# What fallacy accepts P and P → Q but rejects Q (denies modus ponens)?

I am finding that I can get people to agree with my premises, agree that my logic is valid, but then they deny my conclusion. For example, I state the sufficient conditions, P, for an optimal system, Q. They accept P implies Q. I prove P is true under my proposed system. Then they deny Q (they deny the system is optimal).

In short, what is the name of this fallacy: P and P → Q, but not Q?

• And why are my dollar signs not making LaTeX? Apr 21, 2018 at 4:13
• Here's a meta article going into why it doesn't work here as it does on the math SE site: philosophy.meta.stackexchange.com/q/43/29944 Apr 21, 2018 at 4:42
• I will have to check further, but I would say it is not a fallacy since the argument is valid. That someone doesn't accept a valid argument might suggest they are irrational or unwilling to accept the conclusion. Oddly this might not be a bad thing. It forces both sides to come up with even better arguments. Apr 21, 2018 at 4:45
• Thanks for the link, Frank. I'm not saying my argument is invalid. I know it's valid. I'm trying to find the fallacy of people not accepting modus ponens. I wouldn't say people not accepting an airtight argument and having to rely on emotional reasoning to get one's point across is "not a bad thing"; I think it is bad that people don't think logically. (But that's the way the world is, so regardless of whether I like it or not, I'll have to make those emotional arguments in addition to a logical one.) Apr 21, 2018 at 7:47
• It means that the people you are discussing with do not really agree that P is sufficient. Apr 21, 2018 at 8:51

From a purely logical point of view, that inference makes so little sense that there really is no name for it.

However, if you run into a case like this in the real world, i.e. If you see some human agreeing to P, and to If P then Q, but not to Q, there could be several things going on.

For one, we could be dealing with a case of willful Ignorance, or willful denial, which isn't not so much a fallacy in the sense of a logical thinking mistake, but rather a result of other cognitive factors not making one accept what should be accepted. Possible reasons are wishful thinking or just being cognitively stuck on a certain idea or belief.

Another possibility is that the people don't really believe P, or that they believe that P is probably true, or true in most cases, but that in this particular case P does not hold, preventing the inference to Q. Same for If P then Q: maybe they generally believe that If P then Q, but that there are certain exceptions that apply to this particular case. And let's also note that their refusal to acccept Q is not necessarily the same as rejecting Q.

In short, the purely formal logical characterisation of their reasoning might just be too simplistic to capture the nuances that are going on. Indeed, the world is a messy place, so to think that we can make our way around with simple Modus Ponens's is a pipe dream.

Named formal fallacies, like affirming the consequent, usually refer to inferences that can at least superficially be considered valid because they are often used as plausible inferences even when they are invalid (if smoke then fire, etc.). Denying modus ponens is not a mistake of this sort, so it has no name. And sometimes it is not a mistake at all, it is done to make a point that the meaning of terms, implication in this case, depends on the rules we adopt for manipulating them, like modus ponens. Someone may reject modus ponens because they decline to use classical implication, and prefer some alternative logic. Modus ponens fails in Lukasiewicz's 3-valued logic, for example, but this is rare.

Denial of modus ponens was featured prominently in Lewis Carroll's dialogue What the Tortoise Said to Achilles. The argument discussed is:

A: Things that are equal to the same are equal to each other

B: The two sides of this triangle are things that are equal to the same

Z: The two sides of this triangle are equal to each other.

The Tortoise points out to Achilles that one may accept that A and B are true, but not yet that if A and B are true then Z true. Achilles concurs. Then Tortoise agrees to accept

C: If A and B are true then Z is true;

but points out that one can similarly accept A, B and C but not conclude Z. Why? Because she says this needs yet another premise "If A and B and C are true then Z is true", and so on.

The infinite regress comes from the fact that application of a principle, here modus ponens, is different from declaring it, and in effect one application compresses infinitely many declarations. As Wittgenstein put it, "there has to be a way to grasp a rule which is not an interpretation". The Tortoise attempts to replace such practical grasping with endless interpretations, and effectively ends up rejecting modus ponens as a result. She asks for modus ponens to be substantiated before it is applied, instead of taking it on faith, and it turns out that it can not be so substantiated. If the Tortoise fails to grasp the rule there is nothing that can make it do so. Achilles can not make the Tortoise apply modus ponens by simply getting her to accept declarations. Even logic requires something extra-logical.

