Brentano-Venn thesis

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According to what is now[1] called the Brentano-Venn interpretation of propositions, every categorical proposition (one of the form ‘A is B’) is convertible with an existential proposition ‘An A-B exists’.

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Brentano's explanation

The Austrian philosopher Franz Brentano (1838-1917) seems to have been the first to challenge the traditional view that categorical propositions are not necessarily or semantically existential. He argued that we can join the concept represented by a noun phrase "an A" to the concept represented by an adjective "B" to give the concept represented by the noun phrase "a B-A". For example, we can join "a man" to "wise" to give "a wise man". But the noun phrase "a wise man" is not a sentence, whereas "some man is wise" is a sentence. Hence the copula must do more than merely join or separate concepts.

Furthermore, adding "exists" to "a wise man", to give the complete sentence "a wise man exists" has the same effect as joining "some man" to "wise" using the copula. So the copula has the same effect as "exists". Brentano argued that every categorical proposition can be translated into an existential one without change in meaning and that the "exists" and "does not exist" of the existential proposition take the place of the copula. He showed this by the following examples:

The categorical proposition "Some man is sick", has the same meaning as the existential proposition "A sick man exists" or "There is a sick man".
The categorical proposition "No stone is living" has the same meaning as the existential proposition "A living stone does not exist" or "there is no living stone".
The categorical proposition "All men are mortal" has the same meaning as the existential proposition "An immortal man does not exist" or "there is no immortal man".
The categorical proposition "Some man is not learned" has the same meaning as the existential proposition "A non-learned man exists" or "there is a non-learned man"[2].

Modern predicate logic depends on precisely this idea. If we formalise "there is an F" as"E x, Fx", where the E is the so-called "existential quantifier", it is possible to represent all the four traditional forms of categorial proposition by a combination of the existential quantifer with the negation sign ~ as follows

Some man is sick = E x [ man(x) & sick(x) ]
No stone is living = ~ E x [ stone(x) & living(x) ]
All men are mortal = ~ E x [ man(x) & ~ mortal(x) ]
Some man is not learned = E x [ man(x) & ~ learned(x) ]

Using these forms, we can develop a calculus with very simple derivation rules, as Frege did in 1879. It is only on the Brentano interpretation that complex inferences involving generality and existence can be handled, and that is why it has universally found favour among formal logicians (and why the traditional syllogistic ended as a historical curiosity).

Ockham's version of the thesis?

Something remarkably similar can be found in Ockham's Summa Logicae, Book III, part 2, chapter 26. Parallel Latin-English below. The similarity is not exact - Brentano also held that 'every A is B' is equivalent to the negative 'no A is not B', or to 'not: something that is A and not-B exists'. Ockham, by contrast, agreed with Aristotle that 'every A is B' is affirmative, and that it implies the particular 'some A is B'.


Latin English
Si autem propositio dubitabilis in qua praedicatur esse exsistere per propositionem de inesse vel de possibili habeat pro subiecto nomen connotativum vel respectivum vel negativum vel unum compositum ex multis nominibus, quandoque potest demonstrari, quandoque non. Now if there is a dubitable proposition in which existential being is predicated by an assertoric or de possibili proposition, and the proposition has for a subject a connotative, relative, or negative name, or one composed from many names, sometimes it can be demonstrated and sometimes it cannot.
Talis enim propositio semper aequivalet uni propositioni in qua praedicatur passio de subiecto, saltem large sumendo passionem. Sicut ista proposito ‘eclipsis est’ aequivalet isti ‘aliquid eclipsatur’; et ista ‘calefactivum est’ aequivalet isti ‘aliquid est calefactivum’; et ista ‘habens tres angulos aequales duobus rectis’ aequivalet isti ‘aliquid est habens tres angulos aequales duobus rectis’. For such a proposition is always equivalent to a single proposition in which an affection is predicated of a subject, at least when we take "affection" broadly. Thus the proposition, "An eclipse exists" is equivalent to "Something is eclipsed"; and "A heatable thing exists," is equivalent "Something is heatable"; and "Something having three angles equal to two right angles exists," is equivalent to "Something is a thing having three angles equal to two right angles."

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Notes

  1. See Prior: The Doctrine of Propositions and Terms, c.5 sec. 2 ("Existential propositions in Brentano").
  2. Brentano, F., 1874 and 1911, Psychologie vom empirischen Standpunkt, Duncker & Humblot, Leipzig, II ch. Vii
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