Is deduction of use in application to our everyday problems? Aristotle said that in practical matters we cannot use a strictly deductive attitude: "we must be content...in speaking about things which are only for the most part true, with premises of the same kind, to reach conclusions that are no better" (Nic.Eth.I,4 - my underlining). We may content ourselves with conclusions which - according to the usual views - are not true; but what happens when we realize that such conclusions bring us into contradiction with some of our deepest beliefs? We have to pursue our search of rationality in the most compelling manner, and this effort forces us to express our background beliefs in order to check what may be wrong with that. In this paper I will give two classical examples, one form the history of cinema and another from the history of artificial intelligence: in comparing these two examples I claim that logical compulsion - the deep need for convention - is highly valuable where feelings are not enough.

Introduction to the topic: Defeasable beliefs
The topic of today is the importance of rationality in the development of human culture. Philosophy as a discipline is something professional. You have to learn all the right stuff, to keep yourself informed of what's happening today in your specialized field. Philosophy is also the common activity of striving for coherence in our beliefs; when everything is OK, when life proceeds normally, philosophy is not needed, but some philosophical activity is needed when we find a contradiction between some of our beliefs. We have practices, skills, assumptions and beliefs we normally use and accept without acknowledging them; these skills and beliefs - which build up our background - are what give us a certain degree of certainty in our behavior. They are what we rely upon and what we do not put in doubt . But sometimes we have to. When the doubt arises, we have to express the background.
We might summarize the meaning of Aristotle's quotation given in the abstract of this talk: in common-sense and everyday reasoning we cannot have premises that, even if for the most part true, give us a certainty of truth. therefore beliefs we attain with our common-sense ad everyday reasoning cannot be but uncertain.
Which conclusion should we derive from that? Shall we assert, as William K. Clifford says, that "it is wrong always, everywhere, and for anyone to believe anything upon insufficient evidence"? An immediate answer to such a strong claim can be given by the traditional Greek distinction between belief and knowledge: we distinguish belief and knowledge just because belief is what is held true even with insufficient justification, while knowledge is justified belief. Are we therefore free to believe to what we like? That does not follow. But something less drastic than Clifford's claim follows from Aristotle's remark: not attaining to certainty is a characteristic of what we call "belief". Therefore it is mostly right, for most people and for most places to believe something on insufficient evidence2 [taking into account that in most of our beliefs evidence is almost never sufficient, given the possibility of finding new evidence for the contrary. But if beliefs are rightly and correctly held on the basis of insufficient or uncertain ground, we must be ready to change them in case of different and stronger grounds being offered.]
Probably no contemporary mathematician would repeat Clifford's idea as such; even in mathematical logic we find nowadays different formalizations of common-sense reasoning, such as non-monotonic logics which express this important insight: we are justified to believe something on the basis of generally approved premises, with the restriction that the conclusions may be canceled in face on new incoming information.
In what follow I will give some hints on three different problems:
(1) an analysis of the logical structure which is common to most different cases of uncertain reasoning and uncertain deduction - which shows how we are ready to change our conclusions when new information arrives: in order to do that we pick up, from our background, hidden assumptions which we are ready to recognize as true, even if we did not think to them before.
(2) a discussion of the difference between types of uncertain deductions, given by different kinds of justification. Leaving the problem of the possibility of non defeasable reasoning unsolved, it is still possible to distinguish among different kinds of defeasable reasoning depending on the kind of justification which prompt us to accept a new hidden premise, picked up from our background because of new information.
(3) a prospect of different solution of the problem of belief revision in face of contradictory consequences of conflicting beliefs. Conflicting beliefs may be strongly held, and the fact they are competing is shown by the possibility to derive two contradictory statements from them. This rare case may be the beginning of a philosophical work in digging out our background.

Examples
To make these problems clearer I'll make 2 examples, one form the history of cinema ["who is coming for dinner?", a bit old-fashioned, but useful to make the point clear], the other from the history of A.I.[the old fashioned Tweety, still useful to make the point]:

1) from history of cinema
Mr A. has a daughter. She is going to get married. That brings a lot of happiness to Mr.A. John, the future husband, is intelligent, clever, handsome, and rich, everything a new husband has to be. Mr A. invites him for dinner. He is happy, because practical inferences he makes, even without acknowledging them (e.g.. new marriages bring happiness; my daughter is going to get married; therefore that brings happiness). The bell rings. He opens the door; the future husband is black Mr. A is embarrassed. He is no more happy. [This doesn't sound politically correct; on the other hand I take an example from a movie where the term "politically correct" had not yet been invented; you may anyhow substitute "black" with "Italian" or "American", if you like].
2)from history of A.I.
We all know that birds fly. We are told that Tweety is a bird. Therefore we deduce that it flies. But when we go and see Tweety, we realise that it is a penguin. Given the fact the penguins do not fly, we have to revise our previously accepted belief.