Existential import

From The Logic Museum

Jump to: navigation, search

In nineteenth-century logic, the question of existential import was whether a universal or A proposition such as "all buttercups are blue" implies the existence of its subject, i.e. whether it implies the existential proposition "blue buttercups exist". This question depends on two further questions which are frequently conflated, namely

(A) whether a universal proposition like "all dragons are fire-breathing" implies what traditional logicians call the "particular" or I proposition "some dragons are fire-breathing ".

and

(B) whether the I proposition "some dragons are fire-breathing" implies the existential proposition "fire-breathing dragons exist".

Since, in modern logic, the I form and the existential form are completely equivalent, it is often supposed[1] that the nineteenth-century dispute was about question (A). In fact, it was about question (B). Indeed, it was the answer to question (B), that brought about the later confusion about what the original question was.

In traditional logic, the sentence or categorial proposition consists of three parts: the predicate which is "affirmed" or "denied", the subject of the affirmation or denial, and the copula which signifies whether the predicate is affirmed or denied. The copula was thought to be signified by the verb "is", (Latin: est). For example, in the proposition "man is mortal", the verb "is" signifies that the predicate "mortal" is affirmed of the subject "man".

Every proposition consists of three parts: the Subject, the Predicate, and the Copula. The predicate is the name [sic] denoting that which is affirmed or denied. The subject is the name denoting the person or thing which something is affirmed or denied of. The copula is the sign denoting that there an affirmation or denial (Mill, System of Logic).

What is now called a general existential proposition, such as "some men are mortal" was then called a particular proposition. It was not called "existential", because it was not thought to be existential. A proposition of the form "A exists" combines the subject "A" with the verb "exists". Since (according to the traditional theory) every proposition consists of subject, predicate and copula, it follows that "exists" must be a grammatical abbrevation of copula and predicate, and that it really stands for "is existent" or something similar. If so, it is not the copula "is" that signifies existence, but the adjective "existent". Arnauld writes:

Although every proposition contains these three things, it could be composed of only two words, or even one. Wishing to abbreviate their speech, people created an infinity of words all signifying both an affirmation, that is, what is signified by the substantive verb, and in addition a certain attribute to be affirmed. All verbs besides the substantive are like this, such as "God exists", that is, "is existent", "God loves humanity", that is, "God is a lover of humanity". When the substantive verb stands alone, for example when I say, "I think, therefore I am", it ceases to be purely substantive, because then it is united with the most general attribute, namely being. For "I am" means "I am a being", "I am a thing".

Mill defends the same view, and adds the following argument: if "is existent" expresses existence "is non-existent" denies existence, so 'is' cannot assert existence. It merely joins the idea signified by the predicate (such as existence or non-existence) to the idea signified by the subject. Mill writes

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." (System of Logic I.iv.1)

Mill claims his father (James Mill) was the originator of this idea but, as the passage from Arnauld shows, it is older.[2]


Contents

Brentano

The Austrian philosopher Franz Brentano (1838-1917) seems to have been the first to challenge the traditional view.[3] 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".

Modern 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 had has universally found favour among formal logicians (and why the traditional "syllogistic" ended as a historical curiosity).

The predicate calculus sees no difference between the statement "some dragons are fire-breathing", and "there are fire-breathing dragons". Both are represented using the existential quantifier, as E x [ dragon(x) & fire-breathing(x) ]. Hence the question (B) of whether the particular proposition implies the existential proposition, can no longer be asked. It is then natural to suppose that the question of "existential import" that was asked by logicians in the nineteenth century, was the question (A) above of whether the universal proposition implies the particular proposition.

Brentano's Priority

Brentano published his theory in 1874. Was he the originator of the theory? And did the publication have some positive discernible effect on the development of the subject? That he was the first is suggested by a letter from Mill to Brentano dated February 1873, after Brentano wrote to him challenging his theory of the copula. Mill wrote:

You did not, as you seem to suppose, fail to convince me of the invariable convertibility of all categorical affirmative propositions into predications of existence. The suggestion was new to me [my emphasis], but I at once saw its truth when pointed out.

If the suggestion was new to Mill, whose writing on logic shows a detailed and comprehensive knowledge of the early nineteenth literature on logic, it probably was new.

