30 responses John Nolt's Inductive Reasoning Test Howard Kahane University of Maryland Baltimore County It seems to me that John Nolt is right in his re- marks concerning teaching informal logic students about deductive validity/invalidity through the no- tions of valid and invalid deductive forms: that doesn't work very well. And I think he's right that a better way to teach about deductive validity is to get students to "regard an argument as valid just in case the simultaneous truth of its premises and falsity of its conclusion cannot coherently be con- ceived." (I intend to make the 5th edition of my own text conform to his insight.) But Nolt's ideas about assessing the various strengths of inductive reasonings, while very inter- esting, and effective for some kinds of inductive arguments, do not work correctly in generaL' Nolt illustrates his theory by a one premise argu- ment. Roughly, his method is to take the percentage of logically possible worlds in which the premise is true and the conclusion also is true, compare this to all the possible worlds in which the premise is true, and consider the percentage obtained in this way to be the strength of the inductive reasoning. This works all right for the type of arguments he con- siders/ for instance the argument: Everyone believes that Neil Armstrong walked on the moon . . Neil Armstrong walked on the moon. The percentage of worlds in which the premise is true and conclusion true compared to all the possible worlds in which the premise is true is very small, so that on Nolt's theory the inference is weak, which conforms to our intuitions on the matter. But consider another sort of inductive reasoning: All crows examined so far have been black. ••• All crows whatsoever are black. Using Nolt's possible worlds method, we must con- clude that this inference is quite weak, since its conclusion is false in a large percentage of the logic- ally possible worlds in which its premise is true. Yet intuitively, the inference is quite strong. (How strong depends on things like the size of our sample, etc., but not on the number of logically possible worlds that are one way or another.) Note 'The reason is related, interestingly, to disputes over induction years ago between Rudolf Carnap, whose inductive theory is a range theory a la Witt- genstein, and Hans Reichenbach, who rejected such theories as a priori. But a discussion of this point is not relevant to this note. 0 Dr. Howard Kahane, Department of Philosophy, University of Maryland - Baltimore County, Balti· more, MD 21228 31 Degrees of Validity and Ratios of Conceivable Worlds Stephen N. Thomas University of South Florida Professor Nolt interested me with his fascinating article, "Possible Worlds and Imagination in In- formal Logic,"1 because I once considered following a somewhat similar "possible worlds" approach in attempting to formulate a unitary definition of the concept of "validity" applicable to all inferences in natural language. However, I was dissuaded from such an approach by certain difficulties. If I describe some of these, perhaps Professor Nolt will be able to explain to us how these problems are avoided on his approach. Nolt proposes the following definition, or pro- cedure, for determining the degree to which reasons support a conclusion in nonconclusive ("inductive") as well as conclusive inference: ... The fundamental idea is to measure strength of reasoning by the proportion of conceivable worlds in which the conclusion is true among worlds in which the premises are true. If this proportion is 100%, the argu- ment is (deductively) valid. If it is 0%, the premises imply the negation of the conclusion. And there are in- finitely many degrees of strength between these two extremes.2 He tacitly assumes that the various "possible worlds" are all equally probable, so that only the ratio of their numbers need be considered to eval- uate the degree of support of inferences. Nolt claims "that the framework of possible worlds provides a unified scheme for defining and interrelating, not only the usual all-or-nothing logical concepts, but also those that admit of degrees,"3 and he implies that the concept of "degree of validity" (or "degree of support," if one prefers) in natural logic might be among these, at least to an approximation!' The main problem with Nolt's approach is that it seems to be strictly fallacious: although, when ap- plied to some examples, it leads to evaluations that are correct and identical to the evaluations obtained by the methods of natural logic, in infinitely many other instances, it appears to lead to evaluations that are wildly inaccurate. II In some cases, Nolt's approach seems to lead to evaluations of the strength of inferences that are far too high. For convenience, consider the following simple, concocted example: The Earth has at least one moon 1 The Earth