VI1.2&3, Spring&Fall1985 Informal Logic Enthymematic Arguments DAVID HITCHCOCK A recurrent theme in theoretical treatments of argument - such as those of Perelman (1958: 83-99), Toulmin (1958: 98, 100), Hamblin (1970: 235, 238, 245) and van Eemeren and Groot- endorst (1984: 119-149)-is the tendency of most arguers to leave im- plicit an assumption in virtue of which their conclusion follows from their pre- misses. Outside carefully articulated philosophical and mathematical rea- soning, in fact, most arguments are deductively invalid in the sense that the meaning of their constituent state- ments leaves open the possibility that their premisses are true and their con- clusion false. Some of these deductive- ly invalid arguments are appropriately appraised by a non-deductive standard of inference appraisal; they are ilin- ductive" or /I conductive" or 1/ abduc- tive" arguments. Some are obvious non-sequiturs, to be rejected out of hand. The rest are the topic of this pa- per. These arguments, then, are deduc- tively invalid, but not. mere non-sequi- turs and not non-deductive arguments either. Let us call them "enthymemes" or enthymematic arguments, after the name borrowed from Aristotle in tra- ditional 1/ Aristotelian" logic for syllo- gisms in which a premiss (or the con- clusion -but I exclude such cases) is omitted. Two problems arise about such argu- ments. The demarcation problem is to distinguish enthymemes from deduc- tively valid arguments on the one hand and mere non-sequiturs on the other. (I assume for the sake of thi5 discussion that there is some way of separating off arguments whose inference is ap- propriately appraised by a non-deduc- tive standard, and I ignore any problem McMaster University of ilmissing premisses" which may arise among such arguments.) The eva- luation problem is to work out how to evaluate the inference in an enthyme- matic argument. Although enthymemes are common and their recognition goes back at least to Aristotle (Rhetoric I 2 1357a17-19), there is at present no adequate solu- tion to these two problems . To be sure, traditional logic, as represented by such authors as Barker (1965) and Copi (1982), has solved them for incomplete categorical syllogisms, and Duthie (1975) has extended these solutions to a broader logic of terms. And Rolf George (1972, 1983) has, though I believe with insufficient supporting argument, supplied solutions for pro- positional logic and first-order predi- cate logic. But there is no extant general solution for either problem. It is not an adequate solution to the de- marcation problem, for example, to say that an enthymeme is an argument with a missing premiss, for we need criteria for determining when a premiss is missing. Besides, some authorities (Bolzano 1837, Ryle 1954, Toulmin 1958, George 1972, 1983) deny that en- thymemes have missing premisses, and I shall later defend this denial. Nor is it an adequate solution to the evaluation problem to say, as Trudy Govier does, that you add a missing premiss whose addition "you can justi- fy ... with reference to the wording and context that is actually there" (1985: 33) and evaluate the resulting argu- ment. For we need criteria for an ade- quate justification of the additional premiss on the basis of the original argument's wording and context. I intend, therefore, to propose and defend a general solution for natural languages of these two problems. 84 David Hitchcock Deductive Validity in Natural Languages The first step in distinguishing en- thymemes from deductively valid argu- ments on the one hand and mere non- sequiturs on the other will be to define deductive validity for natural lan- guages. The concept of deductive valid- ity is well defined within formal sys- tems constructed using the logistic method (Church 1956: 47-58). An argu- ment expressed in a formal system K is syntactically valid if and only if it is provable within K that its conclusion is a consequence of its premisses. And the argument is semantically valid un- der a specified interpretation of the sys- tem's connectives and operators if and only if no assignment of values to the argument's constants makes its pre- misses true and its conclusion false under the specified interpretation of the connectives and operators. One could therefore define deductive validity for arguments in natural lan- guages with reference to deductive validity within a formal system. On such an approach, a deductively valid argument in a natural language would be an argument which is correctly translated into an argument which is semantically valid in a formal system with a specified interpretation of its connectives and operators. Given the difficulties of such translation, how- ever, and the probable need for as yet undeveloped formal systems, it seems appropriate to advance a conception of deductive validity which can be ap- plied directly to arguments in natural languages Such a conception would have to be a semantic rather than a syn- tactic one, since natural languages do not come equipped with a complete set of primitive syntactically expressed rules for deductively valid inference. The constants in formal systems are the analogues of what we might call atomic content expressions in natural languages. Such expressions, the cate- gorematic terms of medieval logic, can be regarded as referring to or otherwise signifying actual or possible features of the universe: entities, qualities, occurrent states, dispositions, events, relationships, times, places, facts, and so forth. Natural languages thus have a built-in apparent categorial scheme, which could in principle be made explicit. (Revisionary ontologists can reject such apparent categorial schemes by providing functionally equivalent paraphrases into a canonical notation, in the fashion of Quine, or by producing some other account of how language relates to reality.) Content expressions can be defined in terms of this apparent categorial scheme as expressions which in the context of their utterance can be regarded as re- ferring to or otherwise signifying an item in a category. A molecular content expression is a content expression which has as a proper part an expres- sion which is a content expression. An atomic content expression is a content expression which is not molecular. From a given sentence, it is possible to construct a sentence of the same form by substituting for one or more of the content expressions a content ex- pression in the same category. (I ori- ginally introduced the t~rm "category" for categories of items; by extension, one can speak of the category to which belongs an expression signifying an item in a category.) Thus, the sentence liThe dog is on the mat" is of the same form as the sentence' 'The cat is on the mae' because it can be obtained from that sentence by substituting "dog" for "cat". Let us define substitution