Modus Ponens, Modus Tollens, and Likeness Mustafa M. Dagli Middle East Technical University dagli@tubitak.gov.tr ABSTRACT: Modus Ponens (MP) and Modus Tollens (MT) are taught as basic rules of inference related to conditional statements in introductory logic courses. In ordinary reasoning, MP and MT can have important roles in modes of argumentation. However, one can also distinguish counter-examples to such reasoning patterns when considered as ‘strictly’ valid rules (i.e., McGee’s counterattacks for MP, and Adams’ criticisms of MT). I suggest that this problem can be resolved if we revise MP and MT as basic tools of logic, assuming the above-mentioned counter-cases are valid, on the basis of nonmonotonicity. If the only thing that we know is ‘Tweety is a bird,’ we say ‘Tweety flies.’ But, after learning ‘Tweety is an ostrich,’ we (change our minds and) say, ‘Tweety does not fly.’ In actual life, we use ‘rules of logic’ in a limited sense; when we learn new facts, we change some of our beliefs sometimes. The question arises, ‘In which situation, which exception does not violate which rule?’ When reasoning about something, we use some semantic patterns in order to make inferences, or for the sake of argumentation. Two reasoning patterns employed in ordinary life scenes concerning conditional statements will be identified as MP-like and MT-like. These will be exemplified and discussed. The general idea guiding this tableau will be stated as likeness.
 I. Introduction Conditional sentences have attracted concentrated attention of philosophers, although intermittent, since ancient times. Typically, they have a main clause and at least one if-clause (like in "If A, then B). We sometimes say antecedent to the if-clause and consequent to the main clause. In ordinary language uses and daily practice, a conditional statement can be formed by other connective wording than "if/then", and may not begin with an if-clause. Accounts related to conditionals seem also to have tight relations with presuppositions about inference, reasoning, causation, physical existence, truth and validity. Modus Ponens (MP) and Modus Tollens (MT) are considered as basic rules of inference, and we teach them in introductory logic courses, related to conditional statements. In everyday reasoning, MP and MT can also have important roles, in modes of argumentation. II. A Historical Background The Stoics are accredited by historians of logic who did the early work on the nature and the theory of conditionals (in which Chrysippus, Diodorus Cronus, and Philo of Megara can further be distinguished). In Diogenes Laertius or Sextus Empiricus, one can find and read the first inscriptions related to this matter.(1) According to the Stoic logicians, the first kind of indemonstrable statements is as follows: "If the first, then the second; but the first; therefore the second." We call this basic argument form as modus ponendo ponens, in abbreviation modus ponens, the mood that by affirming affirms. The second kind of indemonstrable statements of the Stoics is: "If the first, then the second; but the second is not; therefore the first is not." This basic argument form is called as modus tollendo tollens, in abbreviation modus tollens, the mood that by denying denies, nowadays.(2) III. The Alleged Counterexamples to Modus Ponens and Modus Tollens Vann McGee's first counterexample— which represents the problematic adequately, for modus ponens, I think— is as follows: Opinion polls taken just before the 1980 election showed the Republican Ronald Reagan decisively ahead of the Democrat Jimmy Carter, with the other Republican in the race, John Anderson, a distant third. Those apprised of the poll results believed, with good reason: [1] If a Republican wins the election, then if it's not Reagan who wins it will be Anderson. [2] A Republican will win the election. Yet they did not have reason to believe [3] If it's not Reagan who wins, it will be Anderson.(3) McGee mentions that, in the light of examples resembling the above, modus ponens is not strictly valid. This exemplifying structure is discussed and criticized in Sinnott-Armstrong et al. (1986), Lowe (1987), and Over (1987), defended in Piller (1996). Ernest W. Adams discusses a contextless inference which "looks like" a counter-example to modus tollens:(4) (4) If it rained, it didn't rain hard. (5) It rained hard. (6) So, it didn't rain. As pointed out by Adams, if such a conversation occurs in our everyday lives, the person who uttered (4) will not say (6), after learning (5) from his/her friend related to the current situation outside. The inferences which are similar to the above are criticized in Dale (1989) and Sinnott-Armstrong et al. (1990), discussed in Gillon (1997). IV. What is Nonmonotonicity It is generally acknowledged that standard logics fail to capture the problem of incomplete knowledge. A system is said to be nonmonotonic, if introducing new axioms to the system can invalidate old theorems of it.