The Asymmetry Thesis and the Diversity of "Invalid" A rgumen t-Formsl GEORGE BOWLES Abstract: According to the Asymmetry Thesis, whereas there are many kinds of argument-forms that make at least some of their instances valid, there is none that makes any of its instances invalid. To re- fute this thesis, a counterexample has been produced in the form of an argument-form whose premise-form's instances are all logically true and whose conclusion form's instances are all logically false. The pur- pose of this paper is to show that there are many more kinds of argument-forms that make some of their instances invalid and that, hence, are counterexamples re- futing the Asymmetry Thesis. Resume: Selon la These d' Asymetrie, quoi qu'il y a plusieurs genres des formes-des-arguments qui fassent valide au moins quelques-uns de leurs cas, iI n'y en a aucune qui fait invalide un seul cas. Pour demontrer la faussete de cette These, on a produit comme example contraire une forme-des-arguments dont les cas de la forme de sa premisse sont tous vrais logiquement et dont les cas de la forme de sa conclusion sont tous fausses logiquement. Cette com- munication a Ie but de montrerqu'i1 y a beaucoup plus des genres des formes- des-arguments qui font invalides quelques uns de leurs cas et qui, donc, sont des examples contraires qui demontrent la faussete de la These d 'Asymetrie. Keywords: Asymmetry Thesis, Gerald J. Massey, argument-forms, invalidity, Hasty Generalization. I. The Asymmetry Thesis and its defense In his essay "Are There Any Good Arguments That Bad Arguments Are Bad?", Gerald J. Massey asserted what was later called2 the "asymmetry thesis"- namely, that "at the present stage oflogical theory our ability to prove validity totally eclipses our ability to show invalidity" (1975a: 61-62). The Asymmetry Thesis consists of two claims, one affirmative and un- controversial, the other negative and controversial: the affirmative claim is that we can prove, on the basis oftheir possession of some form or other, that some arguments are valid;3 the negative claim is that we cannot now prove, on the basis of its possession of any form, that any argument is invalid. The reason Professor Massey gave for his controversial negative claim was that no argument-form that we can presently specify renders invalid ant argu- ment that has it. For any argument possessing an "invalid"5 argument-form like ©In/ormalLogic Vol. 19, No.1 (1999): pp.67-76. 68 George Bowles Form 1 or Form 2 Premise 1. p. Premise 2. q. Conclusion. r. Premise 1. lfp, not-q. Premise 2. Not-q. Conclusion. p. may possess another form that makes it valid (Massey, 1975a: 64-65).6 In this paper I shall be concerned with the claim Professor Massey gave as his reason for the negative half of the Asymmetry Thesis-namely, the claim that no presently specifiable argument-form makes any of its instances invalid.? I shall first recount a prior refutation of that claim, then give a new one, and finally offer a possible explanation for what appears to be Professor Massey's error. II. Prior refutation of the claim The question is: Can we specify any argument-form that makes any of its instances invalid? It seems that we can. For, as Gary Iseminger pointed out, we can specify argument-forms that, like Form 3 Premise. p or not-po Conclusion. q and not-q. have a premise-form all of whose instances are logically true and a conclu- sion-form all of whose instances are logically false; and any argument having such a form is invalid (Iseminger, 1989: 35).8 This kind of argument-form (namely, one whose premise-form has only logically true instances and whose conclusion-form has only logically false instances) is the only one the possession of which makes all of its instances invalid. For any other kind of argument-form might have instances whose premises were logically false or whose conclusions were logically true, in either of which cases the argument would be valid because of the paradoxes of strict implication (Lewis and Langford, 1959: 250-251). For example, the form of Affirming the Consequent Form 4 Premise l. Ifp,q. Premise 2. q. Conclusion. p. has at least one instance-namely, Argument 2 Premise 1. Premise 2. Conclusion. The Asymmetry Thesis and "Invalid" Argument-Fonns 69 If either all men are mortal or some men are not mortal, either some snakes are not spotted or all snakes are spotted. Either some snakes are not spotted or all snakes are spotted. Either all men are mortal or some men are not mortal. that is made valid by the logical truth of its conclusion. Only an argument- fonn that excludes the possibility of either a logically false premise or a logi- cally true conclusion can also exclude the possibility of instances that are valid because of the paradoxes of strict implication, and only a fonn like Fonn 3 can do that. III. Present refutation of tbe claim At this point an adherent of the Asymmetry Thesis might object in the follow- ing way: "Even though we can specify one kind of argument-fonn