on a content expression as "replace- ment of that content expression by a content expression in the same cate- gory". (We allow as a degenerate case substitution of a content expression by itself.) Further, by uniform substitu- tion on a content expression let us mean "replacement of all occurrences of a content expression by the same content expression, one in the same category as the original." (It is to be understood that the expression has the same mean- ing at all occurrences; where the ex- pression has different meanings at different occurrences, we treat these as occurrences of different content ex- pressions.) The atomic form of a sen- tence (or set of sentences) can be re- garded either as the set of sentences (or of sets of sentences) obtained by uniform substitution on all the atomic content expressions in the original sentence (or set of sentences) or as a schema which each member of the set instantiates. An argument which is formally de- ductively valid is one whose (atomic) form makes it impossible for its pre- misses to be true and its conclusion false. We can give some precision to the notion of an argument's form ma- king something impossible, in the fol- lowing way: An argument is formally deductively valid if and only if no uni- form substitution on the argument's atomic content expressions produces an argument with true premisses and a false conclusion. It may be objected that an attempt like this to define formal deductive validity for natural languages is bound to fail, because at least some natural languages have grammars which are not logically perspicuous, in the sense that sentences of the same grammatical form have different logical forms. Thus, for example, there follows from the premisses that "That dog is mine" and "That dog is a spaniel" the conclu- sion that "That dog is my spaniel", but from the premisses that "That dog is mine" and "That dog is a father" it does not follow that "That dog is my father" (Cf. Plato's Euthydemus 298d-e.) In general, however, such ap- parent counter-examples to the defini- tion involve the substitution of a con- tent expression of a different category; in the example, "spaniel" signifies a kind of entity but "father" a relation- ship. The proposed definition of formal deductive validity has at least two vir- tues, in addition to its immediate appli- cability to natural languages. First, it brings out the attraction of formal deductive validity as a criterion of ap- praisal for arguments: it is truth- preserving. The conclusion of any formally valid argument with true pre- misses will also be true. In cases where the premisses are not known with cer- Enthymematic Arguments 85 tainty to be true, but merely accepted as true on the basis of more or less adequate evidence or argument, the argument does not give us certain knowledge of the truth of the conclu- sion, but provides some basis for trans- ferring the acceptance we give to the premisses to the conclusion (subject to countervailing considerations from other evidence and arguments). Since one function of arguments is to increase our stock of truths, or at least of well- grounded beliefs, formal validity is a sufficient criterion of inference sound- ness. Secondly, it provides a quick way of showing that an argument is formally invalid. One simply constructs a pa- rallel argument, obtained by substitu- tion on the original argument's atomic content expressions, in which the pre- misses are true and the conclusion false. (Let us call such a parallel argu- ment a counter-example to the original argument.) Suppose, for example, someone argues that nuclear weapons have prevented a war between the superpowers, on the ground that there has not been a war between the super- powers since they both got nuclear weapons. The conclusion does not necessarily follow, one might re- ply: You might as well say that new cars have prevented a fight between the neighbours, on the ground that there has not been a fight between the neighbours since they both got new cars. It is more difficult to de- monstrate formal validity using the proposed definition as a criterion, since the failure to produce a counter-exam- ple may be due to lack of imagination or ingenuity rather than to the absence of a counter-example. I have advanced the proposed defi- nition as a definition of formal deduc- tive validity, rather than of deductive validity in general, in order to allow for arguments which are deductively valid in virtue not only of their form but also of meaning relations among their atomic content expressions. The argu- ment, "Today is Monday, because yes- terday was Sunday," for example, is deductively valid in the sense that the 66 David Hitchcock meanings of the premiss and the con- clusion make it impossible for the pre- miss to be true and the conclusion false, but its validity rests partly on the meaning relations between "today" and "yesterday" on the one hand and between "Monday" and "Sunday" on the other hand. Such arguments can always be made formally deductively valid by adding premisses which are true by definition; in the example, we might add the premisses that yester- day is the day before today, Sunday is the day before Monday, and the days before identical days are identical days. Since the converse proposition is also true (any argument is deductive- ly valid which can be made formally deductively valid by adding definition- ally true premisses), we can define a deductively valid argument as an argu- ment which is either formally deduc- tively valid or can be made so by the addition of one or more definitionally true premisses. Distinguishing Enthymemes from Non-Sequiturs Having separated off non-deductive arguments and deductively valid argu- ments, how are we to distinguish within the rest between enthymemes and mere non-sequiturs? A tempting approach is to regard the enthymemes as the arguments among this set whose authors have omitted one or more premisses. That is, the question would be whether the ar- guer had an additional premiss in mind, but left it unstated, for example because she took it to be common knowledge (see again Aristotle's Rhetoric I 2 1357s17-19) or because she wished to protect it from unwelcome criticism. We should reject this ap- proach, for two reasons. First, we are often not in a position to question the arguer about whether she had another premiss in mind, and so must fall back on textual rather than psycho- logical criteria, which will need to be supplied. Second, and more important- Iy, authors of acknowledged enthy- memes often have no additional pre- miss in mind. To take an everyday autobiographical example, I recently reasoned that it would not be difficult to find a house in a nearby city for which I had been given directions, be- cause the house was just off the main road. This simple piece of reasoning is obviously an