Existential Fallacy

A formal fallacy by which existence is implied in a logical operation that would require it to be proven first.

Take, for example, the following syllogism:

All animals are mortal.   (true)
No unicorn is mortal.   (vacuously true)
Therefore, there exists unicorns which are not animals.     (existential fallacy)

On first glance, this resembles the Modus Camestros Show “Modus Camestros” in Syllogism FInder App, but it lacks the ex­pli­cit existential import that this form nor­mally re­quires. As a result, the con­clu­sion here, which implies existence of unicorns is not valid.

• (Illicit) existential instantiation

Different types of logical state­ments may or may not refer to empty ex­ten­sions of the terms used. Most not­ably, uni­versal quan­ti­fi­ca­tions (like all A are B) do not, wheras exis­tential quan­ti­fi­ca­tions (like some A are B) do imply ex­istence.

This can lead to problems when the lat­ter type is in­fered from earlier. A typ­ical situ­ation where this happens is when an ex­is­tent­ial quan­ti­fi­ca­tion is in­fer­red from a uni­ver­sal one. In prin­ci­ple, this is pos­sible, as the exis­ten­tial quan­ti­fier is the weaker form of the two (Argu­mentum a Forti­ori ), mean­ing that a state­ment like all cars are vehicles ob­vi­ously also im­plies that some cars are vehicles.

What is less obvious, though, is that this inference is only possible because we know that cars actually exist. In other words, by our knowledge (which is external to the logical statements), we implicitly import existence into this trans­formation.

This becomes a problem with state­ments which refer to some­thing that does not exist: “all uni­corns are im­mortal” is vacu­ously true, mean­ing it is true spe­ci­fi­cally be­cause no uni­corns exist that could ac­tu­ally ever die. If we trans­form this into there exist uni­corns which are im­mortal, this im­pli­cit exis­ten­tial import goes wrong and we have ar­rived to a state­ment which is clearly not true.

For this reason, infering from a form that allows re­fer­ring to an empty ex­tens­ion to one that does not, re­quires that the im­port is made ex­plicit.

Such situ­ations arise in a number of syllogisms, where an ex­is­ten­tial con­clu­sion is de­rived from uni­ver­sal pre­mises. In par­ti­cu­lar, this af­fects the modi Bar­bari, Bama­lip, Cale­mos, Cames­tros, Cela­ront, Ce­saro, Da­rap­ti, Fel­ap­ton and Fe­sapo. In all of these, there is an ad­di­tional re­quire­ment to ex­pli­citly vali­date the exis­ten­tial im­port for at least one of the terms used.

However, this fallacy is not limited to syl­log­istic in­fer­ences, but can hap­pen when­ever a state­ment which has an ex­is­ten­tial im­pli­cation is de­rived from a state­ment that does not.

• Existential import
• Existential quantifications
• Extension
• Universal quantifications

• The Existential Fallacy on Fallacy Files