that makes invalid every argument that has it, still that is a 'special case:9 whereas we know many kinds of argument-forms that make valid every argument that has them. Many invalid arguments do not have a fonn like Form 3, so their inva- lidity cannot be proved by reference to any known fonn that they possess. So, there is still a one-versus-many asymmetry in our knowledge of formal detennination of invalidity and validity." The purpose of this paper is to answer this objection, with its weakened version of the Asymmetry Thesis, by pointing out that there are many addi- tional kinds of argument-fonns known to us that make some of their instances invalid-namely, those instances (hereinafter to be called 'standard instances') that are free of such countervailing features as logically false premises or logically true conclusions. 10 Consider Form 5 Premise. Most xs are ys. Conclusion. This x is a y. It does not guarantee that all of its instances are invalid, since some of them are valid either because their premise is logically false or because their conclu- sion is logically true. For instance, Argument 3 Premise. Conclusion. Most logic papers are nonpapers. This logic paper is a nonpaper. is an instance of Fonn 5, and yet it is valid, because its premise is logically false. Still, although Fonn 5 does not guarantee that all of its instances are invalid, it does guarantee that some are. For it detennines that each of its standard instances is such that the conclusion is probable (that is, probable 70 George Bowles but not certain) relative to the premises; and this means that all such instances are invalid. For example, Argument 4 Premise. Conclusion. Most logic papers are papers that are too long. This logic paper is a paper that is too long. is a standard instance of Form 5; its conclusion, then, is probable relative to its premise; II and so it is invalid. 12 Now, compare Form 5 with Form 6 Premise. Few xs are ys. Conclusion. This x is ay. Form 6 guarantees that each of its standard instances is such that the conclu- sion is improbable relative to the premise. Consequently, like Form 5, Form 6 renders invalid all such instances. For example, Argument 7 Premise. Conclusion. Few logic papers are papers that are too long. This logic paper is a paper that is too long. is a standard instance of Form 6; its conclusion, then, is improbable relative to its premise; and so it is invalid. Likewise, Form 7 Premise. No xs are ys. Conclusion. This x is a y. guarantees that each of its standard instances is such that the premise is in- consistent with the conclusion. So, like Forms 5 and 6, Form 7 renders invalid all such instances. 13 For example, Argument 8 Premise. Conclusion. No logic papers are papers that are too long. This logic paper is a paper that is too long. is a standard instance of Form 7; its conclusion, then, is impossible relative to its premise; and therefore it is invalid. Similarly, Form 10 Premise. Some xs are ys. Conclusion. This x is ay. guarantees that each of its standard instances is such that the premise is irrel- evant to the conclusion-i.e., the conclusion is neither certain, probable, im- The Asymmetry Thesis and "Invalid" Argument-Forms 71 probable, nor impossible relative to the premise. '4 Hence, like Forms 5-9, Form 10 renders invalid all such instances. For example, Argument 9 Premise. Conclusion. Some logic papers are papers that are too long. This logic paper is a paper that is too long. is a standard instance of Form 10; its conclusion, then, is neither certain, probable, improbable, nor impossible relative to its premise; and so it is invalid. There are, then, many different kinds of argument-forms that make invalid some of their instances (namely, their standard instances).'5 Some argument- forms (like Form 5) determine that the conclusion in those instances is prob- able relative to the premises, others (like Form 6) that it is improbable, others (like Forms 7, 8, and 9) that it is impossible. Some argument-forms (like Form 10) determine that the premises in those instances are logically irrel- evant to the conclusion. Among those forms that determine that the conclu- sion is probable relative to the premises, some set the probability at one value, others at another; and the same thing is true of those argument-forms that determine that the conclusion is improbable relative to the premises. Forms 5 through 10 are only examples (selected for presentation here on account of their simplicity) of argument-forms that make invalid their standard instances. A complete catalog of such forms is left as an exercise for the reader. These argument-forms severally constitute further refutation of Professor Massey's original claim that we can specify no argument-form the possession of which renders any argument invalid. And collectively they refute the weakened ver- sion of the Asymmetry Thesis that there is a one-versus-many asymmetry in our knowledge of formal determination of invalidity and