, we should
choose the corresponding card.

Doing so, we have enough information on
both components of the rule.

This reasoning guarantees that, on our
particular formulation of The Task, the correct answer consists in choosing
A and 7.

Another way to put this would be to look at another equivalence.

In fact, we know that the material conditional

p⊃q

is also equivalent to

¬(p∧¬q)

This means that the conditional is false when both

p

and

¬q

are
the case.

Consequently, we should choose A and 7 to check for the falsity of
the rule.

So far, so good.

It should now be clear how we should reason in
order to give the right answer.

However, we still have not explained of why
the majority of people choose also irrelevant cards, notably

q.

The next two
sections will be partly concerned with this problem.

A first attempt to explain why people generally perform The Task poorly
came from Wason himself.

Its basic idea is what in psychology and
cognitive science is called Confirmation Bias.

Roughly, this is a tendency to
interpret new information in order to confirm one’s preconceptions, avoiding
information and interpretations which contradict prior beliefs.

Wason was
among the first to investigate this phenomenon.

His experiment consisted
in presenting his subjects the following numerical series: 2, 4, 6, and telling
them that the triple conforms to a particular rule.

The participants were
then asked to discover the rule by generating their own triples and use the
feedback they received from the experimenter.

Every time the subject
generated a triple, the experimenter would indicate whether the triple
conformed to the rule.

Finally, once they were sure of the correctness of
their hypothesized rule, the participants were asked to announce the rule.

The outcome of the experiment showed that, in spite of the fact that the
rule was simply “Any ascending sequence”, the subjects often announced
rules that were far more complex than the correct one.

More interestingly,
the subjects seemed to test only “positive” examples, that is triples that
subjects believed would conform to their rule and confirm their hypothesis.

The point is that they did not attempt to falsify their hypotheses by testing
triples that they believed would not conform to their rule.

Wason referred
to this phenomenon, being a Popperian --and a Londoner Popperian at that --as "confirmation bias".

Subjects systematically seek only
evidence that confirms their hypotheses.

In respect to The Task, this seems
to imply that people would generally choose only those cards which could
confirm the given conditional rather than refute it.

However, it seems that
this account does not fit with the empirical results.

In fact, if "confirmation
bias" was the whole story, one would expect people to choose

¬p

along
with the other “confirmatory” card, that is

q

whereas they choose

p

which
is not “confirmatory”.

Hence, the "confirmation bias" seems insufficient to explain why people deviate from classical logic.

It has been suggested that
this may depend on whether we interpret the keyword "confirmation" in a strict
logical sense, as we did in our previous analysis, or in a broader
psychological sense.

In fact, confirmation bias has been advocated by
several researchers in order to explain many heterogeneous phenomena,
from The Task to people’s belief in pseudo-scientific ideas, from
governmental self-justification to medical diagnosis, from jury-deliberation
processes to conservativism among scientists and even from witch-hunting
to depression (See Nickerson, 1998, on distinguishing different notions of
confirmation and for a very good survey of the interpretations of
confirmation bias).

If, on the one hand, this shows the pervasiveness of the
notion of "confirmation bias", on the other, investigators are left with
further important questions.

Is the "confirmation bias" a consequence of
specific cognitive activities?

Are those activities subject to any constraint?

Does it persist because it has some functional value?

Does it reflect a lack
of understanding of logic?

s it more important to be able to make valid
inferences from positive than from negative premises?

Finally, the foregoing
should be enough to allow us to maintain that the role of confirmation bias
in the understanding of The Task is neither a central nor a privileged one,
because of the presence of other basic issues regarding, for instance, logic
and cognition.

Let us now turn to another point of view on the matter
coming from the philosophy of language.

Herber Paul Grice’s famous account of conversational implicature (predated by his Oxford lectures on Logic and Conversation, where 'implicature' simpliciter is used -- and cf. Sidonius for 'inplicatura' [sic] offers another possible way to
interpret people’s general performance of The Task.

Firstly,
we need to introduce, yet again, the concept of conversational implicature.

Meet Conversational Implicature.

The
underlying idea is that what is literally said is not what is actually meant.

For instance, if the teacher’s reply to the question

“Is she a good student?”

were

"She always attends classes."

we would have some reason to believe
that he implied she is NOT an outstanding student.

Grice specifies this
situation by means of some conversational maxims and a general principle
of cooperation.

