Existential fallacy

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The existential fallacy is an invalid inference from premisses which are not existential to a conclusion which is existential. An existential proposition is one which asserts or implies existence. It is often claimed that the inference in Aristotelian term logic from 'all S are P' to 'some S are P' commits the existential fallacy. However, this depends on the definition of 'existential'. Some term logicians claim that the particular proposition 'some S are P' is not existential.

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Existential propositions

In term logic, a categorical proposition consists of three parts: the predicate which is 'affirmed' or 'denied', the subject of the affirmation or denial, and the copula: the verb such as 'is' which joins subject and predicate. A 'particular' proposition quantifies the subject with a term like 'some', thus 'some hobbits are hairy-footed' or 'some characters in Lord of the Rings are hobbits', a 'universal' proposition quantifies the subject with a term like 'all', thus 'all hobbits are short'. See categorical form. Thus categorical propositions do not include the verb 'exist', unless as a predicate.

An existential proposition, by contrast, is one where existence is predicated. For example 'hobbits exist'. It is not explicitly a part of Aristotelian logic that a categorical proposition, as defined, is existential, since the definition of the categorical proposition does not say whether existence is predicated or not. Whether the copula (which is included in the definition) implies or asserts existence was hotly disputed in the scholastic period, with discussions that filled numerous volumes. Every animal was in Noah's ark, but no animal that was in Noah's ark exists, so does every animal not exist? [1] If Caesar is a man, does that imply 'Caesar is', and does that imply 'Caesar exists'?

Existential fallacy in term logic

In term logic, the universal proposition 'Every S is P' implies the particular proposition 'some S is P'. Some modern textbooks erroneously claim that this is an instance of the fallacy. For example:

The existential fallacy occurs when we draw a conclusion which implies existence from premisses which do not imply that. If our premisses are universals, telling us about 'all' or 'none', and our conclusion is a particular one telling us about 'some', we have have committed the fallacy".[2].

This is only correct if the particular proposition quantified by 'some' is existential. Term logic does not explicitly assume this, nor apparently does ordinary language. For example, in the inference 'All of the characters in Lord of the Rings are fictional, therefore some of the characters in Lord of the Rings are fictional', the consequent is clearly not existential, since it asserts that some of the characters are fictional, i.e. that they do not exist.

That the employment of [the word "is"] as a copula does not necessarily include the affirmation of existence, appears from such a proposition as this: A centaur is a fiction of the poets; where it cannot be possibly implied that a centaur exists, since the proposition itself expressly asserts that the thing has no real existence." [3]

In predicate logic

In modern predicate logic, the universal proposition 'every S is P' is explicitly understood as a negative proposition, of the form 'not Ex: S(x) and not P(x)'. This is not existential, since it merely denies the existence of certain kinds of things (namely Ss that are not Ps). The particular proposition 'some S is P' is explicitly understood as an affirmative proposition of the form 'Ex: S(x) and P(x)'. Under this interpretation, it is clearly incorrect to infer 'some S is P' from 'every S is P'. And since the standard interpretation of the existential quantifier 'Ex' is 'there exists an x such that', it follows that such an invalid inference satisfies the definition of the existential fallacy.

However, this is only under the interpretation of ordinary language imposed by predicate logic. Under the interpretation imposed by term logic, this is not necessarily the case, and it is also questionable whether in ordinary language this is the case, as the 'hobbit' examples suggest. It all depends on whether the ordinary language quantifier 'some' is understood as meaning 'there exists at least one'.[4].

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Notes

  1. See e.g. Authors/Heytesbury/Sophismata/Sophisma_25
  2. How to Win Every Argument: The Use and Abuse of Logic By Madsen Pirie, p.67
  3. (Mill, System of Logic I.iv.1)
  4. See e.g. "The Existential Import of a Proposition in Aristotelian Logic", John J. Morrison, Philosophy and Phenomenological Research 1955, 386-393. "Examples have been offered to demonstrate the obvious absurdity of concluding from a universal with a 'non-existential' sense to a particular which is understood to imply actual existence. … But it is assumed in giving the examples that particular propositions necessarily imply existence and that universals do not, and that Aristotelian logicians do not know this and, consequently, are committing logical sin"
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