(5) Since we continuously add new information to our stock of beliefs when we learn them as new facts (of the world), this epistemic position (being incomplete) seems to correspond to many of the exact situations, when we are pictured randomly from our daily lives. Let's assume a sequence of numbers which are generated by a predetermined function (that is hidden by the game organizer of the computing machine for example), and are screening interactively, after making a guess and entering it to the machine, one by one. If we are given the sequence `994, 996, 998, ?' and required to make a guess, then we normally say `1000'. When we see `994, 996, 998, 1000, ?', and requested to make a guess, then we say `1002'. What happens after seeing the continuation as `1004'? We understand that it is not `1002', it is simply `1004'. And we conclude that, it is not "the series `+2'", maybe "the series `+2' up to 1000 and `+4' thereafter". Then we begin to search if the continuation is really as specified or not. After seeing the continuation as `1004' but not `1002', we understood that the number sequence is not "the series `+2'". We guessed the rule of the number sequence is the rule of "the series `+2'". When we saw the reality we changed our minds. Wittgenstein's response to these kinds of situations is in form of ... It would almost be more correct to say, not that an intuition was needed at every stage, but that a new decision was needed at every stage.(6) In our daily lives, one need not to impose nonmonotonicity to a system; only he/she needs is to observe and understand it. If the only thing that we know is "Tweety is a bird", we say "Tweety flies". But, after learning "Tweety is an ostrich" (or, penguin), we (change our minds and) say "Tweety does not fly." In the actual life, we use the rules of logic in a limited sense; when we learn new facts, we change some of our beliefs sometimes. V. Remarks on Truth, Belief, Acceptability, and Assertability Philosophers consider truth as a basic and an important issue of philosophy, although there exist different conceptions of it.(7) It will be helpful in my opinion, to distinguish truth from belief on the one side, and acceptability from assertability on the other. Acceptability of a proposition is another thing than its truth, since human beings are fallible (although both can have importance in a decision and/or action theory). Not only in our inferences we want to hold truth, but also in our statements, for the sake of sincere assertability. However, there are true but not assertable statements in the real life, in addition to the existence of "true" but not acceptable statements.(8) VI. Applying Nonmonotonicity to Explain Validity of Both Counterexamples, MP and MT Although some conditionals are fainthearted (see, Morreau, 1997), it can be said that a conditional statement has a core which covers the message, and a mystery agent which controls over the form (see, Dudman, 1994). But, we are not omniscient. In our utterings and inscriptions, even if we try to obey the "co-operative principle" of H. P. Grice which yields to conversational implicature,(9) sometimes we cannot predict the current semantic situation beforehand. The difficulties of characterizing English if-then as material (truth-functional) conditional are widely discussed in for example, Adams (1965) and Cooper (1978).(10) Considering the core agent, and by not forgetting that if-then uses in English have the properties of an extended connective, one can still presume conditional applications as "material" (as if truth-functional). For an arbitrary conditional statement, both MP and MT are considered as valid by us (by default), unless having sufficient contrary evidence. However, on a basis of nonmonotonicity, after recognizing a spoiling character in the utterance/inscription, we will leave the above mentioned assumption. Compounds of conditionals can be seen as a source of ambiguity when searching for validity of MP and MT.(11) "Conditionals embedded in conditionals" can further be circumscribed. As mentioned by Christian Piller, McGee's objection to MP amounts to: There are counterexamples to the claim that modus ponens is generally valid. These counterexamples are to be found in the class of indicative conditionals where the consequent of the conditional is itself a conditional. (Piller, 1996: 28.) The attempts to determine the counter-cases, will transform the problem into a well-defined form, on non-monotonic bases. We can apply (for example) "circumscription" to these counter-cases if we can identify them, like in the well-known example "Birds fly / Penguins do not fly / Penguins are birds." (See, e.g., McCarthy, 1980.) When reasoning about something, we use some semantic patterns in order to make inferences, or for the sake of argumentation. McGee counterexamples themselves utter a rule-dependent behavior of us: neither we assert an event with negligibly (or, comparably) small probability as a candidate, nor we reject it as mere possibility.