Roughly, according to the maxims, one’s contribution to the
conversation should be adequately informative, relevant, not believed to be
false by its utterer and generally unambiguous and brief.

The principle
states that participants expect that each of them will make a conversational
contribution such as is required, at the stage at which it occurs, by the
accepted purpose of the talk exchange.

For instance, when a speaker makes
an apparently uninformative remark such as “War is war”, the addressee
assumes that the speaker is being cooperative and looks for the implicature
114
the speaker is making.

We can see how the Gricean implicature is
non-conventional, for it is drawn in accordance with pragmatic principles
only, rather than involving the meaning of a linguistic expression. Finally,
on this view, truth and assertibility crucially part ways.

According to
Grice, in fact, a proposition can be true, without being assertible.

If
spaghetti grow on trees, the pope is a German.

-- is (vacuously) true, but
not assertible, because the antecedent is false and the utterer will mislead his addressee in uttering the corresponding sentence.

This would contradict one
of the maxims.

Consequently, our intuition that that proposition is false is
due, not to its falsity, but to its lack of assertibility.

However, we could
easily think about someone who would assert the sentence of the previous
example and genuinely believe that the antecedent was true, without
violating Grice’s maxim.

Be that as it may, since Grice’s theory of
implicatures has been highly influential among both philosophers and
psychologists interested in the study of natural language, it is beyond the
scope of this essay to attempt any criticism of his ideas worth the name.

Instead, we will just focus on how the foregoing is relevant to our particular
discussion.

So, how would a Griceian account of The Task look?

Briefly, one
might say that when The Task is asserted, its implicatures affect people’s
reply to it.

In particular, people seem to understand the "if" of
the conditional sentence to mean "if and only if"

In other words,
that sentence might not come across as a material implication, but as a
biconditional.

On this account, in order to satisfy both the conditions put
on the sentence by each “direction” of the biconditional, one would expect
the majority of people to choose all the four cards.

Nonetheless, that is
contrary to what the data highlight, namely that the majority goes for

p

and

q

only.

As a result, it seems that also Grice fails to provide a suitable
justification of people’s response to the problem.

At this stage, before moving to the core of the essay, we should draw
attention to the fact that since the literature that sprang from the Wason
selection task is of Gargantuan proportions and it is still growing, our
exposition is by no means an attempt to give an even remotely
comprehensive survey of it.

However, it is worth just mentioning a few of
different approaches that gained a lot of attention in more recent years.

Historically, this had been the case.

When the BBC broadcasted a short documentary
about the crop of spaghetti, hundreds of people contacted the production to ask where
they could buy their own spaghetti trees.

Needless to say, the show was a hoax and was
transmitted on April fools’ day in 1957.

Even if our analysis will not be directly concerned with any of them, this
brief synopsis will be useful later on in order to emphasize the novelty of
Adams’ ideas.

Let us begin our list with some attempts (surveyed by Grice himself in his "Retrospective Epilogue" to WoW) to
develop Grice’s conversational implicature.

Roughly, this means that the inferential component of communication rests
very largely on the ability to work out what is and is not 'related' (Grice relies on Kant's category of Relation, which he (Grice) turns onto a conversational category, as he calls it) in terms
of contextual effect, in what people are saying to you (Cara,
Girotto, 1995, for an application of this idea to the Wason selection task).

Another very influential proposal was given in the framework of
evolutionary psychology.

In particular, Leda Cosmides claimed that people
have not evolved in such a way that would allow them to perform The Task
successfully.

This idea seems to find strong support in experiments
conducted with different versions of The Task involving “social exchange”
scenarios.

In these cases, people’s success in performing the tests highly
increases (Cosmides, 1989).

One more interesting project has been pursued
by Oaksford and Chater.

They have argued in various papers for an
analysis of The Task based on the theory of optimal data selection in
Bayesian statistics (Oaksford and Chater, 1994, for instance).

By applying
such standard, they try to justify the claim that the most frequent card
selections are also the rational ones.

However, so much for the alternatives.

It is now time to start spelling out our suggestion by means of an analogy
between psychology and the philosophy of language.

From what has been
said in this section, it seem that both an English (and Oxford-educated) psychologist, Peter Cathcart Wason, and an English Oxford philosopher, Herbert Paul Grice, invoked pragmatic reasons, in order to clarify people’s
behaviour towards (indicative) conditionals.

In particular, pragmatic
explanations have been given for the results of the Wason selection task
and for the plausibility of the theory of material implication respectively.

