# [FOM] FW: DRAFT: FOM EXISTENTIAL IMPORT THEOREMS

John Corcoran corcoran at buffalo.edu
Sat Apr 15 21:37:37 EDT 2006

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FOM EXISTENTIAL IMPORT THEOREMS
Some basic definitions and facts about existential import are given and
some open questions are asked in a previous communication "EXISTENTIAL
IMPORT", http://www.cs.nyu.edu/pipermail/fom/2006-April/010387.html. The
question of a necessary and sufficient condition for existential import
to hold is not open. It is settled by the Classical Existential Import
Theorem CEI. In particular it is known that a universalized conditional
Ax (S(x) --> P(x)) implies the corresponding existentialized conjunction
Ex (S(x) & P(x)) iff Ex S(x) tautological (logically true).	Consider
a suitable first-order language interpreted in the [natural] numbers as
universe of discourse [ or range of the individual variables].
FACT 0. It is easy to find cases where existential import fails.
"Every odd number is positive" does not have existential import if
'odd' and 'positive' are taken to be primitives, say ODD and POS.
Ax (ODD(x) --> POS(x)) does not imply Ex (ODD(x) & POS(x)). The above
CEI Theorem implies that this is quite general. Most perhaps all logic
texts that discuss the subject emphasize the cases where existential
import fails.
Q0.1. Is there even one logic text that indicates that there is even one
case where existential import holds?
Q0.2. Is there even one logic text that makes any effort to discuss the
question of how wide-spread the failure of existential import is?
Q0.3. Is there any literature on cases where existential import holds in
classical standard first-order logic?

FACT 1: Given any primitive predicate constant ESS and any formula
P(x)), existential import fails for Ax (ESS(x) --> P(x)), i.e., the
universalized conditional Ax (ESS(x) --> P(x)) does not imply the
corresponding existentialized conjunction Ex (ESS(x) & P(x)).

Nevertheless there are many interesting cases where existential import
holds. In fact, if 'odd' is taken as defined by Ey x = (y + y + 1), then
the implication holds: Ax (Ey x = (y + y + 1)) --> POS(x)) does imply Ex
(Ey x = (y + y + 1) & POS(x)).
Ex(Ey x = (y + y + 1)follows from(1 + 1 + 1) = (1 + 1 + 1).

Moreover, it does not matter whether or how POS(x) is defined (this
follows from the CEI Theorem above).
Q1.1. Is there even one logic text or article that makes any effort to
discuss the question of how wide-spread the holding of existential
import is?

FACT 2: In many cases of number-sets S, whenever existential import
fails using a formula S(x) defining the set S, existential import holds
for another formula S'(x) defining the same set.  There are many other
cases where familiar and interesting number-theoretic properties are
like oddness in that they admit of easy defining conditions S(x) whose
existentializations are tautological, thus giving rise to existential
import holding. Here are a few that I noticed: odd, even, square,
oblong, positive, exceeding n, preceding n, being-congruent-to n modulo
m (n and m arbitrary but fixed).
Of course this does not apply in the case of a property such as
being-its-own-successor which belongs to no natural number.
Q2.1. Where in the literature is this fact explicitly stated,
exemplified or discussed?

As peculiar and pedagogically useful as the isolated cases cited in FACT
2 might be, they almost lose all interest as soon as one knows that they
are completely typical of familiar number-theoretic properties.

FACT 3: Every non-empty definable set of numbers is definable by means
of a formula S(x) whose existentialization is tautological.
Q3.1. Where is this proved in the literature?

>From FACT 3 we get the following.
Fact 4: Given any non-null formula F(x), if existential import fails for
Ax (F(x) --> P(x)), i.e. Ax (F(x) --> P(x)) does not imply Ex (F(x) &
P(x)), then there exists a formula S(x) coextensive with F(x) and such
that existential import holds, i.e. Ax (S(x) --> P(x)) does imply Ex
(S(x) & P(x)).

Interpretation in the natural numbers is no limitation. We have the
following.
Existential Import Distribution Theorem: Let L, I, and U be arbitrary.
Every non-empty subset of U definable under I by a formula of L is both
(1) definable by a formula whose existentialization is tautological and
also (2) definable by a formula whose existentialization is not
tautological. Thus, except for a null formula, every formula that does
not give rise to existential import is coextensive with one that does
give rise to existential import and conversely. The holding of
existential import is as widely distributed as its failing.

Q4.1. Where in the literature is this stated or discussed, even
implicitly?
Q4.2. Where in the literature is this proved or given as an exercise to
prove?
Q4.3. Who if anyone is credited for it?
Q4.4. What is this used for?
Q4.5. What is this called?

PEDAGOGICAL OBSERVATION: Let L be an interpreted language. Thus, every
formula F(x) has an extension, i.e. defines a subset of the universe of
discourse.  A pedagogically effective way of looking at the above result
requires defining, for a sentence Ax (S(x) --> P(x)), the extension of
S(x) (respectively P(x)) to be called the material subject (predicate)
of Ax (S(x) --> P(x)). Then the EI Distribution Theorem implies that (1)
for every universalized conditional where existential import fails,
there is another universalized conditional having the same material
subject and the same predicate where existential import hold - unless
the material subject is empty, and (2) for every universalized
conditional where existential import holds, there is another
universalized conditional having the same material subject and the same
predicate where existential import fails.
Q4.1 Is there even one logic text or article that warns students against
jumping to the conclusion that a universalized conditional does not
imply its corresponding existentialized conjunction?
Q4.2 Is there even one logic text or article that tells students that in
order to determine whether a universalized conditional Ax (S(x) -->
P(x)) does or does not imply its corresponding existentialized
conjunction Ex (S(x) & P(x))it is necessary to look at the detail of the
subject S(x)?
Thanks to M.Brown, L. Compton, M. Davis, H. Enderton, J. Friedman, K.
Miettinen, J. Miller and A. Urquhart

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