enthymematic argu- ment, but I was not conscious of having omitted a premiss in articulating it- especially since I articulated it to my- self before later verbalizing it to some- one else. I invite the reader to try the same exercise with her or his own re- cently formulated enthymematic argu- ment; I doubt that you wi II be conscious of having omitted a premiss. This fact, which supports the view that enthy- memes do not have missing premisses, obviously makes it impossible to iden- tify enthymemes as arguments whose authors omitted a premiss. A second tempting strategy is to limit enthymemes to arguments which can be made deductively valid by adding a premiss. This "limitation", however, is no limitation at all, for any argument can be made deductively valid by adding as a premiss the state- ment that, if the premisses are true, the conclusion is true. Let us call this statement the argument's associated conditional. It is the conditional state- ment whose antecedent is the conjunc- tion of the argument's explicit premiss- es and whose consequent is the argu- ment's conclusion. This conditional statement can be regarded, in fact, as making explicit at least part of the claim which the arguer implicitly makes in inferring the conclusion from the premiss(es). To infer a conclusion from given premiss(es) is to assume that the conclusion follows from the premiss(es), and the conditional statement articu- lates this assumption. An unwelcome consequence of the strategy of regarding an argument as an enthymeme if it can be made de- ductively valid by adding a premiss is that arguments whose premisses have no connection to their conclusion turn out to be enthymemes. "Two plus two equals four, so Ulan Bator is the capital of Outer Mongolia," for example, would be an enthymeme, since it can be made deductively valid by adding the premiss, "If two plus two equals four, then Ulan Bator is the Capital of Outer Mongolia." On a truth-func- tional interpretation of the conditional, of course, this added statement is true, and so the expanded argument turns out to be formally valid and have true premisses. But the only way of showing that the assumption is true is to show that its consequent (Le. the conclusion of the original argument) is true, so that the expanded argument is ques- tion-begging. So the argument is not a good one. Rather than going through such an involved discussion, we might prefer simply to say that the conclu- sion does not follow, that the argument is a mere non sequitur. But how are we to distinguish such non-sequiturs from enthymemes? Our example indicates that an argu- ment is a non-sequitur if its associated conditional can only be shown to be true by showing that the conclusion is true. This condition obtains when the argument's premises are irrelevant to its conclusion. An obvious form of such irrelevance is the absence of any con- nection in meaning between the pre- misses and the conclusion. Such a meaning connection is absent when there is no content expression common to a premiss and the conclusion, even implicitly. The presence of a common content expression, or the ability to produce a common content expression by making definitionally equivalent substitutions, would make the pre- miss(es) relevant to the conclusion in this sense. Let us call this sense of relevance topical relevance of the pre- mise(s) to the conclusion. We might also be tempted to regard an argument as a non-sequitur when its premiss is irrelevant to its conclusion in a more substantive sense. That is, there is a common content expression, but the I premis~es\ don't seem to pro- vide any support for the conclusion. Suppose someone argues that Saman- tha is trustworthy because she has red hair. What does having red hair have to Enthymematic Arguments 87 do with being trustworthy, we might respond. The premiss is irrelevant, and the conclusion just does not follow. Although this reaction is natural and ultimately defensible, I prefer to count such arguments as enthymemes and to rest the judgment of their inade- quacy on a substantive verdict about the falsehood of the implicit assump- tion in virtue of which their conclusion follows from their premiss(es). My rea- son for doing so is that irrelevance is a slippery concept, easy to misuse as a term of apparent logical criticism, and I would prefer to confine its ap- plication to cases where the criteria are clear and genuinely logical. We should beware of theories of argument which disguise substantive objections to claims and arguments in termino- logy which sounds purely logical. We also want to count as non-sequi- turs formal fallacies, such as affirming the consequent and denying the ante- cedent. The problem with such argu- ments is that their premisses are too topically relevant to their conclusions. That is, every content expression oc- curs at least twice. To explain why excessive topical relevance is a prob- lem, I need to anticipate the results of the second section of this paper. There I shall argue that an enthymeme implicitly assumes a universal general- ization of its associated conditional over its repeated content expressions, in fact the maximal generalization consis- tent with plausibility. Since a formal fallacy is by definition invalid and con- tains no un repeated content expres- sions, the maximal generalization of its associated conditional will be a purely formal principle which is a logical falsehood. Suppose, for exam- ple, that someone argues that Charles works with graphite on the ground that he has black stains on his hands which people who work with graphite have. The maximal universal general- ization of this argument's associated conditional is that any entity has a pro- perty if that entity has another property and any entity with the first property has the second property. (For any x, F and G, x is F if x is G and whatever 88 David Hitchcock has F has G.) Less maximal general- izations, admittedly, might have some plausibility. It might be that any in- dividual works with graphite if that individual has black stains on his hands and everyone who works with graphite has black stains on his hands. I confess that I do not know how to respond to this problem. If pressed, 1 would allow formal fallacies as enthymemes and evaluate them on the basis of the implicit assumption in virtue of which the conclusion follows from their pre- misses. One way of rejecting some formal fallacies as non-sequiturs is to point out that their associated conditional, if added as a premiss, would make an existing premiss redundant. Thus the conclusion cannot be made to follow deductively from the whole set of original premisses. I once thought this fact made such arguments non-sequi- turs, but have abandoned this view, for three reasons. First, since deduc- tively valid arguments with redundant premisses are still deductively