# Problem with existential import in syllogisms (Barbara)

For a syllogism in Barbara, why is it the case that

All men are mortal.
James’ son is a man.
Therefore James’ son is mortal.

assumes that James has a son but in the case of saying

All men are moral.
All James’ sons are men.
Therefore all James’ sons are mortal.

it is not assumed that he necessarily has sons. All that is asserted is that if he has any sons they are mortal?

• "James’ son" is used as a compound name here, more commonly it is "Socrates". The prevailing convention is that named objects exist. But this does not change much, one can instead interpret both the premise and the conclusion conditionally:"If James's son exists then...". Aug 13 '19 at 5:24
• @virmaior Why is this example not a Barbara? Aug 13 '19 at 5:41
• @MarkAndrews You're are completely right. I misspoke: link.springer.com/chapter/10.1007/978-94-017-0787-9_1 Aug 13 '19 at 6:48
• Oops I need to re edit this Aug 13 '19 at 7:44
• What is assumed depends on a convention, the so-called "existential import". On the modern convention, "all James’ sons are men" does not have the existential import, and "some James’ sons are men" does. In the past, other conventions were used, see In logic, do propositions default to true or false when objects in them do not exist? Aug 13 '19 at 9:34

The problem seems to be more a linguistic than a logical one; in particular, the crucial phenomenon here is that of presupposition.

In "James' son is a man", the possessive construction "James' son" can be seen as acting as a so-called definite description where one particular individual is identified by the description "that individual which is the son of James". That we can apply the definite article "the" is crucial to the semantics of this expression -- it presupposes that such an individual actually exists; so from "James' son is a man" we may infer that John actually has a son. In general, one would say that possessive constructions like "James' son" trigger these so-called existential presuppositions. There is no further going reason as to why this is so; that's just how natural language works -- it is the pragmatic thing for a listener to infer that if someone mentions James' son, then this son actually exists, otherwise the expression would not make sense in the first place.
(Note the difference between presupposition and mere implication: Even if the statement were negated, from "James' son is not a man" we would still infer that James has a son -- this is not the case for logical implication (or entailment), which does not in general survive negation: For example, "John has a red car" logically implies that "John has a car", but the negated proposition "John doesn't have a red car" no longer implies that he has a car at all.)

This presupposition gets (to some extent) lost when we add quantification. (The technical term here is presupposition cancellation.) "All of James' sons are men" in this context is better to be read as "Any of James' sons" -- that is, "for whatever individual, if that individual is a son of James, then it is a man". This paraphrases makes it clearer that the classical logical interpretation of "all" does not come with existential import: If James doesn't have any children at all, then there are simply no individuals to apply the right-hand side of the implication (namely the predication "is a man") to; but this doesn't render the sentence false, instead, "All of James' son are men" becomes vacuously true, without creating any contradiction to the argument.

Hence, the genitive construction "James' son" triggers an existential presupposition suggesting that a son of James actually exists, whereas this presupposition gets cancelled in the context of the quantifier "all", which logically permits for situations where James has no children in the first place.

Of course, in natural language use things are not black and white and these nuances are to some extent ambiguous . For one thing, it would not be unreasonable for a speaker in an actual utterance situation to assume that James has kids when told that "All of James' sons are mortal", the classical logic interpretation of the quantifier "all" and the material implication "if ... then ..." by no means reflects 1:1 the usage of English "all" in natural language. Likewise, the phenomenon of existence presupposition in constructions like "Jame's son" or "the king of France" is not an irrefutable fact, but subject to a great deal of philosophical discussion - but this is a bit out of the scope of this question.