has more than one billion moons. Intuitively, the premise here gives little or no sup- port to this conclusion. What evaluation does Nolt's approach yield? To apply Nolt's evaluative procedure, we must find or estimate, the "proportion of conceivable worlds in which the conclusion is true among worlds in which the premises are true." The conceivable worlds in which the premise is true would be all the conceivable worlds in which the Earth has one or more moons. Now in what proportion of these is the conclusion true? Remember that in Nolt's approach, each conceivable world counts equally; they are not weighted acccording to their relative probabilities. Following the line taken in his illustrations, we would consider the conceivable numbers of moons that the Earth might have, and ask for what propor- tion of these the conclusion would be true. The con- clusion would be true for all numbers of moons greater than one billion; it would be false for all numbers of moons equal to one billion or less. Now what is the ratio of the number of integers greater than one billion to the number equal or less than one billion? There are infinitely many integers greater than one billion, and a finite number equal to or less than one billion, so it is not exactly clear how this question is to be answered, but the ratio would seem to be very high. In almost 100% of the conceivable worlds in which the premise is true, the conclusion also is true. So evidently, this very weak inference would rate a "very strong" (almost deduc- tively valid) rating, according to Nolt's method or procedure. This appears to be a case in which Nolt's approach leads to an evaluation that is far too high. Obviously, infinitely many similar countei-examples could be constructed in which Nolt's approach also leads to ratings that are far too high. So the first problem is that Nolt's method fallaciously gives high ratings to infinitely many weak or nil steps of inference. III The second problem is that Nolt's account fails to generate accurate evaluations even in what one would expect would be its most favorable class of cases, namely, statistical generalizations from samp- lings that can be analyzed in terms of ratios of numbers of equa"y possible cases. Consider balls drawn from an urn at random. Suppose there are fifty balls in the urn, and that the first forty-nine, drawn at random, all have been blue. Normally we would say that the following reasoning deserves a rating of "strong": There were 50 The first 49, all drawn + balls in the urn. at random, have been blue. [strong] The remaining ball is blue If one calculates the probability that the remaining ball is blue given that the first forty-nine drawn at random have been blue, the probability is well in excess of 80%. How does the strength rate on an analysis in accordance with Nolt's approach? On Nolt's approach, one would "measure strength of reasoning by the proportion of conceivable worlds in which the conclusion is true among worlds in which the premises are true." The conceivable worlds in which the premises are true presumably would be the conceivable worlds in which we had drawn forty-nine balls at random from an urn 32 containing fifty, and found that all forty-nine were blue. Now, in what proportion of these is the conclu- sion true? Well, there is the possible world in which the first forty-nine balls were blue and the last ball is blue too; this is the possible world in which the conclusion is true. How many possible worlds are there in which premises are true, but the con- clusion is false? A small problem arises here: Are we to count the possible worlds in which the last ball is white, say, as distinct from the possible worlds in which the last ball is red, green, black, etc., or are we supposed to lump all the non-blue cases together as just one possible world, the possi- ble world (worlds?) in which the last ball is non- blue? The former course seems more reasonable, since all these various ways in which the last ball could be non-blue seem to be distinct possible worlds, and it does not seem reasonable to say that the probability that a given object is blue, rather than any other color, is 50%. How many non-blue colors are there? Again it is difficult to say, but an examination of sample books of color chips from paint manufacturing companies indicates that the number is very large, certainly in excess, say, of 300. So, among the possible worlds in which the premises are true, there is one possible world in which the conclusion is true, and more than 300 in which it is false. We thus arrive at the evaluation that the strength of the inference is less than 1%, very weak indeed. This seems clearly erroneous. On the other approach, if we counted all the pos- sible worlds with a non-blue last ball as just one possible world, the ratio would become 50%, still far less than standard