However, as the debate in the philosophy language regarding conditionals
progressed beyond pragmatics, the material analysis of conditionals became
less appealing, giving way to other explanations, such as, for instance,
Adams’ probabilistic account.

It may be argued then that what happened
in philosophy of language could occur in psychology and thus that
pragmatics would fall short of reasons also in the latter framework.

In
particular, we are interested in investigating whether Adams’ successful
ideas in philosophy of language may cast some light on psychological issues
as well.

This will be the major concern of the next section.

However, one
should notice that there are several accounts based on probability available
apart from Adams’.

Roughly, one of the main reasons to prefer Adams’s view is the insight given by his new logical treatment of probability and its
application to classical logic.

More precisely, unlike the other accounts, e.g.
Chater and Oasksford’s, Adams does not assign weights to objective
probability measures in order to explain selection-task related data.

Instead, his theory sets an alternative normative interpretation of
rationality based on subjective probabilities of patterns of inferences.

Let
us now move on to the main section of the paper, where Adams’ account
will be introduced and discussed in much more detail.

We have seen how both Peter Cathcart Wason and Herbert Paul Grice maintain that
the classical Material Implication provides a good rendering of the English "if".

In other words, both Grice and Wason endorse a classical truth-functional
interpretation of conditionals and focused their respective enquiries on
related pragmatic issues.

We will change our focus here to examine a
different view originating from the philosophy of language, namely Ernest
Adams’ probabilistic semantics for indicative conditionals (Adams, 1975; for
a good overview of Adams’ ideas on this topic, see Bennett, 2003, ch. 9).

Our main goal will be to show how this approach can provide a compelling
normative account for rationality and to confront its explanatory power
with respect to The Task with the one held by classical logic.

Among other things, Adams is credited with taking seriously the idea that
conditionals can be accepted with different degrees of closeness to certainty
and with developing that idea into a fully worked-out formal theory, which
is known also as the Suppositional Theory of Conditionals.

The reason
behind this label is that, if we ask what it is to believe, or to be more or
less certain, that q if p, e.g. that Ellen cooked the dinner if Lauren did not,
that I recover if I sleep more, and so forth, we suppose (assume,
hypothesize) that p, and make a hypothetical judgment about q, under the
supposition that p, in the light of your other beliefs.

That is how we make
judgments such as the ones exemplified above.

Originally, this idea was
expressed by Ramsey in a famous footnote to one of his paper.

“If two people are arguing

If p will q’

and are both in doubt as
to p, they are adding p hypothetically to their stock of
knowledge and arguing on that basis about q.

They are fixing
their degrees of belief in q, given p.

Notably, the first sentence of the previous quote is now known as the
Ramsey Test for the acceptability conditions of conditionals and it has
massively influenced the current debate on conditionals.

One may say that
if Ramsey originated this debate, Adams established some of the
fundamental rules of it.

Let us try to be more precise.

It is important to
notice that so far we have not mentioned truth conditions.

This is not
fortuitous.

In fact, according to the Suppositional Theory, the connective
‘→’, commonly used to express the indicative mode of conditional
sentences, is NOT truth functional and hence indicative conditionals do not
have truth-values.

In particular, the former tenet follows from the axioms
of standard (Kolmogorov) probability calculus, while the latter and most
interesting is proven by Adams (Adams 1975, p. 49-51 and for an
illustration p.4).

If we write the second claim formally as the following

u(A
→ B) = u(A⊃ B)
p(A)

we may notice that this is equivalent to what Adams
called Equation 13, which says that the uncertainty of an indicative
conditional is necessarily greater than the uncertainty of its material
counterpart, unless either both uncertainties equal 0 or their antecedent A
has probability 1.

This seems to highlight a connection between the
equation expressed by the second idea and the standard ratio formula for
conditional probabilities.

Adams is aware of this and adds a very
interesting remark involving the famous triviality result by Lewis.

A
possible way to put this is as follows.

If the conditional probability measure for
conditional’s probabilities is correct, and given other standard assumptions
of probability theory, there is no way of attaching dichotomous truth values
to conditionals in such a way that their probabilities will equal their
probabilities of being true.

As Adams points out, this gives good grounds
to believe that conditional sentences lack truth-values.

However, if this is
so, then a major question arises.

How can arguments using indicative
conditionals be valid?