validity. IV. A possible explanation of Professor Massey's error Supposing that the prior and/or present refutations presented in Sections II and III above are correct, it may be pertinent to ask how Professor Massey came to think that no known argument fonn renders invalid any argument that has it. As already related, Professor Massey seems to have concluded this on the grounds that any argument that has an "invalid" argument-form like Form 1 or 2 may possess another argument-form that makes it valid. These two argument-forms are like the simpler Form 11 Premise. p. Conclusion. q. in that, unlike Forms 5-\ 0, they do not guarantee that each of their standard instances is such that the conclusion is probable, or such that it is improbable, or such that it is impossible, or such that it is none of these, relative to the 72 George Bowles premises. In short, these forms do not determine in what logical relation the premises of such instances stand to the conclusions. Unlike Forms 5-10, then, Forms 1, 2, and 11 do not render invalid any such instances. '6 For example, not only Arguments 4, 7, 8, and 9 but also Argument 10 Premise. Conclusion. All logic papers are papers that are too long. This logic paper is a paper that is too long. are standard instances of Form II. In Argument 10, the conclusion is certain relative to the premise; in Argument 4, it is probable; in Argument 7, it is improbable; in Argument 8, it is impossible; and in Argument 9, it is none of these. Clearly, then, Form 11 does not determine in what logical relation the premises of its standard instances stand to the conclusions. Consequently, it does not render any of those instances invalidY It should now be clear what went wrong. Professor Massey selected, as his examples of "invalid" argument-forms, Forms I and 2, which, like Form II, do not determine, in any of their instances, the logical relation in which the premises stand to the conclusions. Hence, it is not surprising that arguments possessing those two forms may also possess other forms that make them valid. But Forms 1, 2, and II are not representative of the whole class of "invalid" argument-forms, which also includes such forms as Forms 3 and 5- 10. So, to conclude that what is true of Forms I and 2 is also true of all other "invalid" argument-forms is to commit the fallacy of Hasty Generalization. ls Had Professor Massey surveyed the diversity within the class of "invalid" argument-forms, he might never have propounded the Asymmetry Thesis. Notes I An earlier draft ofthis paper was read at Conference 95 on Critical Thinking and Informal Logic at George Mason University, Fairfax, Virginia, on June 17,1995. I have benefited from criticisms offered by the audience and by Informal Logic's referees. Gerald 1. Massey and Maurice A. Finocchiaro have generously provided assistance in this project. 2 By Bencivenga (1979: 249). l For what Professor Massey meant by 'valid', see Massey (1975a: 63, n. 5; 72·73). Cf. Massey (1987: 22-23). 4 Professor Massey said; " ... besides the triviallogic-indiffirent method just mentioned [namely, the method of showing that the premises are all true and the conclusion false], there is at present no way whatsoever to show that an argument is invalid" (I 975a: 64) and " ... since my thesis is a strong universal denial, to show itfalse one need only present one convincing case wherein a bad argument is proved bad by some means other than the trivial logic-indifferent one" (l975b: 46). If even one argument.form rendered invalid even one of its instances, that would constitute a proof ofthe invalidity of that instance, which would be inconsistent with Professor Massey's claim. The Asymmetry Thesis and "Invalid" Argument-Fonns 73 'I think that the tenns 'valid' and 'invalid' should be applied not to argument-fonns but to arguments, and even then only when certain infonnal conditions are taken into account (Bowles, 1991). In this paper, however, I confonn to the usage of the previous partici- pants in the discussion. 6 1 do not here attribute to Professor Massey the claim that no argument-fonn that we can presently specify has any invalid argument as an instance. That claim would have such obviously false consequences as that Argument 1 Premise 1. If Harrisburg is the capital of Pennsylvania, then Pittsburgh is not. Premise 2. Pittsburgh is not the capital of Pennsylvania. Conclusion. Harrisburg is the capital of Pennsylvania. is not both invalid and an instance of Fonn I. Rather, the claim I am here attributing to Professor Massey entails that, although Argument 1 is an instance ofFonn I, and Fonn 1 is an invalid argument-fonn, neither Fonn 1 nor any other presently specifiable argument- fonn of which Argument I is an instance renders, or makes, Argument 1 invalid. 