statistical approaches cal- culate, and far less than the evaluation reached by the methods of natural logic, and far less than our rational intuitions would indicate. Here, then, is a case typical of another infinite set of cases, in which Nolt's "conceivable worlds" approach again appears to be fallacious, this time because it generates evaluations that are inaccurate through being far too low-or have I missed some- thing? IV How are arguments in which reasons give "nil" support to their conclusion evaluated on Nolt's model? Nolt says that if the proportion of conceivable worlds in which the conclusion is true among worlds in which the premises are true "is 0%, the premises imply the negation of the conclusion." This could not be the same as "nil" or "0%" support as represent- ed in the natural-logic model, because reasoning in which the premises give no support to the conclu- sion need not be arguments in which the conclusion is true in none of the possible worlds in which the premises are true. Consider, for example: Some roses are red. + Some violets are blue [nil] The validity of this reasoning IS "nil" or "0%" on the natural-logic approach, because the truth of the premises does nothing to make the truth of the conclusion likely. But the proportion of conceivable worlds in which the conclusion is true among the totality of conceivable worlds in which the premises are true obviously is not 0%, since in some con- ceivable worlds where some roses are red and some violets are blue, Buddy does still love Peggy Sue. Since generally in such cases, the proportion or ratio is greater than 0%, does this mean that on Nolt's model, the only cases in which reasons give no support to a conclusion (a conclusion that is a contingent statement) are the cases in which they entai I its negation? Does Nolt's model conflate pro- viding no support for a conclusion with implying its negation? There seems to be no place in Nolt's model for the situation in which the premises do not imply the negation of the conclusion, but the degree of validity (or degree of support, or strength) of the inference is 0% or nil. What ratio of conceivable worlds in which the reasons are true to conceivable worlds in which the conclusion is true would correspond to the reasons and conclusion being completely irrelevant to each other as far as the relation of logical support or entailment is concerned? Here again the neo-exten- sionalist "conceivable worlds" approach seems to fail really to get at what makes for the "connected- ness" between reasons and conclusions in a valid inference (or the lack thereof). v A further problem is that another argument simil- ar to the one about Buddy above except with the conclusion instead, "Buddy does not still love Peggy Sue" an inference that again should rate close to a "nil" would rate almost 100% (somewhere in the range "strong to deductively valid") on Nolt's model, since the number of conceivable worlds in which Buddy does not still love Peggy Sue is so much larger than the number in which Buddy does still love Peggy Sue. In fact, Nolt's model apparently strictly entails that if reasons Rl , R2 "'" Rn give P% support to conclusion C, then the same set of reasons should give (100 minus P)% support to a conclusion that is the negation of conclusion C (since, if P% is the percentage of conceivable worlds in which C is true (of the worlds in which the reasons 33 are true), then the percentage of those same worlds in which C is false would necessarily be the comple- ment of P%, namely 100% - P%). So, for instance, Nolt's model appears strictly to imply that if certain reasons give 49% support to believing that a given conclusion is true, then the same reasons also give 51% support to its negation. This implication is, in general, of course, false. Examples are everywhere. VI Problems such as these (among others) dissuaded me from a "conceivable worlds" approach, and impelled me to the more radical, but far more accur- ate and much simpler, "natural-logic approach." I think that perhaps the fundamental problem with possible-worlds approaches is that they still are basically extensionalistic (or set-theoretic) accounts that involve impoverished models of the relations of semantic connectedness and entailment in natural languages. Incidentally, the problem of representing the in- fluence of "collateral information" in the natural- logic model (which Professor Nolt indicates contrib- uted to the motivation for his alternative approach) will be addressed, and hopefully clarified, in the forthcoming third edition of Practical Reasoning in Natural language. But I will remark that the rela- tively unrestricted, intentionally loose and free special notation of arrow diagrams and natural logic permit inferences to be evaluated in the context of a totality of presupposed collateral information that is allowed