Adams quick reply would be that ‘→’ expresses a
high conditional (subjective) probability.

Let us see what this means.

One of Adams’ main insights on this topic is given by the notion of
probabilistic validity.

We will introduce this new concept by means of
uncertainty.

In probability theory, a proposition’s uncertainty amounts to
its improbability, which equals 1 minus its probability.

This leads to the
3Ivi. p. 3.
4Ivi. p. 5.

following identities:

u(A) = p(¬A) = 1 − p(A)

where u( ) and p( ) are uncertainty and probability functions respectively
and A is a single proposition, that is A does not stand for an indicative
conditional.

In Adams’ terminology, A is a factual sentence, not a
conditional one.

We will be more precise about this distinction later.

What
is important now is that, according to Adams, indicative conditionals do
not express propositions and, therefore, no probability (of being true) can
be meaningfully assigned to them5.

Now, to get to grips with the notion of
probabilistic validity, one should keep in mind that, as classical validity
does not allow falsity to enter along the way from premises to conclusion, in
the same way, a high degree of uncertainty isn’t allowed to enter in any
probabilistic valid forms of argument.

In other words, in a probabilistically
valid argument, if the probabilities of the premises tend to 1, then the
probability of the conclusion also necessarily tends to 1.

As a consequence,
a probabilistically invalid argument is such that if the probabilities of the
premises tend to 1, then the probability of the conclusion does not
necessarily tend to 1.

To be more precise, Adams has proven the following
result:
Theorem 1 (Adams’ probabilistic definition of validity):

An inference Q1, . . . , Qn ∴ Q is probabilistically valid (p-valid) if it satisfies
the Uncertainty Sum Condition (USC):
u(Q) " u(Q1) + · · · + u(Qn),
for all uncertainty functions u( ).

Hannes Leitgeb offers an alternative to the standard debate on this point. Instead of
assuming that either an indicative conditional expresses a unique proposition or does not
express a proposition at all, he argues for the position that each indicative conditional
corresponds to two propositions, which are the contents of so-called conditional beliefs.
According to this view, this is how we believe in indicative conditionals.

For more details,
see Leitgeb.

In contrast with letter like A and E in italics, which are metaliguistic variables ranging
over factual formulae of our language, capital letters, such as, Q, with or without subscripts,
are metalinguistic variables ranging over all formulae of the language, including
factual and conditional ones. We adopt this convention from Adams.

An important remark is in order. USC is a basic theorem of probability
logic and it is the key for justifying Adams’ application of probability to
deductive logic or, in his own words, the application of deductive logic to
inferences with somewhat uncertain premises, so long as there are not too
many of them.

Most importantly, the fact expressed
by USC, that the uncertainty of the conclusions of an argument cannot be
greater than the sum of the uncertainties of the premises, is a necessary and
sufficient condition for inferences without conditionals to be classically
valid.

This entails that, as long as arguments with only factual sentences
are concerned, the notions of probabilistic and classical validity are
equivalent.

With the following theorem, Adams established that this is true
also for arguments with any number of premises:

Theorem 2

If A1, . . . , An entail C, then u(C) " u(A1) + . . . + u(An), for
all uncertainty functions u( ).

This means that the uncertainty of the conclusion of any classically valid
argument cannot be greater than the sum of the uncertainties of the
premises.

Moreover, this shows that any classically valid argument using
factual sentences is also probabilistically valid.

At this point, it seems useful to concretely show that Theorem 2 holds by
means of an example.

In particular, we will construct a model, which
illustrates that arguments of the classically valid form

A1, . . . , An ∴ A1 ∧ · · · ∧ An

are also probabilistically valid.

Let us consider
the simplest instance of the argument, consisting of the two premises A and
B and the conclusion A∧B.

Our model consists in four possible worlds,
which have been assigned a weight each, according to their respective
probability of being the case, in the following way: p({w1}) = 0.2, p({w2})
= 0.4, p({w3}) = 0.1 and p({w4}) = 0.3

It is important to keep in mind that the results discussed in this section are understood
in terms of subjective probabilities of an idealised thinker who knows all the logical truths
and makes no logical mistakes.

Usually, mathematicians tend to assign probability or uncertainty functions to sets,
while philosophers are inclined to assign them to formulae.

However, the connection
between the two practices seems quite straightforward.

Precisely, when one refers to
formulae, usually assume that each formula express a proposition.