• Hi Lemontree... I don't quite follow your distinction between entailment and presuposition, could you clarify what you mean here? Aug 16 '19 at 9:42
• A proposition A entails a proposition B iff in all situation where A is true, B is true too, i.e. there is no situation such that A is true and B is not. A proposition A is a presupposition of a proposition B iff 1. A entails B (i.e. B is true whenever A is true) and 2. the negation of A entails B (i.e. B is true whenever ¬A is true) and 3. B is not tautological. Aug 16 '19 at 10:22
• That is, if A is a presupposition of B, that's a stronger relation than entailment, since B follows even when A is false, which is not the case for entailment, where if A is false we can't be sure if B is true. If A is a presupposition of B, then also A entails B (because that's part of the definition), but if A entails B that doesn't necessarily mean that A is a presupposition, because other than in presupposition, in entailment we can not necessarily conclude B when A does not hold. Since presupposition always also means entailment but not the other way round, presupp. is a stronger notin. Aug 16 '19 at 10:22
• For example, if we have statement A "James a son and a daughter", this logically entails that B "James has a son". But if we negate A, "James doesn't have a son and daughter", we can no longer conclude that B "James has a son" holds. So if A entails B, that doesn't mean that ¬A also entails B. On the other hand, from the statement A "James loves his son" and the negation of A "James doesn't love his son" it follows both times that B "James has a son", and "James has a son" is not a tautology, hence A doesn't only entail, but also is a presupposition of B. Aug 16 '19 at 10:32
• Hey thanks for elucidating it. I think I follow you Aug 17 '19 at 11:55

Irving Copi calls propositions such as "James' son is a man" or "Socrates is mortal" singular propositions. They are nonstandard-form propositions. They need to be translated into standard form categorical propositions which relate classes before being used in categorical syllogisms. He recommends the following: (page 239)

To every individual object there corresponds a unique unit class (one-membered class) whose only member is that object itself. Then to assert that an object s belongs to a class P is logically equivalent to asserting that the unit class S containing just that object s is wholly included in the class P.

Copi quotes Kant (Critique of Pure Reason, trans. N.K. Smith, p. 107) to justify such a translation: (page 240)

Logicians are justified in saying that, in the employment of judgments in syllogism, singular judgments can be treated like those that are universal.

Treating singular propositions as universal works for Barbara (AAA-1). The issue of existential import does not arise because all of the propositions are universal. It does not work, however, for other forms, such as, Darapi AAI-3 where two universal propositions have a particular conclusion. This breaks Copi's rule prohibiting premises without existential import deducing a conclusion which does have existential import: (page 240)

```s is M          goes into the invalid           All S is M.
s is H          AAI-3 categorical syllogism     All S is M.
∴ Some H is M.                                  ∴ Some H is M.
```

Copi also notes that Bertrand Russell viewed clarifying the status of these singular propositions as the first of two advances made by modern logic over Aristotelian term logic: (page 66)

The first advance consisted in separating propositions of the form 'Socrates is mortal' from propositions of the form 'All Greeks are mortal'. In Aristotle and in the accepted doctrine of the syllogism (which Kant thought forever incapable of improvement), these two forms of proposition are treated as indistinguishable or, at any rate, as not differing in any important way. But, in fact, neither logic nor arithmetic can get far until the two forms have been seen to be completely different. 'Socrates is mortal' attributes a predicate to a subject which is named. 'All Greeks are mortal' expresses a relation of two predicates - viz. 'Greek' and 'mortal'.

Both Copi and Russell, although perhaps not Kant or Aristotle, would have agreed with the OP about the need to treat these singular propositions more carefully.

Copi, I. M. Introduction to Logic. Sixth Edition. (1982) Macmillan.

Russell, B. My Philosophical Development. (1959) George Allen & Unwin. Internet Archive: https://archive.org/details/myphilosophicald0000russ/page/n6

• Hi Frank, I am still not sure why making all the terms universal solves the problem of existential import. Why I say all cars are green I could still be referring to all cars that are in existence, similarly if I said that James' car is green I could still mean and can still be validly interpreted that James' car if it exists is green. Also I am not sure why a proposition with two predicates needs to be treated differently. Also can there be a general 'All x' that is a subject? What is the criterion that distinguishes a subject from a predicate? Aug 16 '19 at 9:32
• Also in the Darapii syllogism you wrote down it looks like there is a mistake as you said 'All S is M' 'All is M' hence 'Some H are M' don't you means 'All S is M' 'All S is H'? Aug 16 '19 at 9:39
• @Deepthinker101 The problem is only when singular propositions are assumed to be universal. Then the s becomes a set S of one element. In Darapi S is used as the middle term. I checked that the quote was correct. Aristotle assumed these predicates (classes, sets) were not empty. Today, one would have to specify that with another proposition and say "James car exists". Using "all" or "none" is fine because the sets could be empty. Darapi is a problem because two "all" (universal) premises lead to a "some" (existential) conclusion. Aug 16 '19 at 13:12