valid, why shouldn't enthymemes with re- dundant premisses still be enthy- memes? Second, the alleged redun- dancy of an existing premiss depends on the controversial truth-functional in- terpretation of the conditional. Third, this criterion does not rule out all for- mal fallacies as non-sequiturs. For ex- ample, if we add as a premiss the condi- tional associated with the argument in the preceding paragraph that Charles works with graphite f none of the original premisses becomes re- dundant. I conclude that enthymemes differ from non-sequiturs in that their pre- misses are partically topically relevant to their conclusions. That is, at least one content expression occurs, perhaps implicitly, in both the premisses and the conclusion. And at least one content expression occurs only once. The reader will be able to think of apparent enthymemes which do not appear to meet this criterion of partial topical relevance. Suppose someone says, "It is cold, so I should put on my coat." (lowe the counter-example to Robert Ennis.) We would count this argument as an enthymeme, but there is no common content expression, even if we substitute definitionally equivalent sentences for the premiss and conclusion. There is, however, a temporal adverb "now" implicit in the present tense of both verbs. This adverb can be regarded as the repeated content expression, and thus the argument is an enthymeme after all . An awkward consequence of this extension of the criterion of partial topical relevance is that some argu- ments which were excluded as non- sequiturs come back in to the class of enthymemes. We can still keep out the argument from a truth of arith- metic to a truth of geography, since truths of arithmetic do not come with an implicit temporal adverb. But an argument, for example, that Washing- ton is the capital of the United States because Ulan Bator is the capital of Outer Mongolia will have to count as an enthymeme. The inadequacy of such an argument will have to rest on the inadequacy of the implicit assumption in virtue of which its con- clusion follows from its premiss. The Universal Generalization Thesis The standard approach to evaluating the inference in an enthymematic argument is to identify and evaluate the implicit assumption in virtue of which the conclusion follows from the premiss(es); if it is true, the enthyme- matic inference is valid, but if false, invalid. A variant allows an en thyme- matic inference to be invalid where the implicit assumption is true but insuffi- cient to make the original argument deductively valid if it is added as a pre- miss. The standard approach typically regards the implicit assumption as an unexpressed, missing, unstated, tacit or even suppressed premiss of the enthymematic argument; for examples of each term, see respectively van Eemeren and Grootendorst (1984), Govier (1985), Scriven (1976), Hitch- cock (1983) and Thomas (1981). I shall argue later that the implicit assump- tion is better regarded as a non-formal rule of inference, but nothing in what immediately follows depends on this position. Since our purpose is evaluation, we should look for an assumption on which the argument depends, regardless of whether the arguer had such an as- sumption in mind, rather than an assumption the author had in mind, which may be neither necessary nor sufficient for the conclusion's follow- ing from the premiss(es). Robert Ennis (1982) used the terms "needed as- sumption" and "used assumption" for these two types. I propose instead to use the terms I, argument's assump- tion" and "arguer's assumption", for two reasons. First, as Ennis holds and I am about to argue, an enthyme- matic argument assumes more than is strictly needed to make the conclusion follow from the premiss(es). Second, an arguer uses the argument's assumption in drawing a conclusion, even if she is not aware of having done so. So in what follows we are looking for a general characterization of the assumption of an enthymematic argument which is im- plicit in inferring its conclusion from its premiss(es). I call the assumption "im- plicit" rather than "unstated" because "unstated" suggests something the arguer had in mind. An enthymematic argument, we have seen, assumes at least the truth of the argument's associated conditional. But, I suggest, it assumes more. Con- sider the argument, " Depo-Provera is safe, because it is an effective contra- ceptive." At the time of writing, this argument's premiss was accepted as true, but its conclusion was contro- versial. Suppose, however, that the conclusion is true. On a truth-functional interpretation of the conditional r the associated conditional "If Depo- Provera is an effective contraceptive, then Depo-Provera is safe" is true. Other interpretations of the conditional either make the associated conditional true or require us to determine whether the consequent follows from the ante- Enthymematic Arguments 89 cedent, which is the question we are trying to answer. So, if we take the argument to be assuming only the truth of the associated conditional, we are driven to say either that the conclusion follows or that we are in the dark as to whether it does. But in fact we know that it does not follow, that the argu- ment is a bad one. The mere fact that something is an effective contraceptive, we might say, does not show that it is safe. We might even be able to cite an example of another drug which is an effective contraceptive but is not safe, say the Dalkon Shield. These responses are irrelevant if an enthymeme as- sumes only its associated conditional. They are relevant, and conclusive, if an enthymeme assumes a un iversal generalization of its associated condi- tional with respect to at least one re- peated content expression. Let us call the thesis that an enthymematic argu- ment implicitly assumes the truth of a universal generalization of its asso- ciated conditional with respect to at least one repeated content expression the universal generalization thesis. The thesis just mentioned is equi- valent to supposing that one can object to an enthymematic argument by pro- ducing a parallel argument with true premiss(es) and a false conclusion, obtained from the original by uniform substitution on one or more repeated content expressions. If we think it legitimate to respond, "You might as well say that the Dalkon Shield is safe because it is an effective contracep- tive" , where it is known that the Dalkon Shield is an effective contraceptive but not safe, then we accept the uni- versal generalization thesis, at least for this argument. The above remarks do not prove the universal generalization thesis. They do, however, make it plausible. My strategy in what follows will be to make it more plausible by showing that the implicit assumption produced by the application of the thesis conforms tolerably well to our intuitive