• I disagree that logic requires "extra-logical". Logic is a different kind of truth to the truth that "Earth exists". Logical truths are defined to be true. I define a set of logical truths to be valid if and only if it does not lead to inconsistencies. And a valid set of logical truths has value if only if it has useful mappings onto the real world: I.e. you can take a set of known real-world truths and apply your logic to reach known or even previously unknown real-world truths. I think we're getting off-topic though: I just wanted to know if there was a name for the fallacy. Apr 21, 2018 at 7:41
• @bEPIK Yes, but that is not the point. Definitions only become effective in the real world, or even just in reasoning, if they are put into action. By refusing to put them into action Tortoise demonstrates that they are impotent even if the "logic" is declaratively accepted. The "extra-logical" is the actional part. As I said, there is no name. Apr 21, 2018 at 7:42
• You can certainly have a valid logic that doesn't contain modus ponens. Take the null set or the set that contains the definitional truth that every X is an X. These are valid logics. Of course, you would be utterly crippled in using existing real-world truths to reach new real-world truths within these logics. Logics are like a more formal kind of language. You can define "pen" however you like. And for some non-English ,language perhaps "pen" means something different. But just because someone is using a different language, doesn't mean I can disregard what they are saying as untrue. Apr 21, 2018 at 7:57
• I need to accept their definitions in order to understand the person, unless I can find a logical inconsistency in their definitions. Apr 21, 2018 at 7:58
• @bEPIK Wittgenstein's favorite response was "and what if I don't?" We are under no obligation to accept other person's conceptual system whether it is consistent or not (and by the way there are logics that allow inconsistencies). Mutual understanding may be better achieved by convincing them to switch to another. But what Tortoise illustrates goes further, people may be incapable of understanding us if they lack some practical ability to apply rules (like modus ponens). Wittgenstein reflected quite a bit on these issues, but this is way off-topic. Apr 21, 2018 at 8:12

What fallacy accepts P and P → Q but rejects Q (denies modus ponens)?

It turns out such reasoning is not necessarily fallacious, but rather reflects a philosophical viewpoint. Wikipedia, "Epistemic closure". The issue is not logical sufficiency, but what the observer can know.

Epistemic closure is a property of some belief systems. It is the principle that if a subject S knows p, and S knows that p entails q, then S can thereby come to know q.

....

While the principle of epistemic closure is generally regarded as intuitive, philosophers such as Robert Nozick and Fred Dretske have argued against it.

I would have thought this question was settled in favor of closure. That's what I like about this forum. You learn something new every day.

• Wikipedia is not a good source, here is SEP's Epistemic Closure. This is a different issue, in epistemic logic K(P) and P → Q does not entail K(Q). One can not settle for it because it is obviously false, take as P the axioms of arithmetic, and as Q any unproved theorem. But people do not "reject" Q when presented with a valid argument, as in OP, because of epistemic non-closure. Apr 26, 2018 at 19:42
• @Conifold. SEP is indeed the better source. I am still surprised that closure exists as a philosophical issue. Apr 27, 2018 at 0:59

Look up "Counterfactual_thinking" on the Wikipedia:

Counterfactual thinking is a concept in psychology that involves the human tendency to create possible alternatives to life events that have already occurred [...] These thoughts consist of the "What if?" and the "If I had only..." that occur when thinking of how things could have turned out differently.

Moreover, according the "Epistemology" section of the Wiki article on Robert Nozick, his:

four conditions for S's knowing that P were:

``````P is true
S believes that P
If it were the case that (not-P), S would not believe that P
If it were the case that P, S would believe that P
``````

Nozick's third and fourth conditions are counterfactuals. He called this the "tracking theory" of knowledge. Nozick believed the counterfactual conditionals bring out an important aspect of our intuitive grasp of knowledge: For any given fact, the believer's method must reliably track the truth despite varying relevant conditions. [...] Nozick believes that the truth tracking conditions are more fundamental to human intuition than the principle of deductive closure.