to remain tacitly understood in the back- ground, without needing to be written down explicit- ly (unles a question arises about some part of it and circumstances require that some key additional assumptions be explicitly articulated, in which case they can be added to the diagram). Some contem- porary philosophers of science and logicians have suggested that many scientific and other infer- ences in natural language ultimately presuppose much of the entire totality of the rest of our know- ledge and beliefs as background assumptions, a totality so large that it would be hopeless ever to try actually to write it down. Since in practice, this totality of background information or assumptions generally exceeds the limit of what could ever be explicitly written down (the same problem arises on Nolt's approach), natural logic thus offers a realistic, genuinely workable method for evaluating such in- ferences- in contrast to the rigid traditional ap- proaches that would (if anyone ever seriously attempted to apply them to actual cases) in effect, make the unrealistic, unsatisfiable, impos~ible demand that all collateral information and back- ground assumptions be written down explicitly, along with the central core of premises, before the inference could be evaluated accurately. Notes 1 John Nolt, "Possible Worlds and Imagination in Informal Logic," Informal Logic, vi, 2 (1984), 14-17. 2lbid., 15. 3lbid., 17. 40n the natural-logic approach, the validity of an inference in a natural language is evaluated by asking the question: Is there any imaginable way (or "conceivable world," if one likes) in which the conclusion could be false, even if (or though) the reason(s) be true? If the an.swer to this question is "No" the inference is rated as "deductively valid.,,'lf the answer is "Yes," one next asks the question: How likely is (are) this (these) possibil- Reply to Englebretsen James F reema n Hunter College, CUNY I want to respond to certain criticisms George Englebretsen has expressed in "Freeman on Deduc- tion/Induction" [2] of certain proposals I make for distinguishing deductive from inductive arguments, essentially in the last section of "Logical Form, Probability Interpretations, and the Inductive/ Deductive Distinction." [5]. Englebretsen seems dissatisfied, first, with the precedence criterion I consider that (*) Explicit prima facie inductive or deductive indicators should take precedence over im- plicit indicators. Second, he is dissatisfied with the lack of any theory to account for why an argument family should be either inductive or deductive. Let me reply to each of these in turn. First, let me put the precedence criterion in the perspective of the entire paper. The purpose of the 34 ities or conceivable world(s)? If all are highly unlikely, the degree of validity rates as "strong," and the inference still rates as "valid." But if some of these are real possibilities-states of affairs of the kind that may actually arise in the real world-then the inference rates as "invalid," and the degree of validity is classified as "mod- erate" or "weak," depending on the likelihood of these conceivable ways, or "nil" if the reason(s) give no support whatsoever to the conclusion. These ideas and procedures are introduced and explained in greater detail in Practical Reasoning in Natural Language (1973); Englewood Cliffs, N . J .: Prentice-Hall, 1981. [ J Dr. Stephen N. Thomas, Visiting Assistant Profes- sor, Department of Philosophy, University of South Florida, Tampa, FL 33620. paper was to defend the inductive/deductive distinc- tion, in particular to defend distinguishing argu- ments as being either inductive or deductive, against the claims of various persons, especially Perry Weddle in [10] and [11]. I proposed that judging whether an argument is inductive or deductive is akin to judging what is an overriding duty when con- fronted with one or more prima facie obligations which mayor may not conflict. If the balance of inductive indicators outweighs the balance of deduc- tive indicators, then the argument should be judged inductive. If the reverse, then deductive. As I see it, two of the strongest prima facie indicators are the explicit modal words like "possibly," "probably," Ulikely," on the inductive side, "necessarity," "must" on the deductive side. which I call explicit prima facie inductive (deductive) argument indica- tors; and the families, such as argument by analogy. causal argument, inductive generalization on the inductive side, and propositional argument, cate- gorical syllogism on the deductive side to which an argument may belong. As I have said, traditionally arguments in the first group of families have been assessed by inductive standards, those in the second by deductive standards, and so membership in these families is a reason for saying that the argument is deductive or inductive. Since