judg- ments, as well as to the theory of enthymemes in traditional logic, and that there are good explanations for 90 David Hitchcock its divergence from our intuitions. Confirmation of the Universal Generalization Thesis Consider first an argument of a common type, in which premiss and conclusion have the same grammatical subject but different grammatical predicates. The logician's favourite example is the sentence, "Socrates is a man, so Socrates is mortaL" The universal generalization of this argu- ment's associated conditional is the sentence, "For any x, if x is a man, x is mortal," or in standard English, "Every man is mortaL/f According to the universal generalization thesis, this is the only possible implicit as- sumption of the argument, since "So- crates" is the only content expression which occurs more than once in the associated conditional. Thus, we can concl ude that someone who advances the argument, "Socrates is a man, so Socrates is mortal," is committed to the proposition that every man is mortal. And this is what we intuitive- ly think. There are arguments where we in- tuitively think that the implicit assump- tion is a particular statement. For example, we would suppose that some- one who argues, "Depo-Provera is safe because any drug is safe which has been approved at all levels of the drug testing procedure in the United States", is implicitly assuming that Depo-Provera has been approved at all levels of the drug testing procedure in the United States. Since the uni- versal generalization thesis holds that the implicit assumption is always a universal generalization, our intuitive judgments about these arguments might seem to conflict with the thesis. But, surprisingly, in cases of this kind the universal generalization in ques- tion is equivalent to a particular state- ment. In abbreviated form, the asso- ciated conditional of the above argu- ment is the sentence, "If any consist- ently approved drug is safe, then Depo- Provera is safe." Its universal general- ization is the sentence, "For any F, if any consistently approved drug is F, then Depo-Provera is F," or, in somewhat more standard English, "Depo-Provera has every property which every consistently approved drug has off But this sentence is equivalent to the sentence, "Depo-Provera is a conSistently approved drug" We can demonstrate this equivalence by deducing each sentence from the other. One property which every consistently approved drug has is that it is a con- sistently approved drug. So, if Depo- Provera has every property which every consistently approved drug has, then it is a consistently approved drug. But, conversely, if it is a consistently approved drug, then it will have every property that every consistently ap- proved drug has, since it is one of the consistently approved drugs, Consider next an enthymeme of the kind recognized by traditional logic, that is, an argument which can be filled out so as to become a two-premiss syllogism in one of the moods recog- nized as valid by the Aristotelian tradition. Consider the argument, "No man has feathers, so no man is a bird. If Since just one content expres- sion, "man", appears in the associated conditional, "If no man has feathers, then no man is a bird," the universal generalization thesis implies that the implicit assumption of this argument is the sentence, "For any F, if no F has feathers, then no F is a bird," that is, "Any non-feathered thing is not a bird," or, contraposing, "Every bird has feathers," Th i sis exactly the as- sumption which "traditional logic" would supply on the basis of its recog- nition of the argument as an incomplete second-figure assertoric syllogism. As can be verified by complete enu- meration, this coincidence of results obtains for all incomplete assertoric syllogisms. We find the same coincidence of results for arguments which we would intuitively recognize as incomplete instances of arguments deductively valid in virtue of the sentence-forming expressions "not," "and," "or" and "if". Consider, for example, the argu- ment. "John is asleep, because he's asleep when the television is off." We would intuitively recognize an in- complete modus ponens argument of the form, "lf p then q, and p, so q." The implicit assumption is intuitively that the television is off. The argu- ment's associated conditional is the sentence, "J ohn is asleep, provided that, if the television is off, he's asleep." Here the universal generaliza- tion thesis allows us to generalize on the words 1/ J ohn" or /I asleep," but, as I shall argue later, we are entitled to generalize on the molecular content expression, II John is asleep." The re- sulting sentence is, "For any p, p, provided that, if the television set is off, p," or, in slightly more standard English, "Any proposition at all is true if this proposition follows from the proposition that the television set is off." But this sentence is equivalent to the proposition that the television set is off. (The equivalence can be demonstrated by assuming each sen- tence in turn and proving the other on its basis. To prove the particular state- ment, instantiate the generalization with the sentence "The television set is off" and detach the logically true antecedent, "If the television set is off, the television set is off." To prove the universally generalized conditional, assume its antecedent for an arbitrary sentence q and use the particular state- ment to detach the antecedent of this antecedent, thus deriving the conse- quent of the larger conditional; then conditional ize and general ize over q.) Similar coincidences of results between our intuitive judgments and the application of the universal general- ization thesis apply to other incomplete examples of forms of argument which are deductively valid in virtue of the meanings of "not," "and," "or" and "if." For some arguments, however, the universal generalization thesis gives a result different from our intuitions. As far as I have been able to determine, the intuitively supplied assumption is either a stronger assumption from Enthymematic Arguments 91 which we can deduce the universal generalization of the associated con- ditional or a weaker assumption which can be deduced from the uni- versal generalization of the associated conditional. An example of the first type of discrepancy, supplied by Mary Ri- chardson, occurs with the enthyme- matic argument, "x and y have started wars, so some generals have started wars./I We would intuitively suppose that this argument assumes that x and yare generals. The universal generalization of the associated condi- tional-that some generals have every property which x and y have-is a weaker statement which follows from the intuitively supplied assumption that x and yare generals. For, if x and y are generals, then some generals- namely x and y - have every property which x and