7 Bencivenga (1979), McKay (1984), Finocchiaro (1994), and Krabbe (1995) offer criticisms of the Asymmetry Thesis that do not challenge this claim. s Massey (1987: 1) credited Oliver (1967) with the anticipation, by eight years, of his Asymmetry Thesis. If so, it is remarkable that Oliver also anticipates, by twenty-two years, Iseminger's refutation of Massey's defense of the same Thesis: Argument-fonns that are not universally valid are of two kinds: (1) those that have both a premiss-fonn all of whose instances are logically true and a conclusion-fonn all of whose instances are logically false, and (2) all others. Those of kind (I) can be used to show that arguments which are instances of them are invalid .... (Oliver, 1967: 478) 9 Iseminger (1989: 36). 10 (a) Any suspicion with which this restriction might be received may be diminished by the following two observations. First, most non-mathematical everyday arguments confonn to it. And second, similar restrictions are commonly made in discussions of the traditional square of opposition and the relevance offonn to validity and invalidity (e.g., in Copi and Cohen, 1990: 169-170, 195). (b) Although Oliver (1967: 477) suggests that the only proof that a premise is not logically false is that it is logically true, and that the only proof that a conclusion is not logically true is that it is logically false, this cannot be the whole story; for sometimes we can ascertain (but perhaps not prove) that a proposition is neither logically true nor logically false. II Objection. Fonn 5 does not detennine that each of its instances whose premise is not logically false and whose conclusion is not logically true is such that the conclusion is probable relative to the premise. For Argument 5 Premise. Most men are dissimilar in hair color to someone who shares most of the properties of this man. Conclusion. This man is dissimilar in hair color to someone who shares most of the properties of this man. has Fonn 5, and yet its conclusion is not probable relative to the premise, since the conclusion is intrinsically improbable: because of analogy, this man is probably similar, not dissimilar, to someone who shares most of his properties. (Adapted from Powers, 1995: 2.) Reply. The Objection assumes that an intrinsically improbable proposition cannot be probable relative to a premise, and this assumption is made plausible by the probability calculus' definition of conditional probability. I have argued elsewhere (Bowles, 1990: 67- 74 George Bowles 68, and Bowles and Gilbert, 1993: 256-257; 258, n. 2) against employing that as a defini- tion of a conclusion's probability relative to its premise. 12 That there are other countervailing features besides possessing a logically false premise or a logically true conclusion is shown by Argument 6 Premise. Most swans have a color which is shared by most swans and by this swan. Conclusion. This swan has a color which is shared by most swans and by this swan. which possesses Form 5, lacks a logically false premise and a logically true conclusion, and yet has a premise that entails its conclusion (adapted from Powers, 1995: 3). How many other kinds of countervailing features there are, beyond those mentioned in this paragraph, is unknown to me. But that Form 5, unlike, say, Form I, determines that each of its standard instances is such that the conclusion is probable relative to the premise can be seen in the case of Argument 4: its conclusion is probable relative to its premise, and it is so because o/the argument's form. 13 Forms similar to Form 7 include Form 8 Premise. Form 9 Conclusion. Premise L Premise 2. pandq. It is false that p. All Mare P. All S areM. Conclusion. Some S are not P. 14 Bowles (1990: 65-67). That every instance of'Somexs are ys' is relevantto an instance of 'This x is a y' in the sense that they have shared subject matter may be conceded. But that it is relevant to it in the sense that it makes it at least more likely to be true than false or at least more likely to be false than true is refuted by the fact that' Some philosophers are rich people' makes 'This philosopher is a rich person' neither certain, probable, improbable, nor impossible. Granted, 'Some philosophers are rich people' tells against 'No philoso- phers are rich people', which in tum tells against 'This philosopher is a rich person'. But this does not mean that' Some philosophers are rich people' tells in favor of This philoso- pher is a rich person', for the same reason that a refutation of an objection to a position does not constitute an argument in favor ofthe position (Bowles and Gilbert, 1993: 260, n. 3). Moreover, although 'Some philosophers are rich people' is compatible with This philosopher is a rich person', that does not mean that the former is logically relevant to the latter, since irrelevance entails compatibility. IS This conclusion implies not merely that there are many instances of invalid argument- forms that are invalid but also that (1) at least some of those instances are invalid because they are instances of invalid argument-forms, and (2) there are many kinds of invalid argument-forms that make some oftheir instances invalid. 