(12) (A similar case for Adams' counterexamples. When asserting (4), one remembers that "it was not raining outside when he/she was in, and the weather was not promising a hard rain".) Related to other potential counterexamples, in my opinion, there can be found close connections to an inquiry such that "In which situation, which exception does not violate which rule?" According to non-monotonic reasoning, which can be claimed as a formalization of commonsense— or, everyday— reasoning, the answer is as follows: (EX) An exception (or, some exceptions) does not (/do not) violate the rule, if the rule itself accepts exceptions. Thus, if we know that "If A, then B" states a general law which can have exceptions in certain situations, we can also say "If A and D, then not-B", in addition to "If A, then B", without being inconsistent. VII. Conditional Structures for Strengthening Assertions, and 'Likeness' as a Concept Concerning MP and MT An astronomer who believes in Copernicus', Kepler's, or Galileo's theses can say: (7) If the earth is at the center of the universe, then I am Greta Garbo. But he does not say: (8) If the earth is at the center of the universe, then 2 x 2 = 4. Additionally, (7) cannot be transferred appropriately into the argument form: (9) The earth is at the center of the universe. (10) Therefore, I am Greta Garbo. (7) seems as containing a counterfactual structure in simple subjunctive mood, and falsity of the consequent (which is an observation report) strengthens the claim that "the antecedent of (7) is false." In (7), the astronomer claims that "If A were the case, then B would be the case"; or, "If B were true, then A would be true"; or, "A is not true, as much as B is not", which is not the case in (8). When considering merely their truths, the consequent of the statement (7) could also be any true statement, if we accept (9) as false. However, I suppose, one can rationally think that, a widely accepted true statement (such as 2 x 2 = 4) will not be employed properly as the consequent of the conditional, in sentences which resemble the structure of (7). In this example, one can observe two phenomena: first, a relation between arguments and conditional statements.(13) Second, an indication of inferrability as a factor of assertability of a statement. In my opinion, beside some other facts, we say an hypothesis H of the form "If A, then B" is assertable, if B is inferrible from A. For a conditional statement, if we know that the antecedent is false, we can immediately say that "the whole statement is true", regarding its truth-conditions. However, when considering assertability, the epistemic value of a piece of information will act on the hypothesis H such that, "a false consequent" will be preferred to "a true consequent" for counterfactuals, from an inferrability point of view. This is mainly due to the "truth preservation" principle. Not only in our inferences we want to hold truth, but also in our statements, for the sake of sincere assertability. Today we think that we know many scientific facts. In these days, an astronomer (or, a sufficiently clever primary school student) can say: (11) If 2 by 2 makes 4, then the earth is not at the center of the universe. Stemming from the obvious mathematical knowledge that "2 x 2 = 4", in (11), he/she strengthens an observation report type opinion about the universe. If we believe that Q, naturally we can assert it. Assuming a medieval astronomer thought and was convinced that the earth is not at the center of the universe, (s)he could state this belief as: "The earth is not at the center of the universe." Or, if (s)he firmly believes that the earth's being not at the center of the universe is an important physical truth, considering that "2 x 2 = 4" is a mathematical (a priori) truth, for strengthening purposes (s)he could also say (11). On a strong belief that P, sometimes asserting P->Q is stronger than asserting only Q. (This principle can be called as "strengthening the consequent".) Similarly, a strong belief (or, an observation report) that NOT-Q, may strengthen the belief that NOT-P, more than stating only NOT-P, when asserting P->Q. (This principle can be called as "strengthening the antecedent".) The reasoning pattern employed in the strengthening process of (11) will be called as MP-like. The reasoning pattern used in (7) will be called as MT-like. The general idea guiding me in this analysis can be identified as likeness.