This book has its origins in conversations I had with André Fuhrmann at the meetings of the International Congress of Logic, Methodology, and Philosphy of Science in Uppsala in 1991. These exchanges led to a joint essay (Fuhrmann and Levi, 1994) that discussed some peculiarities of conditional reasoning when induction is taken into consideration. One sin begets another and I began contemplating the writing of a long paper combining the version of Ramsey test conditionals I had already advocated with ideas on inductive inference I had discussed ever since I had published Gambling with Truth (Levi, 1967).

As a preliminary, however, it seemed desirable to spell out somewhat more elaborately than I had done before the view of Ramsey test conditionals I favor. At around the same time, John Collins and I started a reading group at Columbia that included as regular participants Markko Ahtisaari, Horacio Arlo Costa, John Danaher, Scott Shapiro, and, for a brief period when he was visiting Columbia, André Fuhrmann. John Collins had presented to the group his account of the structural differences between revision of belief as understood by Alchourrón, Gardenfors, and Makinson (1985) and revision by imaging in a sense parasitic on the possible-worlds semantics for conditionals pioneered by D. Lewis (1973). In the course of his presentations, Collins defended the view that imaging was best suited to characterizing suppositional reasoning whereas AGM revision is suited to changing beliefs. I had already argued (Levi, 1991) that the AGM formalism was inadequate as an account of rational belief change. And I was quite convinced that imaging was ill suited to capture suppositional reasoning – especially in applications to practical deliberations.

Supposing and Believing

Keeping a firm grip on the difference between fact and fiction entails much more than distinguishing between what is judged true, what is judged false, and what hangs in suspense. Any agent who has a clear “sense of reality” distinguishes between what he or she fully believes to be true and what he or she supposes to be true for the sake of the argument. But both what is fully believed and what is supposed furnish the basis for a tripartite distinction between what is judged true, what is judged false, or what hangs in suspense.

Given an agent's state of full belief at a given time, some propositions are judged possibly true because they are not ruled out by the state of full belief and others are judged impossible. In this epistemic sense of serious possibility, h is judged true relative to the agent's state of full belief K if and only if ∼h is not a serious possibility, h is judged false if and only if ∼h is judged true, and the question of the truth of h hangs in suspense if and only if both h and ∼h count as serious possibilities.

If a would-be investor is uncertain as to whether the government will propose an investment tax credit for long-term investments in U.S. firms, the uncertainty will have some relevance to the investor's conclusion as to how to make investments.

Truth, Information, and Full Belief

When an inquirer seeks to improve his current state of full belief, the legitimacy of the alteration made depends on the aims of the inquirer. There are many kinds of aims inquirers might and do have in altering their full beliefs. These aims need not be economic, political, moral, or aesthetic. Cognitive aims may be pursued as well. The kind of cognitive aim that, in my opinion, does best in rationalizing scientific practice is one that seeks, on the one hand, to avoid error and, on the other, to obtain valuable information. Whether inquirers always seek error-free information or not need not concern us here. I rest content for the present with making the claim that agents can coherently pursue cognitive aims of this kind.

A consequence of this view is that states of full belief should be classifiable as error free or erroneous. Otherwise it makes little sense for an inquirer to seek to avoid error in changing his or her state of full belief. Likewise states of full belief should be classifiable as stronger or weaker; for those who seek valuable information should never prefer weaker states of full belief to stronger ones.

The two classifications are interrelated. If state 1 is stronger than state 2 and is error free, so is state 2. If state 2 is erroneous, so is state 1.

Rational agents and inquirers are sometimes capable of evaluating their beliefs, values, and actions with respect to their coherence or consistency – that is, with respect to prescriptive standards of rationality. But this is not always so. Often the predicaments faced are too complex or the time available before a judgment is to be made too short or the cost of deliberation too great for even the most intelligent and well balanced agents to engage in such self-criticism. And even when these considerations present no obstacle, emotional difficulties and indolence may impede this type of activity.

Because agents sometimes can and sometimes cannot fulfill the demands of reason by themselves, they need help. Education in logic and mathematics can contribute. Practical training in various types of deliberation is often useful. Whether current forms of psychotherapy are of value is a matter of dispute but their chief importance to us is their claim or promise to do so. The same is true of reading good literature.

We also use prosthetic devices when they become available. We write notes to ourselves when we don't trust our memories. We consult handbooks as stores of information and resources for how to make exact or approximate calculations. And more recently we have become fond of using the products of the burgeoning technologies that furnish us with calculators and other automata that enhance our capacity to engage in self-criticism.

I make these banal observations to emphasize a simple but important point.

Default Conclusions as Degrees of Belief

The conclusion of an inductive inference is a coming to full belief.1 It entails a change in the agent's state of full belief and, in this respect, in one's doxastic commitment. Or, if the reasoning is suppositional, it is a full belief conditional on an inductively extended supposition. In both cases, the induction is a transformation of a corpus of full belief into another one. If default reasoning is a species of nonmonotonic inductive inference, the conclusion of a default inference must be representable by such a transformation.

It is far from clear, however, that students of default reasoning always understand the conclusions of default inference to be held so confidently. The inquirer might judge the conclusion as worthy of belief to some degree without coming to endorse it fully. In this sense, nonmonotonic reasoning need not involve inductive expansion to a new corpus of full beliefs.

What, however, is to be meant by a degree of belief or beliefworthiness? The currently fashionable view is that degrees of belief are to be interpreted as credal probabilities used in evaluating the expected values of options in decision problems.

There are alternative proposals. In particular, there is the interpretation as degrees of belief of measures having the formal properties first identified by G. L. S. Shackle (1949) and embedded in the b-functions described in section 6.7. Moreover, there are diverse proposals for ordinal or quantitative indices of beliefworthiness or inductive support such as the difference between a “posterior” probability and a “prior” probability, the ratio of a posterior and a prior, or the logarithm of such a ratio.