# Dictionary Definition

vacuous adj

2 devoid of significance or point; "empty
promises"; "a hollow victory"; "vacuous comments" [syn: empty, hollow]

# User Contributed Dictionary

## English

### Etymology

vacuus, vacant### Adjective

vacuous- Showing a lack of thought or intelligence; vacant

### Derived terms

- vacuity (noun)
- vacuously (adverb)
- vacuousness (noun)

# Extensive Definition

A vacuous truth is a truth that is devoid of
content. It may refer pejoratively to
the truth of a tautological statement, such
as that "all rectangles are rectangles" or less obviously to a
statement such as that "all squares are rectangles" (Sanford,
1999). It can refer to a property true by virtue of its logical
structure, not because it refers to an existing entity (Blackburn,
1994), say the whiteness of a white unicorn.

More formally, a relatively well-defined usage
refers to a conditional
statement with a false antecedent.
The statement is considered vacuously true because the falsity of
the antecedent prevents us from using the conditional to infer the
consequent.

This notion has relevance in pure
mathematics, as well as in any other field which uses classical
logic.

Outside of mathematics, statements which can be
characterized informally as vacuously true can be misleading. Such
statements make reasonable assertions about qualified objects which
do not actually exist. For example, a child might tell his parents
"I ate every vegetable on my plate," when there were no vegetables
on the child’s plate.

## Scope of the concept

The term "vacuously true" is generally applied to
a statement S if S has a form similar to:

The first instance is the most basic one; the
other three can be reduced to the first with suitable
transformations.

Vacuous truth is usually applied in classical
logic, which in particular is two-valued, and most of the
arguments in the next section will be based on this assumption.
However, vacuous truth also appears in, for example, intuitionistic
logic in the same situations given above. Indeed, the first 2
forms above will yield vacuous truth in any logic that uses
material
conditional, but there are other logics which do not.

## Arguments of the semantic "truth" of vacuously true logical statements

This is a complex question and, for simplicity of
exposition, we will here consider only vacuous truth as concerns
logical implication, i.e., the case when S has the form P
\Rightarrow Q, and P is false. This case strikes many people as
odd, and it’s not immediately obvious whether all such statements
are true, all such statements are false, some are true while others
are false, or what.

### Arguments that at least some vacuously true statements are true

Consider the implication "if I am in
Massachusetts, then I am in North America", which we might
alternatively express as, "if I were in Massachusetts, then I would
be in North America". There is something inherently reasonable
about this claim, even if one is not currently in Massachusetts. It
seems that someone in Europe, for example, would still have good
reason to assert this proposition. Thus at least one vacuously true
statement seems to actually be true.

#### Arguments against taking all vacuously true statements to be false

##### Making implies and logical AND logically equivalent

Second, the most obvious alternative to taking
all vacuously true statements to be true — i.e., taking all
vacuously true statements to be false — has some unsavory
consequences. Suppose we are willing to accept that P \Rightarrow Q
should be true when both P and Q are true, and false when P is true
but Q is false. That is, suppose we accept this as a partial
truth
table for implies:

Suppose we decide that the unknown values should
be F. In this case, then implies turns out to be logically
equivalent to logical AND (\land), as we can see in the following
table:

Intuitively this is odd, because it certainly
seems like "if" and "and" ought to have different meanings; if they
didn’t, then it’s confusing why we should have a separate logical
symbol for each one.

Perhaps more disturbing, we must also accept that
the following arguments are logically valid:

- P \Rightarrow Q
- P \land Q
- P

and

- P \Rightarrow Q
- P \land Q
- Q

That is, we can conclude that P is true (or that
Q is true) based solely on the logical connection of the two.

##### Intuition from mathematical arguments

Picking "true" as the truth value makes many
mathematical propositions that people tend to think are true come
out as true. For example, most people would say that the statement

- For all integers x, if x is even, then x+2 is even.

- If 3 is even, then 3 + 2 is even

- if x is even, then x+2 is even

##### A linguistic argument

First, calling vacuously true sentences false may
extend the term "lying" to too many different situations. Note that
lying could be defined as knowingly making a false statement. Now
suppose two male friends, Peter and Ned, read this very article on
some June 4, and both (perhaps unwisely) concluded that "vacuously
true" sentences, despite their name, are actually false. Suppose
the same day, Peter tells Ned the following statement S:

- If I am female today, i.e., June 4, then I will buy you a new house tomorrow, i.e., June 5.

### Arguments for taking all vacuously true statements to be true

The main argument that all vacuously true
statements are true is as follows: As explained in the article on
logical
conditionals, the axioms of propositional
logic entail that if P is false, then P \Rightarrow Q is true.
That is, if we accept those axioms, we must accept that vacuously
true statements are indeed true. For many people, the axioms of
propositional logic are obviously truth-preserving. These people,
then, really ought to accept that vacuously true statements are
indeed true. On the other hand, if one is willing to question
whether all vacuously true statements are indeed true, one may also
be quite willing to question the validity of the propositional
calculus, in which case this argument begs the
question.

### Arguments that only some vacuously true statements are true

One objection to saying that all vacuously true
statements are true is that this makes the following deduction
valid:

- \neg P
- P \Rightarrow Q

Many people have trouble with or are bothered by
this because, unless we know about some
a priori connection between P and Q, what should the truth of P
have to do with the implication of P and Q? Shouldn’t the truth
value of P in this situation be irrelevant? Logicians bothered by
this have developed alternative logics (e.g. relevant
logic) where this sort of deduction is valid only when P is
known a priori to be relevant to the truth of Q.

