On Proving Too Much Moti Mizrahi St. John's University [Forthcoming in Acta Analytica] Abstract: It is quite common to object to an argument by saying that it "proves too much." In this paper, I argue that the "proving too much" charge can be understood in at least three different ways. I explain these three interpretations of the "proving too much" charge. I urge anyone who is inclined to level the "proving too much" charge against an argument to think about which interpretation of that charge one has in mind. Keywords: argument, objection, proves too much, proving too much 1. Introduction In argumentative contexts, one often hears the following complaint being made: It seems to me that, if the author were right, then s/he would prove too much. Here are three examples: First example. It is often said that G. E. Moore's (1903) Open Question argument "proves too much." Moore's Open Question argument can be reconstructed as follows: 1. Suppose that 'x is good' is equivalent to 'x is pleasure'. 2. If 'x is good' is equivalent to 'x is pleasure', then 'pleasure is good' is equivalent to 'pleasure is pleasure'. 3. But 'pleasure is pleasure' is an uninformative tautology. 4. If those who assert 'good is pleasure' mean to express more than a mere tautology, then the question whether pleasure is good remains "open." 5. Those who assert 'good is pleasure' mean to express more than an uninformative tautology. 6. Therefore, the question whether pleasure is good remains "open." Those who charge Moore's Open Question argument with "proving too much" do so because they think that, by substituting 'water' and 'H2O' for 'good' and 'pleasure', a parallel argument can be made to prove that the question whether water is H2O is open. And since that is obviously false, since water is H2O, Moore's Open Question argument "proves too much," i.e., it can be used to prove something false, which means that there must be something wrong with Moore's Open Question argument. Second example. It is often said that Anselm's (1965) Ontological Argument "proves too much." Anselm's Ontological Argument can be reconstructed as follows: 1. By definition, God is the greatest conceivable being. 2. If God does not exist, then God is not the greatest conceivable being, since to exist is greater than not to exist. 2 3. Therefore, God exists. Gaunilo's critique in On Behalf of the Fool is that Anselm's Ontological Argument "proves too much," since a parallel argument can be made, by substituting 'God' for 'the perfect island', to prove that the perfect island exists. And since that is obviously false, since there is no such thing as a perfect island, Anselm's Ontological Argument "proves too much," i.e., it can be used to prove something false, which means that there must be something wrong with Anselm's Ontological Argument. Third example. It is often said that Pascal's Wager (1910) "proves too much." Pascal's Wager can be represented as a decision matrix: Table 1. Pascal's Wager God exists God does not exist Believe Eternal salvation You've been duped Do not believe Eternal damnation You got it right Accordingly, the argument goes, the rational decision is to believe, since one has more to gain by believing than by not believing, and more to lose by not believing than by believing. Those who charge Pascal's Wager with "proving too much" do so because they think that, by substituting 'God' for any other deity that promises eternal rewards or punishments, a parallel argument can be made to prove that one should believe in a rival god. And since that is obviously false, since one cannot (or, at least, is not supposed to) have incompatible theistic beliefs, Pascal's Wager "proves too much," i.e., it can be used to prove something false, which means that there must be something wrong with Pascal's Wager. Surprisingly enough, very little has been written about the "proving too much" charge. Most of what has been written about the "proving too much" charge seems to date back to the nineteenth century. For instance, Walter (1879) mentions the "proving too much" charge in his discussion of fallacies. Walter (1879, p. 57) writes: To the "ignoratio elenchi" [i.e., ignoring the question] may be referred the error which consists in "proving too much:" of which it is justly said, "he that proves too much, proves nothing;" v.g., if one should attempt to demonstrate that the human soul is a substance, and his arguments went to show that man's soul has the properties of matter, he would then commit the error of "proving too much." (original emphasis) Accordingly, Walter (1879) seems to be construing the "proving too much" charge as a kind of reductio ad absurdum. That is to say, an argument can be said to "prove too much" when its conclusion has absurd implications. Similarly, Gilbart (1857, p. 90) writes: Akin to the reduction ad absurdum is an exposure of the fallacy called "proving too much." This fallacy is an argument that, if admitted to prove the point in dispute, would, if carried out to all its legitimate consequences, also prove other points that neither of the disputants admit to be true. (original emphasis) 3 And Coppée (1850, p. 234) says that "sometimes an argument which seems to prove the proposition in question, may also prove another and an absurd conclusion; and thus the first conclusion is invalidated." Likewise, Whately (1963) has characterized the "proving too much" charge more recently as follows: It is evident that either the Premiss of an opponent, or his Conclusion, may be disproved, either in the Direct, or in the Indirect method; i.e. either by proving the truth of the Contradictory, or by showing that an absurd conclusion may fairly be deduced from the proposition you are combating. When this latter mode of refutation is adopted with respect to the Premiss, the phrase by which this procedure is usually designated, is, that the "Argument proves too much;" i.e. that it proves, besides the conclusion drawn, another, which is manifestly inadmissible (original emphasis). 