y have. The intuitively supplied assumption here supplies the most obvious backing for the mechanic- ally derived assumption. Curious Iy our earliest explicitly labelled enthy- meme-Aristotle's example of the argument, "Doreius has won a crowned contest, for he has won in the Olympic games" -is of this type. (It is also not an incomplete categorical syllogism, unless one recasts the argument very awkwardly.) The universal gen- eralization of the associated condi- tional is that anyone who has won in the Olympic games has won a crowned contest, a claim compatible with the crowned contest in question being different from the Olympic games. The intuitively supplied assumption, which Aristotle regards as unexpressed because everybody knows it, is that the Olympics is a crowned contest. This claim is stronger than the univer- sal generalization, and again supplies the most obvious backing for it. In these cases, then, the universal gen- eralization thesis conforms to our in- tuitions to the extent that the assump- tion it supplies is at least part of what our intuitions tell us the argument assumes. Without background knowl- edge, it can be argued, our intuitions could play us false in such cases. 92 David Hitchcock An example of the second type of dis- crepancy, in which the intuitively sup- plied assumption is weaker than the associated conditional's universal generalization, arises with the argu- ment, "All socialists support trade unions, so you are a socialist." (This example too comes from Mary Richard- son ,) We would intuitively supply as the implicit assumption the claim that you support trade unions. And we would then go on to criticize the resul- ting argument as invalid, since it is an example of affirming the consequent which is not valid on other grounds. The universal generalization thesis, however, tells us that the argument assumes that, for any property F, if everyone who has F supports trade unions, then you have F. In somewhat more standard English: You have every property whose possessors all support trade unions. Taking the property of supporting trade unions as one such property, we can derive by instantia- tion the intuitively supplied assump- tion that you support trade unions. Since we cannot make a converse deri- vation, the assumption postulated by the universal generalization thesis is stronger than the intuitively supplied assumption. Is the universal general- ization thesis therefore too strong? I think not. The intuitively supplied assumption is a reasonable conjecture about the arguer's assumption, what the arguer thought licensed his in- ference of the conclusion from the pre- miss. But this reasonable conjecture is an assumption which is insufficient to make the conclusion follow, and which therefore cannot serve as the argument's assumption, the principle in virtue of which the conclusion follows from the premiss, to which the arguer implicitly commits himself in drawing the conclusion, Since our purpose is to evaluate the inference in an enthymematic argument, we should supply an assumption which is sufficient to make the conclusion follow, and investigate the truth of that assumption, The universal gene- ralization thesis gives us such an assumption, whereas our pre-theoretic- al intuitions do not, Qualifications of the Universal Generalization Thesis In discussing the enthymematic argument that John is asleep, because he is asleep when the television set is off, I mentioned that, where an argu- ment contains repeated molecular content expressions, the universal generalization thesis is inderterminate as to whether one should generalize over the molecular repeated content expressions, over atomic repeated con- tent expressions separately, or only over some of them, and if so which ones. Here the intuitively correct reso- lution of the indeterminacy seems to occur if one generalizes over content expressions which are as molecular as is plausible, In other words, if a molecular content expression is re- peated, one generalizes over the entire expression rather than over one of its constituent content expressions, or over each constituent content expres- sion separately-unless it would be implausible to do so. The universal generalization thesis is indeterminate in a second respect. If the associated conditional contains more than one repeated content ex- pression, where these are not part of a single molecular content expression, the thesis does not tell us which of these expressions we are to general- ize over; an argument which reveals this indeterminacy is the argument, "Marijuana should be legalized, be- cause it is no more dangerous than al- cohol, which is already legal," where we are not sure whether to generalize over all or only some of the repeated content expressions "marijuana," "alcohol" and "legaL" The intuitively correct resolution of this indeterminacy is to generalize over each of the re- peated content expressions - unless it would be implausible to do so. For example, the argument for legalizing marijuana assumes that any substance which is no more dangerous than an already legal substance should be legal- ized. If we generalize only with respect to "marijuana," we get the implicit assumption, "If alcohol is already legal, anything which is no more dangerous than alcohol should be legal- ized." If the argument depended only on this assumption, then it would be irrelevant to object that by the same reasoning one would have argued in the nineteenth century that heroin should be legalized, since it is no more dangerous than opium, which is already legal. But this objection seems rele- vant. So it seems justifiable to general- ize the associated conditional with res- pect to both "marijuana" and "alco- hol," producing the result that con- forms to our intuitive judgment. Note that it is not so plausible to general- ize with respect to the content expres- sion "legal." If we did so, we would attribute to the argument the assump- tion that any substance which is no more dangerous than another sub- stance should be given all the proper- ties which the other substance has. Such an assumption is absurd, because, for example, it is impossible to give marijuana the chemical properties of alcohol. Apart from these confirmations by our intuitive judgment, the justifica- tion for broad and multiple general- ization is that arguments are implicitly general, so that any repeated content expression is a candidate for general- ization. The justification for making exceptions on grounds of implausibility is the principle of charity: in case of ambiguity, interpret a passage in the way in which it makes the best possible case. The last example exhibits a third and final indeterminacy in the universal generalization thesis. Over what class or category should we generalize the repeated content expression(s)? For some arguments, the class