Note that this "relevance" objection really
applies to logical implication as a whole, and not merely to the
case of vacuous truth. For example, it’s commonly accepted that the
sun is made of gas, on one hand, and that 3 is a prime number, on
the other. By the standard definition of implication, we can
conclude that: the sun’s being made of gas implies that 3 is a
prime number. Note that since the premise is indeed true, this is
not a case of vacuous truth. Nonetheless, there seems to be
something fishy about this assertion.

### Summary

So there are a number of justifications for
saying that vacuously true statements are indeed true. Nonetheless,
there is still something odd about the choice. There seems to be no
direct reason to pick true; it’s just that things blow up in our
face if we don’t. Thus we say S is vacuously true; it is true, but
in a way that doesn’t seem entirely free from arbitrariness.
Furthermore, the fact that S is true doesn’t really provide us with
any information, nor can we make useful deductions from it; it is
only a choice we made about how our logical
system works, and can’t represent any fact of the real
world.

## Difficulties with the use of vacuous truth

- All pink rhinoceros are carnivores.
- All pink rhinoceros are herbivores.

Both of these seemingly contradictory statements
are true using classical or two-valued logic – so long as the set
of pink rhinoceros remains empty. (See also Present
King of France.)

Certainly, one would think it should be easy to
avoid falling into the trap of employing vacuously true statements
in rigorous proofs, but the history of mathematics contains many
‘proofs’ based on the negation of some accepted truth and
subsequently demonstrating how this leads to a contradiction
(proof
by contradiction).

One fundamental problem with such
‘demonstrations’ is the uncertainty of the truth-value of any of
the statements which follow (or even whether they do follow) when
our initial supposition is false. Stated another way, we should ask
ourselves which rules of mathematics or inference should still be
applicable if we suppose that pi is an integer (which it is
not).

The problem occurs when it is not immediately
obvious that we are dealing with a vacuous truth. For example, if
we have two propositions, neither of which implies the other, then
we can reasonably conclude that they are different;
counter-intuitively, we can also conclude that the two propositions
are the same. The reason for this is that (P \Rightarrow Q)\lor(Q
\Rightarrow P) is a tautology
in classical logic, so every assertion that is made about "two
propositions, neither of which implies the other" is an assertion
about nothing, hence vacuously true. Although such a fact that "two
propositions, neither of which implies the other, are both
different and the same" poses no theoretical problems, it can
easily be disturbing to the human mind.

Avoidance of such paradox is the impetus behind
the development of non-classical systems of logic relevant
logic and paraconsistent
logic which refuse to admit the validity of one or two of the
axioms of classical
logic. Unfortunately the resulting systems are often too weak
to prove anything but the most trivial of truths.

## Vacuous truths in mathematics

Vacuous truths occur commonly in mathematics. For instance,
when making a general statement about arbitrary sets, said statement ought to hold
for all sets including the empty set. But
for the empty set the statement may very well reduce to a vacuous
truth. So by taking this vacuous truth to be true, our general
statement stands and we are not forced to make an exception for the
empty set.

For example, consider the property of being an
antisymmetric
relation. A relation R on a set S is antisymmetric if, for any
a and b in S with a\mathrelb and b\mathrela, it is true that a=b.
The less-than-or-equal-to relation \leq on the real numbers
is an example of an antisymmetric relation, because whenever a\leq
b and b\leq a, it is true that a=b. The less-than relation is also
antisymmetric, and vacuously so, because there are no numbers a and
b for which both a and b, and so the conclusion, that a=b whenever
this occurs, is vacuously true.

An even simpler example concerns the theorem that
says that for any set X, the empty set \varnothing is a subset of
X. This is equivalent to asserting that every element of
\varnothing is an element of X, which is vacuously true since there
are no elements of \varnothing.

There are however vacuous truths that even most
mathematicians will outright dismiss as "nonsense" and would never
publish in a mathematical journal (even if grudgingly admitting
that they are true). An example would be the true statement
More disturbing are
generalizations of obviously "nonsensical" statements which are
likewise true, but not vacuously so:

- There exists a set S such that every infinite subset of S has precisely seven elements.

## References

- Blackburn, Simon (1994). "vacuous," The Oxford Dictionary of Philosophy. Oxford: Oxford University Press, p. 388.

- David H. Sanford (1999). "implication." The Cambridge Dictionary of Philosophy, 2nd. ed., p. 420.

## External links

vacuous in Hebrew: באופן ריק

vacuous in Hungarian: Igazhalmaz

# Synonyms, Antonyms and Related Words

airy,
asinine, awkward, bare, barren, bland, blank, blankminded, bleached, callow, calm, catchpenny, characterless, clear, devoid, dumb, empty, empty-headed, empty-minded,
empty-pated, empty-skulled, existless, fatuous, featureless, flimsy, foolish, fribble, fribbling, frivolous, frothy, futile, gauche, green, groping, hollow, idle, ignorant, inane, incogitant, inexperienced, innocent, insipid, jejune, know-nothing, lacking, light, minus, missing, naive, negative, nescient, nirvanic, nonexistent, nugacious, nugatory, null, null and void, oblivious, otiose, passive, quietistic, rattlebrained, rattleheaded, raw, relaxed, scatterbrained, shallow, silly, simple, slender, slight, strange to, superficial, tentative, thoughtfree, thoughtless, tranquil, trifling, trite, trivial, unacquainted, unapprized, uncomprehending,
unconversant,
unenlightened,
unexisting, unfamiliar, unideaed, unilluminated, uninformed, uninitiated, unintellectual, unintelligent, unknowing, unoccupied, unposted, unreasoning, unrelieved, unripe, unsure, unthinking, unversed, vacant, vain, vapid, void, white, windy, with nothing inside,
without being, without content