Now, since possible
worlds may be understood as sets of propositions, it is easy to see why our notation
should make mathematicians happy as well.
120
✤
✣
✜
✢
★
✧
✥
✦
A
B
w1 0,2 0,4 w2
w3 0,1 0,3 w4

Figure 2: A possible worlds model of a p-valid argument of the form A, B ∴
A ∧ B.

Now, it is obvious what the probabilities of the premises and the conclusion
are.

Remembering that, for factual sentences, ‘probability’ means
‘probability of being true’, premise A has probability 0.6 of being true,
premise B has probability 0.7 and the conclusion A ∧ B has probability
0.4.

One should notice that, in spite of the fact that the probability of each
premise is greater than the probability of the conclusion, the argument is
valid.

Let us spell this out in terms of uncertainty: premise A has
uncertainty 0.4 of being true, premise B has uncertainty 0.3 and the
conclusion A ∧ B has uncertainty 0.6.

This amounts to say that the
uncertainty of the conclusion is less than the sum of the uncertainties of the
premises.

Therefore, the argument satisfies Adams’ probabilistic criterion of
validity and, since it deals with factual sentences only, it is both classically
and probabilistically valid.

This reasoning shows another important
consequence.

P-validity differs from more traditional concepts of
validity, e.g. that a conclusion must be ‘at least as true’ as its
premises.

Another way of putting the foregoing
appeals to the mathematical notion of limit.

In particular, what Adams
found is that it is impossible to make the probability of each premise tend to
1 at the limit, without their conjunction (in this case, the conclusion)
tending to 1 as well.

After having made the acquaintance of p-validity relative to arguments with
only factual sentences, it is time to have a look at how this notion allows us
to analyze arguments with indicative conditionals, in spite of their lack of
truth-values.

Suppose we have fixed the semantics for factual sentences A,
B, C, D, E, . . . of a propositional language L in the standard classical way,
121
and for indicative conditionals of the form A → B, where A and B are
members of L.

Now, consider arguments of either of the following forms:
A1 A1
.
.
. .
.
.
Am Am
B1 → C1 B1 → C1
.
.
. .
.
.
Bn → Cn
D
Bn → Cn
E → F

According to Adams’ semantics such arguments are probabilistically valid if
and only if, for all sequences p1, p2, p3, . . . pi, . . . , where i tends to infinity,
of subjective probability measures pi on L, the following holds:
If If
pi(A1) tends towards 1, pi(A1) tends towards 1, .
.
. .
.
.
pi(Am) tends towards 1, pi(Am) tends towards 1,
pi(C 1|B1) tends towards 1, pi(C 1|B1) tends towards 1, .
.
. .
.
.
pi(C n|Bn) tends towards 1, pi(C n|Bn) tends towards 1,

Then Then
pi(D) tends towards 1 pi(F |E) tends towards 1
(where if pi(ϕ) = 0, the conditional probability pi(ψ|ϕ) of the
corresponding conditional, by stipulation, equals 1).

One could encapsulate
this idea in the following slogan.

The more certain the premises, the more
certain the conclusion.

In summary, Adams found that arguments using factual sentences are
classically valid if and only if they are probabilistically valid.

On the other
hand, arguments using indicative conditionals can be evaluated only on the
basis of probabilistic validity, because no truth-values can be meaningfully
assigned to conditional sentences. In the next subsection, we shall give an
122
example of how that works, and we will show why this is particularly
relevant for our general discussion.

1 A → B ∴ ¬B → ¬A

As the above section title indicates, the main concern of this section will be
the classically valid rule of inference called Contraposition.

This already
seems to require justification. As anyone familiar with the psychological
literature on the Wason selection task would know, the reason why people
generally fail (according to the classical view) to give the correct answer to
the task is not clear.

Aside from invoking pragmatic factors,
there are two possible explanations for this failure.

We could say that the
outcome is due to the fact that people fail to apply the classical valid form
of Modus Tollendo Tollens (MTT) or we could say that they fail to employ the rule of
contraposition.

Let us spell out this distinction.

Suppose we were to
consider the upshot of The Task in terms of MTT.

Then our concern would
be why people are not drawing an inference from the two premises A → B
(the given rule of The Task) and ¬B to the conclusion ¬A.

Instead, if we
see the issue in terms of contraposition, we should explain why people do
not reason from A → B (the given rule) to ¬B → ¬A and then, by Modus Ponendo Ponens (MPP), from the latter and ¬B to ¬A.