membership in a fam- ily does not make an explicit claim about the argu- ment's being inductive or deductive, we regard it as an implicit prima facie inductive (deductive) argu- ment indicator. Trouble begins when our indicators conflict. Sup- pose we have an argument which, by virtue of its family membership, is clearly inductive, but which contains an explicit deductive indicator; or where the argument clearly belongs to a deductive family, but contains an explicit inductive indicator. Engle- bretsen's examples illustrate this clash nicely. (1) Washington was rational. Lincoln was rational. Kennedy was rational. So, all U.S. presidents must be rational. (2) All men are rational. All U.S. presidents are men. So, probably all U.S. presidents are rational. ([2], p. 26) according to (*), (1) should be judged deductive, "must" taking precedence over the argu- ment's being an inductive generalization; and (2) inductive, "probably" taking precedence over the argument's being a categorical syllogism. Notice first that rejecting the proposed precedence rule, judging in this case (1) inductive and (2) deductive, would in no way affect the basic criterion that arguments should be judged inductive or deduc- tive by whether the balance of indicators is deductive or inductive. The precedence rule is one way of determining that balance in certain cases. But, as pointed out in [5], p. 10, there can be disagree- ment over precedence rules and yet basic agreement on the criterion for distinguishing inductive from deductive arguments by the balance of indicators. Hence (*) is not part of my central thesis and can be modified or replaced without affecting my basic view. In this connection, I should say there is something misleading in Englebretsen's saying I have proposed a "recipe" for distinguishing inductive from deduc- tive arguments. As I understand it, a recipe is a procedure which can be applied rather· mechanically to get a correct result. The thrust of my position in [5] is that decisions as to whether an argument is inductive or deductive are not mechanical, but require sensitivity to the argument and its context. Englebretsen rejects the precedence rule, count- ing (1) as inductive and (2) as deductive. He offers 35 justification for this in his suggestion for distinguish- ing inductive from deductive arguments. We shall examine this below. Englebretsen expects that most other teachers of logic would agree with his judg- ment about (1) and (2). This is an empirical claim, and I have no hard empirical data to either support or refute Englebretsen here. But I am not sure that all logic teachers would judge (1) and (2) this way. First, consider Irving M. Copi's definitions of "deductive argument" and "inductive argument" in the latest edition of Introduction to logic: We characterize a deductive argument as one whose conclusion is claimed to follow from its premisses with absolute necessity, this necessity not being a matter of degree and not depending in any way upon whatever else may be the case. And in sharp contrast we char- acterize an inductive argument as one whose conclu- sion is claimed to follow from its premisses only with probability, this probability being a matter of degree and dependent upon what else may be the case. ([1], p. 54) Again: Although every argument involves the claim that its premisses provide some grounds for the truth of its conclusion, only a deductive argument involves the claim that its premisses provide conclusive grounds. ... An inductive argument, on the other hand, involves the claim, not that its premisses give conclusive grounds for the truth of its conclusion, but only that they provide some support for it. ([1], p. 51) Now I do not know how Copi would react to Englebretsen's examples, but it seems to me that (1), by virtue of containing the modality "must" does claim that the conclusion follows with absolute necessity, and (2), by virtue of containing the modal- ity "probably" claims that the conclusion follows only with probability. So, unless Copi has some other account of what it is for an argument to make these claims about how strongly the premises support the conclusion, to be consistent with his definition he should judge (1) deductive and (2) inductive. Of course, if it is a consequence of Copi's definition that (1) is deductive and (2) inductive, it is open to anyone who does not like this consequence to say "So much the worse for Copi's definition!" I present this merely as indirect empi~ical evidence that there are teachers of logic who would accept (*). I think Copi is not the only logic teacher who would accept this precedence rule either. In [5], I point out that the precedence rule seems to capture the intuitions of people like Sam Fohr [4] and Fred Johnson [7] who want to take intentions of arguers seriously. But this is just additonal indirect