Further evidence is a paper published by Land in 1876, attacking Brentano's account of the copula. The language used by Land suggests the theory is new. The title of the paper is "Brentano's Logical Innovations", for example. He says that Brentano's discoveries in Logic are "well calculated to awaken the most lively curiosity" and says that "there is no need to dwell upon his anticipations of the horror and dismay with which his doctrines will be received among logicians of the older school.

It is commonly asserted that it was Boole who has priority, and that he was the first to reject the inference from the A to the I proposition. Indeed, Brentano's interpretation of the categorical propositions is sometimes called the "Boolean" interpretation.[4] This is a complete mistake. From the chapter "On syllogisms", in Boole (1847) it is clear that Boole's system supports the traditional reading of the propositions, such as the inference from A to I, and other related conversions. Research by Stanley Burris (2003) also supports this view, as does Wu (1969) and Church 1965, (p. 422) who finds the interpretation first (implicitly) in Cayley 1871, explicitly in Brentano 1874, and in Peirce 1880, 15-57.

Whether Brentano's interpretation influenced the development of the subject is not so clear. Brentano's name, although well known to historians and students of phenomenology, is not usually associated with the development of symbolic logic. (For example, when I mentioned Brentano to a mathematical historian, he said he had never heard the name). Within ten year's of Brentano's publication, other competing theories had appeared in print, such as Frege (1879), Peirce (1880) and Venn (1881). These all use ideas that Brentano published in 1876. Did these writers borrow from his work?

How much did Frege know of Brentano's work? This is difficult to answer, as Frege rarely acknowledged his sources. He corresponded with one of Brentano's students, Benno Kerry, which led to a famous paper "On Concept and Object". But this was in the 1880's, after Kerry noticed the Grundlagen (1884). In any case, it is controversial whether Frege had much impact on the development of symbolic logic in the nineteenth century. The development of set theory by Zermelo and others depended on the notation developed by the C.S.Peirce, and the German logician Ernst Schröder. As Putnam (1982) writes:

… While, to my knowledge, no one except Frege ever published a single paper in Frege's notation, many famous logicians adopted Peirce-Schroeder notation, and famous results and systems were published in it. Loewenheim stated and proved the Loewenheim theorem (later reproved and strengthened by Skolem, whose name became attached to it together with Loewenheim's) in Peircian notation. In fact, there is no reference in Loewenheim's paper to any logic other than Peirce's. To cite another example, Zermelo presented his axioms for set theory in Peirce-Schroeder notation, and not, as one might have expected, in Russell-Whitehead notation.

Before the 1950's, after which he had a tremendous influence, Frege enjoyed the same invisibility[5] as Brentano.

Concerning Venn, the fact that Land's paper appeared in 1876, in a major journal, means that Brentano's work attracted attention early. It is hardly plausible that it escaped the attention of Venn, who was a regular contributor to Mind throughout the 1870's and 1880's (he reviewed Frege's Begriffschift for the journal in 1880).

Concerning Peirce, recent research (see e.g. Grattan-Guinness 2000) confirms Putnam's view about his influence on mathematical logic. But was he also familiar with Brentano's theory?


Another misconception

Another misconception is that the question of existential import is about how to translate sentences of traditional logic, such as "all men are mortal" into sentences of the predicate calculus. According to Keynes and Prior, it is simply a question of interpretation.

If we take the form A to mean "Nothing is X without being Y", E to mean "Nothing is both X and Y", and O to mean "Something is X without being Y", it can hardly be questioned that their implications are as Leibniz, Brentano and Venn describe them. (Prior 1976 p. 122)

But, as Prior says, we are under no compulsion to assign these meanings to these forms. For example, if the A proposition "every A is B" means "something is A and not: something is A and not B", then the A proposition does imply the I proposition, and the A proposition does have existential import in the sense of question (A) above. According to Prior, the question is whether we formally translate the English sentence "every A is B" as

~ E x [ A(x) & ~B(x) ]

or as

E x [ A(x) & B(x) ]& ~ E x [ A(x) & ~B(x) ] .

But that is not the real question. The question is is whether we can translate English sentences containing the word "exists" into sentences of first order calculus without defining some predicate "exists(x)" or not. On the Brentano interpretation, we do not need to define such a predicate. The word "exist" is analysed completely away. By contrast, on the traditional interpretation, "exist" is a genuine predicate, and we cannot do without it.

Possibly "all the books in this room are about philosophy" means "some books are in this room, and no book in this room is not about philosophy", and so the universal proposition is existential in sense (A). But is it existential in sense (B)? Does "some books are in this room" means "books in this room exist"? Is the I proposition existential? Traditional logicians did not accept this.