or category is a matter of indifference, since it drops out in the simplification of the generalized conditional. For other argu- ments, the class or category makes a difference. I generalized "alcohol" and "marijuana" over the class of substances. If one generalized over Enthymematic Arguments 93 kinds of entities, or over any item whatever, one could easily find objec- tions to the implicit assumption thus generated. For example, driving a car without a seat belt is no more dan- gerous than hang-gliding, which is legal/but not everyone accepts the pro- position that driving a car without a seat belt should be legalized. But such an objection seems unfair. The argu- ment involves a comparison of the danger of two substances, in parti- cular, of two mood-altering drugs, and it seems unreasonable to extend the principle on which the argument is relying beyond this subcategory. In some cases the context will impose restrictions on the class over which to generalize. Robert Ennis (1969) gives an example of a teacher asking a group of elementary school pupils to say whether words ending in "-ing" are participles or gerunds in given sen- tences. Asked to justify his claim that a given word is a gerund, a pupil replies, "Because it is the subject of a sentence." This is a good justifica- tion, but the generalization, "Every subject of a sentence is a gerund," is false. The context of utterance of the pupil's argument indicates that we should generalize the associated condi- tional only over the class of words ending in "-ing". Doing so, we get the sentence, "For any word ending in '-ing', if it is the subject of a sentence, it is a gerund," or, in more standard English, "Every word ending in '-ing' which is the subject of a sentence is a gerund." In short, the appropriate qualifica- tion of the universal generalization thesis seems to be that each general- ized content expression should be generalized over the entire category to which it belongs, unless the context or considerations of plausibility indi- cate a restriction on this category. We are now in a position to articu- late the fully qualified version of the universal generalization thesis: The author of an enthymematic argument implicitly assumes the truth of a universal generalization of 94 David Hitchcock the argument's associated condition- al with respect to one or more con- tent expressions which occur more than once. Unless it would be im- plausible, where a molecular content expression is repeated, this general- ization is over the most molecular repeated content expression. If more than one distinct content expression is repeated, this generalization is over all such distinct content ex- pressions except those over which it would be implausible to general- ize. Unless the context of utterance of the argument or considerations of plausibility indicate a restriction, the generalization is over the entire category of items within which the content expression's significatum occurs. Missing Premisses or Rules? I now turn to the question hinted at earlier of whether we should regard the implicit assumption in virtue of which an enthymeme's conclusion fol- lows from its explicit premiss(es) as a missing premiss of the enthymeme. Although this interpretation of the im- plicit assumption, is the usual one, it is problematic. First, we ordinarily define an argu- ment as a set of statements, one of which, the conclusion, is advanced on the basis of the other(s), the pre- miss(es). To say that an argument has a given premiss is to say that that state- ment is a member of the set. But by definition a missing premiss is not a member of the set; it is not a state- ment, because it is not stated. So, in saying that an argument has a mis- sing (or unexpressed, or tacit, or un- stated, or suppressed) premiss, we seem to be saying that an argument has a premiss which it does not have. One can avoid the self-contradiction just expressed by redefining the con- cept of argument to include among the premisses sentences which the arguer had in mind but left unstated. A second problem, however, arises. To regard an enthymeme's implicit assumption as a mlssmg premiss is to regard the argument as somehow defective or in- complete. But most deductive argu- ments, I would guess, are enthyme- matic, and even the most logically acute among us are prone to utter en- thymematic arguments. We should therefore be suspicious about a theory which regard enthymematic arguments as incomplete. A common response to this problem is to explain the frequency of such alle- gedly logically defective arguments by their superior rhetorical effectiveness. We have the authority of Aristotle, in the aforementioned passage from the Rhetoric, for the view that orators, in order to make their arguments brief enough for audiences to follow, will omit premisses which the hearer can supply because everybody knows them, The trouble with this explana- tion, and in my view the most serious objection to regarding an enthymeme's implicit assumption as a mlssmg premiss, is that we are unaware of having omitted a premiss when we advance an enthymeme, especially when we do so to convi nce ourselves. We should, I conclude, be skeptical of the claim that enthymemes are logic- ally incomplete, with a missing pre- miss. The standard alternative to the mis- sing premiss approach is to take the implicit assumption of an enthyme- matic argument as the articulation of a rule of inference in virtue of which the conclusion follows from the pre- miss(es). This rule approach can be found in Toulmin (1958), who seems to have got it from Gilbert Ryle (1954). It is also adopted by Rolf George (1972, 1983), who gets it from the nine- teenth century logician Bernard Bolzano (1837). The rule in question will be a non-formal rule of inference, in the sense that the statement of the rule will include at least one content expression. If this rule is implicit, nothing is missing from the enthymeme which ought from a logical point of view to be stated, just as there is no omis- sion if a formal rule of inference like modus ponens is not stated when a conclusion is drawn in accordance with it. Regarding the implicit assumption as a rule makes it possible to evaluate an enthymematic inference without stating the implicit assumption. The procedure is a modification of the pro- cedure of counter-exampling described above as a method of testing for formal deductive validity. Just as a substi- tution on the atomic content expres- sions of an argument which produces an argument with true premisses and a false conclusion will show that the for- mal rule of inference in accordance with which the original argument's conclusion follows from its premiss(es) is invalid, so a substitution on the re- peated content expressions which pro- duces an argument with true premisses and a false conclusion will show