empir- ical evidence that there are teachers of logic who accept (*). On the other hand, Englebretsen is not alone either in rejecting (*). As I point out in [5], David Hitchcock would surely oppose it. For all I know, there may be a case of what Robert Fogelin calls "deep disagreement" between these two groups of philosophers. What I have shown, contra Englebretsen, is that there may very well be two genuine groups of philosophers here, and so also genuine philosophical disagreement. To try to develop a conclusive argument for one side or the other is beyond the scope of this reply, if indeed it can be done. What I should point out, how- ever, is that Englebretsen is just plain wrong when he claims that I have given no argument for (*) in [5]. I have presented reasons for (*), although I have also been careful to point out that they are not conclusive or decisive. Besides pointing out that (*) captures the intuitions of certain philosophers, and that surely is a reason favoring a philosophical view, principle, or rule, I also offer the following reasoning which I am here adapting to Englebret- sen's example and elaborating further. Suppose the person putting (1) forward r~ally used "must" in the formal, technical, logical, or philosophical sense of this word to explicitly claim that the conclusion follows from the premises with absolute necessity. Indeed, we may expect this is the case with (1) since Englebretsen reports that it was given by a student in a logic class as an example of a deductive argument. But clearly there is something wrong with (1) as an example of a deductive argument, as Engle- bretsen agrees. What is wrong? (1) is a bad argu- ment. It is bad precisely because it claims that its premises conclusively support the conclusion when in fact they do not-the argument is not deductively valid. In criticizing the argument, we take it as a deductive argument, criticize it as such, finding it to be an invalid deductive argument. But why did we take it as a deductive argument? Precisely be- cause of the presence of the modality "must" in the argument. But why did we take that as the decisive factor in judging the argument deductive? We did so precisely because of (*). (*) then is an underlying assumption in our criticism of (1), that (1) is not a good argument, which it surely is not. (*) ultimately helps us to explain why this argument is bad. This fact, as suggested in [5], is a reason for (*). If a hypothesis (together with background information) yields correct explanations and predictions, those explanations and predictions constitute evidence for the hypothesis. Likewise, if a philosophical principle yields a certain judgment, say that an action is right or an argument bad, and that judgment is correct, that is evidence for the principle. How would Englebretsen criticize (1)? I gather from [2] that Englebretson's criticism would be more linguistic rather than logical, pointing out that "must" is inappropriate in arguments like (1), which are inductive. As he says, "Indeed, our task as logic teachers is to bring b's [the student's] implicit claims and uses of illative signs into line" with traditional use. ([2], p. 27.) But this seems to me just wrong. Anyone who puts (1) forward as a good argument has made a logical mistake and has not just used the word "must" inappropriately. 36 His argument, not just its presentation, is faulty. What about (2)? According to (*), (2) is to be counted inductive. Does this constitute a reason for or against (*)? Now (2) is a distinctly odd argument. Unlike (1), which involves a logical mistake by claiming the premises give more support than they actually do, (2) claims they give less, which is not obviously wrong. To make a weaker claim when a stronger claim is warranted, unless one asserts that this is the most that can be claimed, is not wrong. Hence, with (2) we do not have the negative criticism that we did with (1), which led us to provide a reason for (*). Does anyone outside the logic classroom ever construct arguments like (2)? Since (2) without the word "probably" is an obviously valid deductive argument, the insertion of "probab- Iy," although not logically wrong, is certainly ill- motivated. One wonders whether anyone seriously reasons according to (2). It is for these reasons that in [5] I called arguments like (2) "freaks." I mention this because Englebretsen seems to take exception to this word, and, more importantly, states that one recourse open to me regarding arguments like (2) (and (1)!) is "to avoid classification all together by admitting them as recalcitrant Ifreaks,'" ([2], p. 27). I vigorously reject this approach, and want to set the record straight that I never endorsed it in [5] nor said anything which could be construed as such an endorsement, should one suppose this from Englebretsen's remark. What I did say was that I failerl .