(14) What I can say for likeness related to MP and MT is that, it states that modus ponens and modus tollens are valid patterns for making inferences, in a limited sense. We accept and use them by default, unless having strong counter-evidence. When considering practical life scenes, we are living as if in a realm of logic; sometimes we suffer from the dictatorship of it, sometimes we like it. VIII. CONCLUSION In ordinary reasoning, MP and MT can have important roles in modes of argumentation. However, one can also distinguish that there are counterexamples to such reasoning patterns, when considered as "strictly" valid rules (i.e., McGee counterattacks for MP, and Adams' criticisms of MT). In my opinion, this problem can be resolved; and, it will still be correct to educate MP and MT as basic tools of logic, assuming the above mentioned counter-cases are valid, on a basis of nonmonotonicity.
 Notes(1) See also, Mates (1953) and Bochenski (1957). (2) Propositional variables are designated by "first" and "second" in the Stoic terminology. For example, an argument can be in form of "If it is day, it is light; but it is day, therefore it is light." according to the Stoics. Or, "If it is day, it is light; but it is night, therefore it is not day." (See, e.g., Diogenes Laertius, 185-189.) (3) See, McGee (1985): 462. (4) See, Adams (1988): 122. (5) There can be found a lot of work done (and accumulated in the course of time) related to non-monotonic reasoning and incomplete knowledge beginning from late 70's. See, for example, Minsky (1975), McCarthy (1980), Reiter (1980), Turner (1984), Moore (1985), Hobbs and Moore (1985), Ginsberg (1987), and Lukaszewicz (1990). (6) See, Wittgenstein (1953), parag. 186. (7) See, e.g., Haack (1978): 86-134. Susan Haack mentions correspondence, coherence, pragmatic, semantic, and redundancy theories of truth. There existed also a dispute on the truth-conditions of conditionals, among the Stoic logicians. (According to Philo, a true conditional is one which does not have a true antecedent and a false consequent. However, a true conditional is one "which neither is nor ever was capable of having a true antecedent and a false consequent" in Philo's teacher Diodorus Cronus' account. A connection (or, coherence) view suggests that a conditional holds "whenever the denial of its consequent is incompatible with its antecedent" which is sometimes attributed to Chrysippus. A fourth, "suggestion" view also states that a conditional is true "if its consequent is in effect included in its antecedent.") (8) On the one side, one has to prove truth of his/her proposition in certain situations for somebody else to accept it. But, in some cases, at least for an interval of time (maybe for quite a long period of time) he/she will not be able to prove his/her argument by evidence. On the other side, mere truth or acceptability of a proposition, will not suffice to state its assertability. (It must not be meaningless to utter the true/acceptable proposition.) (9) See, Grice (1975). Four components of the `co-operative principle' are: 1. Quantity: The message must be as informative as is required, but not more than the need. 2. Quality: Truth (or, not falsity) is important for the contribution. 3. Relation: Be relevant. 4. Manner: Avoid obscurity, ambiguity. Be brief, orderly. (10) As mentioned by William S. Cooper, an argument such as "If it is the case that if Jones passes logic he will graduate, then he will graduate." therefore, "If Jones does not pass logic he will graduate." would have to be valid classically, which is highly suspicious. (See, Cooper, 1978: 203.) {When symbolizing, (P->Q)->Q. Therefore, (NOT-P->Q).} (11) See, e.g., Adams (1975), Appiah (1985), and Jackson (1987) on this subject. (12) In McGee's original examples, we can find a propositional structure similar to P->(Q->R). Although we (strongly) believe that it is not (/will not be) the case that NOT-P, still we do not prefer to assert Q->R seperately. This is only for the reason that, — although we (strongly) believe that— the probability of seeing NOT-P in the real domain is very very small; we still cannot assign a zero probability to NOT-P, in (what can be said) the set of sentences that we believe their truths. There still remains a possible world in which Q is true, but the truth of R is not a must. If we could sacrifice such a precaution, we could pronounce Q->R as a seperate statement. (We ignore side margins of the Bayesian curve, when articulating the probabilistic distribution of a factuality. In a manner, we see only a "sombrero". But, when we are asked about the— negligible— rest, we probably will accept it.) 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