This distinction could seem
rather irrelevant to psychologists.

The reason is that they usually endorse
classical validity, according to which both contraposition and MTT are valid
and, therefore, it does not matter which inferential step people actually
carry out.

Consequently, they would conclude that logic is insufficient to
explain the data, mainly because they understand ‘logic’ as ‘classical logic’,
and therefore they would look for an explanation elsewhere, e.g. pragmatic
assumptions.

On the other hand, we can avoid that situation by
considering, as it were, the right logic for the right kind of conditionals. In
particular, it seems that Adams’ normative standard (which is usually
overlooked by psychologists) can better accommodate the data without
charge of irrationality.

More precisely, since MTT is probabilistically valid,
but contraposition is not (as we will see later on), it does matter whether
people use the former or the latter.

This amounts to saying that, roughly
speaking, if people use MTT, they are rational, but if they use
contraposition, they are irrational. From the foregoing, we can envisage two
interrelated reasons for preferring the explanation in terms of
contraposition.

Firstly, since contraposition is not probabilistically valid, the fact that people do not implement that pattern seems perfectly
rational.

So, perhaps, unless further empirical evidence is provided, we
should prefer explanations according to which people turn out to be
rational rather than irrational (following some sort of Principle of Charity).

Thus, if the possible answers to “Why do people not turn the ¬q-card?”are

(i) because they think MTT is invalid, which is irrational, or

(ii) because
they think contraposition is invalid, which is rational, we should opt for the
latter explanation.

Secondly, in the reasoning involving contraposition, the
only inference from a conditional and a factual premise that people need is
MPP.

At this point, we may even speculate that it would not be surprising if
research on the human brain uncovered evidence that MPP is the most basic
inference that we can use.

Be that as it may, it is important to keep in
mind that psychologists should not overlook the previous distinction,
because of its bearing on questions about rationality.

We can now go back
to our analysis of contraposition.

What we really want to explore here is what happens to Contraposition
when we plug indicative conditional signs into the places of material
implication signs.

In other words, how can we evaluate the argument

A →
B ∴ ¬B → ¬A?

Needless to say, our main tool to spell that out will be
probabilistic validity.

Let us consider the following equivalent version of
Contraposition:

B → ¬A ∴ A → ¬B

where ‘→’ is the non-truth-functional connective for indicative conditionals.

Our next move is showing that Contraposition is a probabilistically invalid
rule of inference.

Let us look at an example of it.

If we substitute the
sentences

Jane is drinking vodka.

and

Jane is at least 16 years old.

for A and
B respectively, we obtain

Premise: If Jane is at least 16 years old, then she is not drinking
vodka
∴

Conclusion: If Jane is drinking vodka, then she is less than 16
years old.

Now, recalling that according to Adams’ view, ‘→’ expresses a high
conditional probability, it should be intuitive that it might be reasonable not to draw the above inference.

The reason is that the probability of the
premise might be high, while at the same time the probability of the
conclusion might be low.

Let us assume that our example refers to a
country where people are allowed to drink if and only if they are at least 16
years old, and where people usually abide the law.

Then we would assign a
high conditional probability to the premise, because most people might not
drink vodka at all, but not to the conclusion, because in this country people
very rarely break the law.

This means that we have reasons to believe that
the argument does not accomplish the definition of probabilistic validity.

Contraposition is probabilistically invalid.

Nonetheless, we may
still feel some skepticism about this reasoning.

Conveniently, Adams is in
the position of giving us some more reason to buy his explanation.

He does
so through an ingenious application of Venn diagrams.

Firstly, we want to
characterize the already-mentioned distinction between factual and
conditional sentences by means of Venn diagrams.

Let us assume that all
possible worlds are represented by points enclosed in a rectangle D.

Factual
sentences such as A and B are represented by the corresponding
sub-regions of D containing the possible worlds in which the propositions
are true.

This is shown in Figure 3 below:
D
✬
✫
✩
✪
✬
✫
✩
✪
A B

Figure 3: The diagrammatic representation of a highly probable conditional
A → B.

In this case, the advantage of the Venn diagram is representing all the
truth-functional combinations of A and B, such as ¬B and A ∧ B.

On the
other hand and most importantly, the diagram doesn’t give us a
representation for conditional sentences in terms of region, which is in
accordance with the assumption that they lack truth-values.