that the non-formal rule of inference in accordance with which the original argument's conclusion follows from its premiss(es) is invalid. In the first case we say that the argument is not formal- ly deductively valid. Let us say in the second case that the argument is not enthymematically valid. This concept of enthymematic validity is due to Rolf George (1972, 1983), following Bolzano (1837). To define this concept, we need the concept of an enthymeme's variable content expressions, the repeated con- tent expressions over which one generalizes in articulating its implicit assumption; the criteria for their iden- tification appear in the qualified ver- sion of the universal generalization thesis. We also need the concept of a permissible substitution, the sub- stitution for a variable content expres- sion of a content expression which belongs to the class or category over which that variable content expression is generalized in articulating an enthy- meme's implicit assumption; the crite- ria for delimiting this class or category also appear in the qualified version of the universal generalization thesis. With these concepts, we can define an argument as enthymematically valid if and only if no uniform permissible substitution on its variable content ex- pressions produces an argument with Enthymematic Arguments 95 true premisses and a false conclusion. To show that an enthymeme is (en- thymematically) invalid, therefore, we simply need to construct an appro- priately parallel argument with true premises and a false conclusion. I have already given some examples of this procedure. Thus, the desirability of legalizing marijuana does not follow from the fact that marijuana is no more dangerous than alcohol, which is al- ready legal: opium was legal in the nineteenth century and is no more dangerous than heroin, but it was not desirable at that time to legalize heroin. And to the argument for the safety of Depo-Provera, one can reply that you might as well say that the Dalkon Shield is safe because it is an effective contraceptive. As with formal deductive validity, inability to construct such a counter- example does not prove enthymematic validity, since the inability might be due simply to a failure of imagina- tion. To prove enthymematic validity, one needs to make the implicit assump- tion explicit and if necessary to support it with argument. The premisses of such supporting arguments are what Toulmin (1958) calls backing and Ennis (1982) backups, in this case for an im- plicit assumption rather than an ex- plicit premiss. They are what Scriven (1976) refers to by the expression "optimal assumptions": the best basic support one can find for the drawing of the stated conclusion from the stated premisses. Other Purposes for Identifying Enthymeme's Assumptions So far I have been discussing the task of identifying an enthymeme's impli- cit assumption for the purpose of evalu- ating the enthymeme's inference. A survey of recent philosophical literature shows that philosophers at least some- times have more specific and pointed reasons for identifying such implicit assumptions. It is instructive to con- sider the variety of such purposes and the way they modify the criteria for 96 David Hitchcock identifying the assumption. Barnes (1975), for example, routine- ly fills in Aristotle's arguments, relying on the entire Aristotelian corpus, to try to understand why Aristotle thought his conclusions followed from his pre- misses. For this purpose, which is that of identifying the arguer's assumption rather than the argument's assump- tion, evidence of the arguer's beliefs will help to resolve ambiguities about the argument's implicit assumption, and may furnish backing for the as- sumption one can reconstruct using just the argument itself. Reconstruc- tion of the arguer's assumptions will be guided, on the basis of the prin- ciple of charity, by a presumption that they do genuinely license the inference involved-that is, that the argument will become deductively valid if the arguer's assumptions are added as extra premisses. Philosophers such as Bertrand Russell (1948), David Palmer (1972), David Bryant (1972), Norman Geisler (1973, 1978) and Stefan Nowak (1978) supply an additional premiss to strengthen an apparently flawed argu- ment by showing that the addition of a plausible premiss makes it a good argument. For this purpose, a premiss somewhat stronger than the assump- tion implicit in the argument itself may be appropriate. Geisler (1978) and R. A. Fumerton (1980) supply an extra premiss in order to seek support for their own position by showing that the author of an argu- ment implicitly supports that position. For this purpose, one needs to be as charitable as possible to the author of the argument, since one needs to claim that any defensible filling out of the argument commits its author to one's own position. Lewis Ford (1975) supplies addi- tional premisses to discredit an argu- ment by showing that any added pre- misses sufficient to make it deductively valid are false. This purpose also re- quires as charitable as possible a filling out of the argument's premisses. Note I presented earlier versions of portions of this paper at the June 1983 Second International Symposium on Informal Logic in Windsor, Ontario; at the De- cember 1983 session of the Association for Informal Logic and Critical Thinking held in conjunction with the Eastern Division meetings of the American Philosophical Association in New York City; at the Third International Confer- ence on Critical Thinking and Educa- tional Reform in July 1985 at Sonoma State University in Rohnert Park, California; at Conference '86 on Criti- cal Thinking at Christopher Newport College, Newport News, Virginia, in April 1986; and at the 30th annual congress of the Canadian Philosophical Association in Winnipeg, Manitoba in May 1986. I am grateful for these opportun ities to receive feedback on my ideas. 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Logical and em- pirical assumptions of validity of inductions. Synthese 37: 321-349. Palmer, David. (1972). What the tor- toise said to Aristotle (about the practical syllogism). New Scholas- ticism 46: 449-460. Perelman, Chaim, and L. Olbrechts- Tyteca. (1958). The New Rhetoric: A Treatise on Argumentation. Trans- lated by John Wilkinson and Purcell Weaver. Notre Dame. This paper- back edition published in 1971. Russell, Bertrand. (1948). Human Knowledge: Its Scope and Limits. Allen & Unwin. Ryle, Gilbert. (1954). Dilemmas. Cambridge. Scriven, Michael. (1976). Reasoning. McG raw-H i II. Thomas, Stephen N. (1981). Practical Reasoning in Natural Language. Second Edition. Prentice-Hall. Toulmin, Stephen. (1958). The Uses of Argument. Cambridge. Professor David Hitchcock, Department of Philosophy, McMaster University, 1280 Main Street West, Hamilton, Ontario l8S 4K1 0