Modal Scope Fallacy

Taxonomy: Logical Fallacy > Formal Fallacy > Fallacy of Modal Logic > Modal Scope Fallacy < Scope Fallacy < Amphiboly < Ambiguity < Informal Fallacy < Logical Fallacy

Note: In the Taxonomy, above, it will be seen that Modal Scope Fallacy is at the end of two branches, the first of which traces back to Formal Fallacy and the second to Informal Fallacy. This is because the fallacy has both a formal and informal aspect. The formal aspect is the fact that the scope of a modality is a feature of the logical form of a proposition―see the Exposition, below, for more on the scopes of modalities. The informal aspect comes from the fact that modal scope fallacies typically occur in arguments in natural languages, such as English, and are concealed by the ambiguity of sentences involving modalities in such languages.


If you know something then you cannot be mistaken. But if you cannot be mistaken, then you are certain. Thus, certainty is a necessary condition of knowledge. If you're uncertain, then you don't really know, but the only things that you can be truly certain of are logic and mathematics. Therefore, there is no knowledge of anything outside of logic and math.



If a man is a bachelor then he cannot be married, but if he cannot be married then he must be a priest. Therefore, all bachelors are priests.


Modalities, like other logical concepts such as negation, have scope, that is, they logically influence a part of any sentence in which they occur. Moreover, in English grammar the scope of modalities is often ambiguous, or more precisely "amphibolous". Compare the following sentences with their modal terms highlighted:

Though these two propositions have different subject matter, they seem to have the same form: both appear to be conditional propositions, and the modal words―"must" and "necessarily"―in them are embedded in their consequents. However, on the most plausible interpretations, the modalities differ in scope. Here are the most plausible readings of the examples:

In these two examples it is clear what the scope of the modality is, but in other sentences it is not clear whether the modality has a broad or narrow scope. The modal scope fallacy occurs when this amphiboly is exploited to create the false appearance of a sound argument: this happens when under one scope interpretation the argument is valid, while under the other the premisses are all true. It is by confusing these inconsistent interpretations that such an argument can appear sound, when in reality it is either invalid or has at least false one premiss.

Analysis of the Example:

Simplified a little, the example argument is the following:

  1. If you know something, then you cannot be mistaken about it.
  2. If you cannot be mistaken, then you are certain.
  3. Therefore, if you know something, then you are certain about it.

The scope ambiguity is found in the first premiss, where the alethic modality "cannot" may have two scopes:

  1. Narrow Scope: If you know something, then it is impossible for you to be mistaken about it.
  2. Wide Scope: It is impossible to both know something and be mistaken about it.

The modality in the first premiss must have narrow scope in order for the argument to be valid, but the modality must have wide scope in order for the premiss to be obviously true. The wide scope reading is uncontroversially true: it is impossible to know a falsehood. However, the narrow scope reading is at least controversial, and probably false: knowledge does not require the impossibility of error, merely its lack.

The above example is invented, but it's an attempt to explain a tenacious confusion in epistemology that goes back at least to Plato, namely, that between knowledge and certainty. It's hard to think of a philosophical mistake that has done more intellectual damage, and some of the blame for it may be due to modal scope confusion.


Norman Swartz, "'The' Modal Fallacy". "The modal fallacy", as Swartz refers to it, is the modal scope fallacy. There are other fallacies involving modalities, but this is probably the most common one. Swartz provides many examples of the fallacy using different types of modalities. The article is moderately technical, and uses some logical symbolism without explanation.

Resource: Raymond Bradley & Norman Swartz, Possible Worlds: An Introduction to Logic and Its Philosophy (Hackett, 1979), pp. 330-332. This book may be downloaded for free.

Acknowledgment: Thanks to Alberto Mio for pointing out a broken link in the Resources and a broken sentence in the text.