However, we
did not say anything about probabilities yet.

Adams’s move is identifying
125
the areas of the sub-regions corresponding to factual propositions with the
probabilities of the propositions. It’s should be noted that this makes sense
if the area of D is assumed to equal 1.

For this reason, we can say that the
bigger the region corresponding to a proposition, the bigger the area, and
therefore the bigger the probability of the proposition.

The diagram represents a possible probabilistic state of affairs, where
propositions corresponding to large regions are represented as probable
while propositions corresponding to small regions are represented as
improbable.

The case of conditional sentences is
unfortunately more complicated.

In fact, neither their corresponding
propositions are represented by regions of D, nor their probabilities by
areas of D.

Nonetheless, this seems to fit nicely with the fact that, unlike
unconditional probabilities, conditionals’ probabilities are not probabilities
of truth.

The probability of a conditional

A → B

is instead identified in the
diagram with the proportion of sub-region A which lies inside sub-region B.

But this is just the geometric analogue of saying that the probability of a
conditional A → B is the ratio of the probability of the conjunction A ∧ B
to the probability of its antecedent A, that is the standard definition of
conditional probability:
p(B|A) = p(A ∧ B)
p(A)
On this view, if most of region

A corresponding to the antecedent of a
conditional lies inside region B corresponding to the consequent, the
probability of the conditional A → B should be understood as high.

Conversely, if most of region A lies outside region B, the probability of the
conditional A → B is low.

Thus, it should be easy to see how Figure 3
shows the diagrammatic representation of a highly probable conditional A
→ B.

In fact, most of region A lies inside region B, therefore the probability
of the corresponding conditional A → B is high. At this point, a couple of
important limitations concerning probabilistically interpreted Venn
diagrams should be noted.

The first is that Adams’ view deliberately ignore
the case in which the probability p(A) of the antecedent of the conditional
A → B equals 0.

This is because, when it is taken into account, it may
cause problems to the given picture of conditionals and their probabilities.
In fact, whenever p(A) = 0, the probability of A → B is not defined, since
the corresponding proportion is not defined.

The second constraint is that
probabilistic Venn diagrams can represent factual and simple conditional
126
propositions, but not more complex constructions, such as conjunctions,
disjunctions and negations of conditionals.

This is the diagrammatic
expression of the fact that it is highly problematic to assign probabilities to
such constructions.

However, dealing with this problem would be beyond
the scope of this paper, so let’s go back to our main concern.

We have now the necessary informations to turn to our example involving
Contraposition.

With the help of Venn diagrams, we want to show
Contraposition to be probabilistically invalid.

Let us start with its premise
B → ¬A and probabilities being given by Figure 4 below. Its probability is
high, because most of region B lies inside region ¬A.

A concrete instance
that fits with the previous diagram is our previous

If Jill is at least 16
years old, then she is not drinking vodka.

D
✬
✫
✩
✪
✬
✫
✩
✪
A B

Figure 4: The diagrammatic representation of a highly probable conditional
B → ¬A.

On the other hand, Figure 4 also shows that the probability of the
corresponding conclusion A → ¬B is low. In fact, most of region A lies
inside region ¬B.

In this case, a concrete example is If Jane is drinking
vodka, then she is less than 16 years old.

Finally, on the one hand the
probability of the premise of our argument is highly probable, on the other
the conclusion is assigned a low probability.

But, if this informal reasoning
in terms of Venn diagrams is turned into more formal reasoning in terms of
sequences of probability measures, then one can see that it is possible to
construct a counterexample to our definition of probabilistic validity.

In
particular, we could construct a model, similar in kind to the one of Figure
3 above, that shows that the uncertainty of the conclusion is actually
greater than the sum of the uncertainties of the premises (which in our
previous example consist of only one).

Therefore we can conclude that
127
contraposition is a probabilistically invalid rule of inference.

However, one
might still feel somewhat baffled by this reasoning, maybe because the
previous example was given for the less familiar (but nonetheless
equivalent) version of contraposition, that is B → ¬A ∴ A → ¬B.

Hence,
at the cost of being redundant, I will provide one more example showing
how Contraposition, in its standard formulation, that is A → B ∴ ¬B →
¬A fails to be probabilistically valid. Here is a counterexample to
Contraposition illustrated by means of Adams’ adaptation of Venn
diagrams.
A
✬
✫
✩
✪
B
not-A
Figure 5: The probability of the conditional A → B is high, while the probability
of ¬B → ¬A is low.