16 Of course, some instances of Form II are invalid; but they are so accidentally and not because they are instances of that form. 17 Much remains to be discovered and clarified concerning "standard instances". But the effort seems worth while because (a) Forms 5, 6, 7, and 10 do not make all of their instances in-valid, and yet (b) the possession of those forms by Arguments 4, 7, 8, and 9 is not irrelevant to the invalidity of those arguments (as is, say, their possession of Form 11). In order satisfactorily to reconcile (a) and (b), we must be able to say what exactly is the difference between those arguments (here called 'standard instances') that are made invalid by such forms as Forms 5, 6, 7, and 10 and those that are not. IS Cf Johnson (1989: 423). The Asymmetry Thesis and "Invalid" Argument-Forms 75 Objection. Professor Massey has not committed the fallacy of Hasty Generaliza- tion, because he could give the same argument concerning Forms 5-10 that he gave concern- ing Forms 1 and 2. Take Form 5 as an example. Professor Massey could have argued thus: "Form 5 has some instances which are valid (namely, those in which such countervailing features as a logically false premise or a logically true conclusion are present). Therefore, possession of Form 5 is not a sufficient cause of invalidity. Therefore, Form 5 by itself cannot make any instance invalid." Reply. (1) Whether or not Professor Massey could have given the same argument concerning Forms 5-10 that he gave concerning Forms 1 and 2, the fact remains that he did not. He apparently argued from what is true of Forms 1 and 2 to a conclusion about all invalid argument-forms-a diverse class among whose members not only Forms 5-10 but also Form 3 have relevant logical characteristics different from those of Forms 1 and 2. Even if it is not necessary to commit the fallacy of Hasty Generalization in order to defend the Asymmetry Thesis, it stilI appears that Professor Massey did commit it. (2) Even if Professor Massey had given arguments concerning Forms 6-10 like the one suggested above concerning Form 5, they would not have supported even the weak- ened version ofthe Asymmetry Thesis (namely, that there is a one-versus-many asymme- try in our knowledge of formal determination of invalidity and validity) stated at the beginning of Section III. For, to begin again with'Form 5, although that form may have some valid instances, it does by itself make others of its instances (namely, its standard instances, like Argument 4) invalid. The same can also be said of Forms 6-10 and ofthe many other kinds offorms like them. This tells against, not for, the weakened Asymmetry Thesis. References Bencivenga, Ermanno. "On Good and Bad Arguments", Journal of Philosophical Logic, Vol. 8, No.3 (August 1979),247-259. Bowles, George. "Evaluating Arguments: The Premise-Conclusion Relation", In- formal Logic, Vol. XIII, No.1 (Winter 1991),1-20. Bowles, George. "Propositional Relevance", Informal Logic, Vol. XII, No.2 (Spring 1990),65-77. Bowles, George and Gilbert, Thomas E. "The Probabilistic Import of Illatives", Ar- gumentation, Vol. 7, No.3 (1993), 247-262. Copi, Irving M. and Cohen, Carl. Introduction to Logic, Eighth Edition, (New York: Macmillan Publishing Company, 1990). Finocchiaro, Maurice A. "The Positive Versus the Negative Evaluation of Argu- ments", in New Essays in Informal Logic, eds. Ralph H. Johnson and J. A, Blair (Windsor, Ontario: Informal Logic Publications, 1994),21-35. Iseminger, Gary. "The Asymmetry Thesis", Monist, Vol. 72 (1989),25-39. Johnson, Ralph H. "Massey on Fallacy and Informal Logic: A Reply", Synthese, Vol. 80 (1989), 407426. Krabbe, Erik C. W. "Can we ever pin one down to a formal fallacy?", in Analysis and Evaluation: Proceedings of the Third ISS A Conference on Argumentation, Uni- versity of Amsterdam, June 21-24, 1994, Vol. 2, eds. Frans H. van Eemeren et al. (Amsterdam: International Center for the Study of Argumentation, 1995),333-344. 76 George Bowles Lewis, Clarence Irving and Langford, Cooper Harold. Symbolic Logic, Second Edi- tion (New York: Dover Publications, Inc., 1959). McKay, Thomas J. "On Showing Invalidity", Canadian Journal oj Philosophy, Vol. XN, No.1 (March 1984),97-101. Massey, Gerald J. "Are There Any Good Arguments That Bad Arguments Are Bad?", Philosophy in Context, Vol. 4 (1975a), 61-77. Massey, Gerald 1. "In Defense of the Asymmetry", Philosophy in Context, Supple- mentto Vol. 4 (1975b), 44-56. Massey, Gerald J. "Tom, Dick, and Harry, and All the King's Men", American Philosophical Quarterly, Vol. l3, No.2 (April 1976), 89-107. Massey, Gerald J. "Asymmetry, fallacy, and indeterminacy", paper presented at the Symposium on "Informal Logic: Asymmetry and Fallacy", American Philosophi- cal Association, Pacific Division, San Francisco, March, 1987. Oliver, James Willard. "Formal Fallacies and Other Invalid Arguments", Mind, Vol. LXXVI (1967), 463-478. Powers, Larry. Private communication, June, 1995. George Bowles, 4466 Arglington Blvd. Arlington, VA 22204-134066 U.S.A.