But does this mean that they did not know the truth of their conclusions? That verdict is too severe. In many cases, they did know. Their conclusions were not just 'lucky guesses'.
Nor were they merely 'educated opinions'. The same lesson may be learned outside of mathematics, and possibly with greater clarity. In our ordinary day to day living in the world, many of our inferences which seem to us perfectly compelling and some of which are undoubtedly deductively valid do not proceed in accord with any known inference rules. Many of the inferences which we make daily, in a flash, in a twinkling of an eye, are so complex that when we come to try to analyze them, they defy canonization; even less are they able to have their validity demonstrated by anyone's citing known inference rules with which they accord.
A great many formal logic textbooks perpetrate and perpetuate a myth: that ordinary inference exhibits the logic of the examples so carefully chosen in the textbooks. No one who has taught informal logic — and experienced the frustration of students who crave rules but whose yearning cannot be satisfied by the instructor — can have the least doubt about this.
We learn to make valid inferences, neither by learning rules nor by testing our inferences against the opinions of experts in rules — but by hard knocks and by apprenticeship. Many of us have been beguiled by our collective success in creating logic textbooks for our students. Particularly since the s, with the development of so-called 'Natural Deduction', many have come to believe that the formal arguments of these textbooks portray our psychological operations, if not quite perfectly, then at least normatively.
But to the extent that we believe this, to that extent we have been playing at empirical science in a Rationalist manner, more specifically, we have been trying to do psychology in an a priori fashion.
What are human inferences really like? Having studied and taught for so long what we assume inference ought to be viz. How often do we ever stop to ask our students how they reasoned to a mistaken conclusion? Recently, I tried this exercise myself.
In my introductory class, I asked my students to tell me shortly 1 whether they had ever before seen the question I was about to pose them; 2 what their first response was when they thought they had the answer; and 3 as best as they could, how they arrived at that answer or the reasons which prompted that answer.
The question was adapted from one appearing in A. Ironically, the puzzle as posed in Latcha's book is unwittingly subtly self-contradictory. It takes twenty days for a certain tree to lose all of its leaves. On each day after the first day, the tree loses twice as many leaves as on the previous day. At some point, the tree is exactly half bare. When is that, i. Eight students reported that they had seen the puzzle before. I have eliminated their responses from the tally although not one of them got the right answer.
The remaining 95 responses are categorized in Table 1. Of the 66 students who offered an answer, 37 i. No student, either among the ones listed or in the group whose answers were eliminated, gave the correct answer: "During the 20th day". But what had motivated the exercise was my wanting to examine the reasoning lying behind the wrong responses. Typical explanations among those who gave the overwhelmingly most frequent answer "day 19" read: "If the tree loses twice as many leaves per day, then it loses half as many each previous day.
So if it loses all its leaves by day 20, it must have lost at least half the day before, day The reasoning is, of course, fallacious. But the wonder of it is that so many persons reason in the identical, fallacious matter. There are, I suppose, an infinite number of ways to reason fallaciously.
But evidence such as this suggests that there are common patterns of inference — even in cases of fallacious reasoning. To be sure, the evidence produced by this little lapse into empirical research is not the stuff articles are made of in professional journals in psychology or cognitive science. There is much to criticize in the design protocol, and I shall not pretend otherwise.
My data are meant only to be illustrative and suggestive. Data such as I have just reported fit well a theory that a number of cognitive scientists and researchers in artificial intelligence have been urging in recent years: that human beings reason, not so much in accord with valid inference rules of the sort identified and sanctioned by formal logicians, but rather in accord with perceived similarities with former cases and rules of thumb, so-called 'heuristics' see e.
Langley et al. The reasoning my students used to produce and justify their answer "day 19" is the sort of reasoning which can be used successfully in some deceptively similar cases. For example, that reasoning would work if applied to this sort of case: "A silo is filled with grain over a twenty-hour period. At the end of each hour after the first, the total amount of grain in the silo is twice that present at the end of the previous hour.
When is the silo half filled? The crucial issue is this: "Suppose a person were to use in this latter case filling the silo much the same sort of reasoning used by the majority of persons in the previous instance the defoliation of the tree. In this latter case, the person would arrive at the right answer.
But if so, then knowledge can proceed from inferences drawn in accord with logically invalid heuristics. Even if we were to learn of the logically fallacious heuristic involved, inasmuch as that heuristic gives the right answer for this case, we would probably still, I think, want to say that the person does know the right answer.