Traditional Logic never reads the I proposition as existential and Symbolic Logic must read it as existential in order to have an I proposition. Thus, in Traditional Logic some men are white means that the quality white modifies some cases of man; it does not mean, nor even imply, that there are also some who are not white. Nor does it mean that some white men exist [my emphasis]. To get this said requires an existential proposition, e.g. some white men are, where there is no strict logical predicate. The point of the I proposition is not the existence of some men but the whiteness of some men. (Wade 1955)


References

  • Arnauld, A. & Nicole P., Logic, or the Art of Thinking, transl. J. Buroker, Cambridge 1996
  • Boole, G., The Mathematical Analysis of Logic, 1847
  • Brentano, F., 1874 and 1911, Psychologie vom empirischen Standpunkt. Duncker & Humblot, Leipzig.
  • Burris, Contributions of The Logicians Part I ms, 2003.
  • Church, Alonzo. 1965 "The History of the Question of Existential Import of Categorical Propositions," in Bar-Hillel, Yehoshua (ed.) Logic, Methodology, and Philosophy of Science: Proceedings of the 1964 International Congress. North-Holland, Amsterdam, 417-24.
  • Frege, G. (1879) Begriffschift.
  • Grattan-Guinness, I. The Search for Mathematical Roots, 1870-1940, Princeton 2000
  • Joyce, G.H., Principles of Logic, London 1908
  • Land, J.P.N. "Brentano's Logical Innovations", Mind Vol. 1 (April 1876)
  • Peirce, C.S., On the algebra of logic. Amer. Journal of Math. 3 (1880), 15. Reprinted in Vol. III of Collected Papers, 104.
  • Poli, R., ed. The Brentano Puzzle. Ashgate 1998.

Prior, A.N. The Doctrine of Propositions& Terms London 1976

  • Strawson, P.F., Introduction to Logical Theory, London 1952.
  • Venn, J. 1880 - review of Frege's Begriffschift Mind 5 p. 297.
  • Venn, J. Symbolic Logic, London 1881
  • Wade, F.C. Introduction to John of St Thomas' Outline of Formal Logic Milwaukee 1955.
  • Wu, J.S., "The Problem Of Existential Import" Notre Dame Journal of Formal Logic Volume X, Number 4, October 1969.

See also

Notes

  1. For example, Stanley Burris writes

    In modern semantics we can interpret a symbol as any subclass of the universe, including the empty class. But this semantics is in apparent conflict with the traditional conversion per accidens

    All X is Y
    :. Some Y is X

    which clearly gives existential import [my emphasis] to the universal premiss.
    This is true, only if "some Y is X" has existential import.
  2. In Aristotle (Met. E .4 1027b20 – 30), and in Aquinas:

    Sed sciendum, quod esse dicitur [tripliciter]. Uno modo dicitur esse ipsa quidditas vel natura rei, sicut dicitur quod definitio est oratio significans quid est esse; definitio enim quidditatem rei significat. Alio modo dicitur esse ipse actus essentiae; sicut vivere, quod est esse viventibus, est animae actus; non actus secundus, qui est operatio, sed actus primus. Tertio modo dicitur esse quod significat veritatem compositionis in propositionibus, secundum quod est dicitur copula: et secundum hoc est in intellectu componente et dividente quantum ad sui complementum; sed fundatur in esse rei, quod est actus essentiae. I Sent. dist 33, Q1 Art. 1 ad.1

  3. Brentano (1874) II ch. vii.
  4. A cursory Google on "Boolean interpretation" shows how widespread this idea is. For example

    Logicians of the nineteenth century dropped the traditional assumption of non-emptiness, and adopted what is called the "Boolean interpretation"—after logician George Boole—of universal quantifiers. Under the Boolean interpretation, A- and E-type propositions lack existential import, while both I- and O-type have it."
    On the Boolean interpretation of categorical syllogisms, we cannot assume the existence of individuals mentioned in universal statements. If our language, if we want to assert that individuals exist, we must say so by adding a particular statement.

    For "Boolean" interpretation, read "Brentano" interpretation. Prior correctly calls it the Venn-Brentano interpretation, after Brentano (1874) and Venn (1881). See Prior 1976, chapter V.
  5. See Poli (1998) for discussions of the "invisibility problem".
  • [[]]
Personal tools