1 I think that these accounts of the "proving too much" charge are incomplete. In what follows, then, I seek to remedy this lacuna. I will show that the "proving too much" charge can be understood in at least three different ways. I will explain these three interpretations of the "proving too much" charge. I will urge anyone who is inclined to level the "proving too much" charge against an argument to think about which interpretation of that charge one has in mind. 2. Three different ways of charging an argument with "proving too much" The three different ways of charging an argument with "proving too much" can be summed up in the following diagram (Figure 1). It will be useful to refer to this diagram as I explain these different ways of accusing an argument of "proving too much." Figure 1. Three different interpretations of the "proving too much" charge 1 Cf. "prove too much" entry in A Dictionary of Modern Legal Usage: "to make an overboard argument; (of an argument) to be overboard." 4 When one levels the charge of "proving too much" against an argument, one might have one of the following in mind: (A) Here is an argument for p. Assuming that the argument for p is valid, by saying that the argument "proves too much" I mean that one of the premises must be false, since a parallel argument with the same form can be used to prove that q, and q is obviously false. (Since a valid argument cannot have true premises and a false conclusion, if the conclusion is false, one of the premises has to be false as well.) (B) Here is an argument for p. By saying that the argument "proves too much" I mean that the argument is inconclusive, since a parallel argument can be used to prove that q, and q is implausible. In this case, the fact that there is a target argument for p and a parallel argument for q, which is implausible, shows that the premises of the argument for p do not provide conclusive support for p. (C) Here is an argument for p. Assuming that the premises in an argument for p are true, by saying that an argument for p "proves too much" I mean that the argument is invalid, since a parallel argument with the same form can be used to prove that q, and q is obviously false. (Since a valid argument with true premises must have a true conclusion, if the conclusion is false, and the premises are true, the argument must be invalid.) Here is how (A) and (C) would work (see Figure 1). Take Anselm's Ontological Argument, for example. If we assume that Anselm's Ontological Argument is valid, and we also have Gaunilo's parallel argument, which has the same logical form, we could reason as follows: 1. Suppose that Anselm's Ontological Argument is valid. 2. Gaunilo's parallel argument has the same logical form as Anselm's Ontological Argument. 3. So, Gaunilo's parallel argument is also valid. By making a parallel argument for q, which is false/implausible, the charge purports to show that The charge is meant as an attack on An argument for p is charged with "proving too much" the premises of the argument for p the argument for p is unsound (A) the argument for p is inconclusive (B) the form of the argument for p the argument for p is invalid (C) 5 4. But the conclusion of Gaunilo's parallel argument is obviously false. 5. So, one of the premises of Gaunilo's parallel argument must be false. 6. So, one of the premises of Anselm's Ontological Argument must be false as well. Alternatively, if we think that the premises of Anselm's Ontological Argument are true, and we have Gaunilo's parallel argument, we could reason as follows: 1. Suppose that the premises of Anselm's Ontological Argument are true. 2. So, the premises of Gaunilo's parallel argument are also true. 3. But the conclusion of Gaunilo's parallel argument is obviously false. 4. So, Gaunilo's parallel argument must be invalid. 5. So, Anselm's Ontological Argument must be invalid as well. The basic point, then, is this: the relationship between the target argument and the parallel argument will tell us how the "proving too much" charge should be interpreted. More generally, for any target argument for p and a parallel argument for q, one could reason as follows: 1. Suppose that the argument for p is valid. 2. A parallel argument for q has the same logical form as the argument for p. 3. So, the parallel argument for q is also valid. 4. But q (i.e., the conclusion of the parallel argument) is obviously false. 5. So, one of the premises of the parallel argument for q must be false. 6. So, one of the premises of the target argument for p must be false as well. This is the (A) reading of the "proving too much" charge. Alternatively, for any target argument for p and a parallel argument for q, one could reason as follows: 1. Suppose that the premises of the target argument for p are true. 2. So, the premises of the parallel argument for q are also true. 3. But q (i.e., the conclusion of the parallel argument) is obviously false. 4. So, the parallel argument for q must be invalid. 5. So, the target argument for p must be invalid as well. This is the (C) reading of the "proving too much" charge (see Figure 1). As for the (B) reading of the "proving too much charge," one could reason as follows (see Figure 1): 1. The target argument for p provides non-conclusive reasons to believe that p. 2. So, the parallel argument for q also provides non-conclusive reasons to believe that q. 3. But q (i.e., the conclusion of the parallel argument) is implausible. 4. So, the target argument provides inconclusive reasons to believe that p. (In other words, the parallel argument for q, which is implausible, is a defeater vis-à-vis p.) 3. Conclusion I have argued that the "proving too much" charge is much more nuanced than a simple reductio ad absurdum. The "proving too much" charge can be understood in at least three different ways. I urge anyone who is inclined to level the "proving too much" charge against an argument to think about which interpretation-(A), (B), or (C)-one has in mind. 6 Acknowledgments I would like to thank Clayton Littlejohn, David Morrow, Marcus Arvan, and an anonymous reviewer of Acta Analytica for helpful comments on earlier drafts. References Anselm, St., (1965). Proslogion. In St. Anselm's Proslogion, M. Charlesworth (Ed.). New York: Oxford University Press. Coppée, H. (1850). Elements of Rhetoric: Designed as a Manual of Instruction. J. H. Butler & Co. Gilbart, J. W. (1857). Logic for the Million. Longmans & Roberts. Moore, G. E. (1903). Principia Ethica. New York: Cambridge University Press. Pascal, B. (1910). Pascal's Pensées. W. F. Trotter (Trans.). New York: Collier & Son. Walter, H. H. (1879). Elements of Philosophy, Comprising Logic and Ontology, or General Metaphysics. Fifth Revised Edition. Baltimore: John Murphy & Co. Whately, R. (1963). Elements of Rhetoric: Comprising an Analysis of the Laws of Moral Evidence and of Persuasion, with Rules for Argumentative Composition and Elocution. Southern Illinois University Press.