If, and I want to underline that this is a big if, human reasoning does not typically proceed by taking account of valid rules of inference, but instead issues from our perceiving real or apparent similarities with previous cases i. It may well be that the conceptual ingredients needed to reconstruct our actual concept of knowledge have less to do with valid inference than we have hitherto supposed.
We may need to broaden our historical, normative, model of knowledge by asking ourselves both how persons actually go about acquiring knowledge and what we are prepared to recognize as being genuine cases of knowledge. Doubtless some philosophers are altogether too prone to answer this latter question a priori and with a normative fervor.
It may well be that what we and, in particular, the vast majority of human beings who are not philosophers or persons trained in the niceties of formal logic, take to be knowledge includes a very great deal of that which results from inferences which proceed causally in accord with heuristics.
This is just a sketch. I need not share your heuristic in order to believe that your inference gives you knowledge. Perhaps I need only assume that you have some such heuristic.
But if anything like this is so, if, that is, the very concept of knowledge that we are trying to analyze allows for 'heuristic inference', then, as philosophers, we have gone pretty much as far as we can go. To give a proper analysis of the conditions under which someone can come to know as a result of having inferred from will require a conjoint effort of philosophy and cognitive psychology.
Neither discipline is going to be able to produce this reconstruction on its own. This article is a revised version of an essay which originally appeared in a privately published festschrift for Raymond D. I will leave aside questions whether machines, in particular computers, also perform or could perform acts of inference. Wittgenstein had asked " We can ask a parallel question about inference: "what is left over if I subtract the fact that I-come-to-believe- q -after-having-believed- p from the fact that I-infer- q -from- p?
Brown challenges this characterization as well a, pp. Again his criticism invokes the fact that inferring seems to take no time; that one can stop halfway through a journey, but there is no stopping halfway through an inference. And again, I will grant Brown his insight, but persist with the standard terminology. Some philosophers, of course, eschew the theory that beliefs have objects.
They would, thus, object that nothing whatsoever answers to the description I have just given, i. I will not attempt to rebut such objections here. Recall that a moment ago I suggested that construing inferences solely as relations between statements is overly narrow.
As a matter of fact, the sentence just written suggests an inference proceeding, not from a statement to a statement, but from a concept to a concept, viz. Notice that this definition is recursive: it invokes the very concept it is intent to define. Yet it is not circular, since it has as its so-called 'basis' a condition viz. None of the points I argue below depends on the actual example used. My using the Goldbach Conjecture is a mere convenience and is meant only to be illustrative of a particular infinite class of propositions, viz.
If it should happen that the Goldbach Conjecture comes to be proven, one way or the other, then one need only substitute " " in place of "the Goldbach Conjecture", where " " is an arbitrary name for an unproven necessary truth.
Shortly below, again for convenience, I will drop the reminder about the proviso of Cathy's conclusion being true. Nidditch has argued that proofs are — in fact — relatively rare within mathematics. The number of original books or papers on mathematics in the course of the last years is of the order of 10 6 ; in these, the number of even close approximations to really valid proofs is of the order of 10 1. Thus the chances of finding a mathematical publication that contains even one genuine proof are less than 1 in 10, Mathematics is continually asserted to be a deductive science.
Yet, with the extremely rare exceptions mentioned above, there is no piece of deduction in extant mathematics. None of the so-called proofs forms a deductive chain. They are at best outlines of proofs, not themselves proofs. Many of the necessary details are omitted; but if a detail is necessary for the validity of a proof, then its absence involves the absence of a valid proof. Nidditch , pp. Note that the evidential base pertains to logical operations on sets of actual material objects, not to physical operations — such as placing objects in spatiotemporal proximity.
The results of physical operations may be, and sometimes are, vastly different from the results of logical operations. On one account, the inference just made — from experientially sampling the features of sets of actual physical objects — to a universal conclusion, would appear to be no more than inductively valid; but on the account under consideration — insofar as the conclusion is a necessary truth — that inference turns out to be deductively valid. I will let this curiosity pass. Actively scan device characteristics for identification.
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Select personalised ads. Apply market research to generate audience insights. Measure content performance. Develop and improve products. List of Partners vendors. Share Flipboard Email. Richard Nordquist. English and Rhetoric Professor. Richard Nordquist is professor emeritus of rhetoric and English at Georgia Southern University and the author of several university-level grammar and composition textbooks.
Updated January 10, Featured Video. Cite this Article Format. Nordquist, Richard. These two relations in question are that of antecedent to consequent in a conditional proposition and that of premiss to conclusion in an inference or argument.
This chapter discusses inference and kinds of valid inference. Keywords: inference , valid inference , propositions , logical sequence , logic , antecedent , consequent , premiss , conclusion.
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