Intro to: Modus Ponens

By Ben Andrew

Modus Ponens (more on what this is in a minute) belongs to the domain of a field of philosophy called logic. Logic concerns itself with arguments, which are series of propositions (called premises) that aim to support a conclusion, and, more specifically, what makes arguments good. In good arguments, according to most logicians, if the premises are true, then the conclusions must be true. Though there might not be good arguments that meet this criterion, in this explainer we will confine ourselves to arguments that are good in this way.

One way (but certainly not the only way) of doing logic is by giving inference rules, which tell you what conclusions follow from a premise or a collection of premises of a certain form. Good arguments (which, remember, are the explanandum of logic) are those which utilize only valid inference rules. Modus Ponens is said to be one such inference rule. It states that if one has premises of the form “if P, then Q” (where “P” and “Q” stand for any declarative sentence), and “P”, then you can conclude “Q.” It’s pretty intuitive, but like all seemingly common sense ideas, philosophers have raised some thought provoking objections to it. So without further ado, here are 3 alleged counterexamples to Modus Ponens that will make you say “wow, that’s puzzling” (a note: I will present the counterexamples here but I won’t even begin to consider how one might respond to them. However, for the curious reader, I have attached some suggested reading which does just that).

 

  • Vann Mcgee’s Counterexample

 

Take the following argument: “If Hillary Clinton didn’t win the 2016 election, then if Donald trump didn’t win, Gary Johnson would have.” Our premise – that Hillary Clinton didn’t win the 2016 election – is true. Therefore, according to the argument, if Donald Trump didn’t win, Gary Johnson would have.” This argument, according to Modus Ponens, is a good one. We have a premise of the form “if P, then Q,” a premise of the form “P,” and we’ve concluded “Q.” But this isn’t a good argument. Even though the premises are true, as Clinton did not win, the conclusion is false – if Trump didn’t win, Clinton almost certainly would have. This argument can’t be one where the truth of the premises guarantees the truth of the conclusion, because the premises are true, but the conclusion is false.

 

  • Seth Yalcin’s Counterexample

 

Yalcin’s counterexample is presented as a counterexample to another inference rule, the rule Modus Tollens. However, we can derive Modus Tollens from Modus Ponens by using Negation Introduction, which is the rule that if assuming “P” leads us to a contradiction, we can conclude “not P.”  We can take this as a counterexample to Modus Ponens as well, since one of these rules will also have to be given up if we find a compelling contradiction.

Modus Tollens is the rule that from “if P, then Q” and “not Q,” then we can derive “not P.” Modus Ponens, combined with Negation Introduction gives us Modus Tollens. To see how, assume we have “if P, then Q” and “not Q.” For the sake of Negation Introduction, assume “P.” Then, by Modus Ponens, we get “Q.” But this contradicts “not Q.” So by Negation Introduction we can conclude “not P.”

So a counterexample to Modus Tollens is a counterexample either to Modus Ponens or Negation Introduction. Either way, it’s bad news, and Yalcin has what he thinks is one. He asks us to imagine we have drawn a marble from a jar of marbles, some large, some small, some blue, some red. The exact breakdown is this:

BLUE RED
BIG 10 30
SMALL 30 5

According to Yalcin, it is true of the marble we have drawn that “if the marble is big, then it is likely red,” since most of the big marbles are red. It is also true that “the marble is not likely red” since most of the marbles are blue. Therefore, we should conclude using Modus Tollens that “the marble is not big”. But this is false since some blue marbles are big. So we have a counterexample to Modus Tollens.

 

  • Malte Willer’s Counterexample

 

A final alleged counterexample to Modus Ponens comes from (UChicago’s own) Malte Willer, although once again this could be a counterexample to either Modus Ponens or Negation Introduction. Willer’s argument asks us to begin with someone who considers their friend Sally very crafty, and so who accepts “if Sally is deceiving me, I do not believe it.” The person, in order to reason by Negation Introduction, assumes “Sally is deceiving me,” and from Modus Ponens concludes “I do not believe Sally is deceiving me.” But, recognizing the absurdity of believing both that Sally is deceiving them and that they do not believe that Sally is deceiving them, our reasoner concludes that “Sally is not deceiving me.” This seems like a bad argument, and so since it relied only on Modus Ponens and Negation Introduction, we must give up one or the other (at least for these sorts of weird cases), and indeed, Willer claims we should give up the former.

 

  • Suggested reading:

 

    1. Vann McGee – A Counterexample to Modus Ponens
    2. Seth Yalcin – A Counterexample to Modus Tollens
    3. Malte Willer – New Surprises for the Ramsey Test
    4. A response to McGee and Willer: Justin Bledin – Modus Ponens Defended

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