On quantifiers and bound variables
We now know how to deal with any formula of
that does not have any quantifiers occurring in it. As we just saw, in such cases, the assignment function does not do much, and you may very well wonder why there is even an assignment function distinct from an interpretation function when they function very much the same.
Well, the reason why we do need a separate variable assignment function over and above the interpretation function for constants and predicates is because sometimes, we want to actually vary the variable assignment, without varying the interpretation of the constants and the predicates. And indeed, using quantifiers and binding variables is a way to systematically incorporate these variations into the meaning of complex expressions.
How to do this was already briefly sketched above. Let’s look at it in the simplest case, now from a more formal point of view. First, take
, where
is a a variable and
is a one-place predicate. You can read this as “There exists an
such that
”, or “There is an
such that
is
”, or some variation of the above. At any rate, what this formula (and indeed, sentence) says is that there is a way of assigning a value to the variable
(there is an assignment) such that
is satisfied under that assignment. In other words, it says that there is a member of the domain that is in the interpretation of
. Or in yet other words, the interpretation of
is non-empty. Put as before, it might also be taken to say that
has a solution.
Suppose
is a structure with (non-empty) domain
being the set of all living things on the planet (at the moment of writing this book), and
is a subset of the domain consisting of all pandas. Then, we can ask: is the sentence
true in
? Well, the answer is yes if there is an assignment
that renders
satisfied under
and
, and otherwise, the answer is no. Thankfully, at the moment of writing this book (and hopefully, at the moment of reading it), there are pandas among the living things on the planet. Thus,
is a non-empty set, and so there should be an assignment that manages to assign an actual panda to
, thereby satisfying
in
.

For example, in the above figure, it is noted that the panda
. Thus, we can find an assignment that makes
satisfied in
, namely, any assignment that assigns Tián Tián to
. In fact, making use of
-variance, we can say that given any assignment
, the
-variant assignment
satisfies
in
. In other words,
, since
and
.
Now remember that what we were evaluating originally is whether
, i.e., whether
is true in
. And we said that it is true if we can find and appropriate assignment under which
is satisfied in
. Since we have found such an assignment,
. Note that this is really just a precise way of saying: there is a panda!
Clearly, this is a lot of words, and as we have seen, in formal logic, it is possible to say a lot of things in very few words (symbols). In general, what we can say is the following:
iff there is an
-variant assignment
such that
.
Clearly, our previous reasoning does conform to this definition, since
is such an assignment, as we have shown.
Now let’s see the same type of reasoning with the universal quantifier
. In fact, we can take the formula
relative to the structure
, as before. In this case, the sentence can be read “For every
,
”, or “Every
is
”, or some variant of this. Now if
meant that there is an assignment under which
is satisfied in
, then
says under every assignment,
is satisfied in
. Again, we can express what
says in various ways. For example, it says that every member of the domain is in the interpretation of
, or that the domain
, or that
has only solutions in
. Or again, it says: everything is a panda!
Clearly, not everything is a panda in general, but focusing on
, it is also not true that every living thing is a panda. Thus, it should be the case that
comes out false in
. And indeed, this is the case. First, we have the general case where:
iff for every
-variant assignment
,
.
But again, is it true that for every
-variant assignment
,
? Clearly not. For example, among the currently living things in the world, there is Kanzi the bonobo – that is,
. Now Kanzi is not a panda, so
. So if we take an
-variant assignment
(no matter what
was initially), then
is clearly not satisfied in
. That is,
. So it is not the case that every
-variant assignment
is such that
. So
, as expected. In other words,
is false – not every living thing is a panda.
To reiterate what we have seen here, when we have
true in a structure, it means we can find at least one assignment under which
is satisfied, while if we have
true in a structure, it means we can find that every assignment of a value to
satisfies
.
Exercise 7.4. For now, let’s stay with our structure
of living things in the world with the predicate
standing for pandas as specified. Then, think of how you would decide whether the following are true:
-
;
-
;
-
;
-
;
-
;
-
.