Now that we have hopefully cleared any doubt about Contraposition, we
will try to gauge whether at this stage we are in any position to attempt an
answer to our initial question, are people irrational, or more specifically, was Wason rational?

We started this paper presenting the following view.

Since people, when
confronted with simple selection tasks, fail to apply basic, classically valid
rules of inference, such as contraposition, one needs to look for an
explanation of the data outside logic.

As examples of pursuing this route,
we suggested the approaches by Peter Cathcart Wason and Herbert Paul Grice..

It may seem thta both failed
to recognize that logic can still tell us something important regarding the
notion of rationality, provided that we use it in a suitable way.

But of course Grice does not fail, if Wason does.

We tried to show that Adams’ theory of conditionals
provides such a way.

The reason Adams’ point of view seems compelling is that, by means of a general probabilistic perspective, it takes into account a
fundamental feature of human rationality, that is subjective uncertainty.

Furthermore, it provides a new normative interpretation of conditionals,
according to which contraposition, whose failed application was held
responsible by psychologists for people’s poor performance of The Task, is
invalid.

Hence, we gained a new perspective on how to define rationality.

In
doing so, Adams can justify why people fail to apply Contraposition on
normative grounds.

In other words, he can explain, unlike Peter Cathcart Wason and Herbert Paul Grice, our failure to apply a logical principle of inference within logic,
without endorsing any ‘merely’ pragmatic justification.

In fact, we can infer
that, since contraposition is probabilistically invalid, people’s failure of
applying contraposition to the solution of The Task may well be considered
as a rational response.

However, if on the one hand, Adams’ account, unlike
Wason’s and Grice’s, gives a purely logical explanation of why people don’t
generally choose the card corresponding to not-q, on the other, it still does
not justify people’s choice of the q-card.

At this point, the best that we can
say is that, according to Adams’ view, people are rational in not drawing
the inference based on contraposition, but they still are irrational in not
drawing the inference on Modus Tollendo Tollens, which is probabilistically valid.

On
these grounds, we can conclude that even if our suggestion does not
completely rescue people from irrationality, it seems to support at least the
idea that rationality is not an “on-off” phenomenon, but a partial one.

It is
plausible that some help may come from psychology itself. In fact, recent
studies in the field seem to give empirical evidence for Adams’ view (Pfeifer
and Kleiter).

Further research in both psychology and logic may
corroborate the idea that people actually do what they ought to do when
confronted with The Task 9.

If this is the case, rationality would be an
essential property of what we may tentatively call ‘probabilistic cognizers’.

Adams himself is not exactly crystal clear on the matter.

For instance, when he raises
the question about the status of every day inferences which are not probabilistically valid,
he says that contraposition is rational, but it is not rational in virtue of being of the
contraposition form.

According to him, in fact, further conditions
need to be met for that reasoning to be rational.

These conditions are not part of the
meaning of propositions such as B → ¬ A, but they obtain when people are told such
propositions. Adams goes further and makes some comments worthy of being reported
here.

He calls the conditions under which a pattern is probabilistically valid its conditions
of partial rationality.

References

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probability to deductive logic, Reidel, Dordrecht.

Adams, E.W. A primer of probability logic. Stanford, Ca.:
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Bennett, J. A philosophical guide to conditionals. Oxford:
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Cosmides, L. The logic of social exchange: Has natural selection
shaped how humans reason? Studies with the Wason selection task.
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Grice, H. P. Logic and Conversation, Oxford.

-- Logic and conversation. In P. Cole and J. Morgan
(Eds.) Syntax and semantics (vol. 3). New York: Academic Press.

-- Way of Words, Harvard.

-- Aspects of Reason.

-- Indicative Conditionals.

Leitgeb, H. Beliefs in conditionals vs. conditional beliefs.
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Oaksford, M. and Chater, N. A rational analysis of the selection
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Pfeifer, N.and Kleiter, G. Towards a mental probability logic.
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of the 5th International Symposium on Imprecise Probability: Theories
and Applications (pp. 347-356). Prague, Czech Republic.

Ramsey, F. P. General Propositions and Causality. In D.
H. Mellor (Ed.) Philosophical papers (pp. 145-63). Cambridge, UK:
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Speranza, Join the Grice Club!

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Wason, P.C. and Johnson-Laird, P.N. Psychology of reasoning:
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