74
Chapter 3: Semantics for Sentential Logic
7
Expressive completeness
At the end of §1 in Chapter 2 we claimed that our five sentential connectives
‘
~
’, ‘
∨
’, ‘
&
’, ‘
→
’ and ‘
↔
’ are all we need in sentential logic, since other sentential
connectives are either
definable
in terms of these five or else
beyond the scope
of sentential logic. When we say that a connective is beyond the scope of clas-
sical sentential logic, what we mean is that it is
non-truth-functional;
in other
words, there is no truth-function that it expresses (see §1 of this chapter for a
discussion of expressing a truth-function). In the next section we will consider
various connectives of this sort. Meanwhile, we will concern ourselves with the
definability of other truth-functional connectives.
An example of a truth-functional connective which is definable in terms of
our five is ‘neither…nor…’, since for any English sentences
p
and
q
,
[
neither
p
nor
q
\
is correctly paraphrased as
[
not
p
and not
q
\
(see (11) on page 18). But
this is just one example. How can we be confident that
every
truth-functional
connective can be defined in terms of ‘
~
’, ‘
∨
’, ‘
&
’, ‘
→
’ and ‘
↔
’? Our confidence
is based in the fact that our collection of connectives has a property called
expressive completeness
, which we now explain.
At the end of §1 of this chapter, we listed the function-tables for the one-
place, or
unary
, function expressed by ‘
~
’, and the four two-place, or
binary
,
functions expressed by the other connectives. However, there are many more
unary and binary truth-functions than are expressed by the five connectives
individually. For example, there are three other unary truth-functions:
>
⇒
>>
⇒
⇒
⊥
⊥
⇒
>⊥
⇒
⊥⊥
⇒
⊥
(a)
(b)
(c)
To show that all unary truth-functional connectives are definable in terms of
our five basic connectives, we establish the stronger result that all unary truth-
functions are definable, whether or not they are expressed by some English
connective. (While (b) is expressed by ‘it is true that’, neither (a) nor (c) has an
uncontrived rendering.) Our question is therefore whether we can express all
of (a), (b) and (c) in terms of our five chosen connectives. And in this case it is
easy to see that (a) is captured by ‘…
∨
~
…’, (b) by ‘
~~
…’, and (c) by ‘…
&
~
…’,
where in (a) and (c) the same formula fills both ellipses.
What about the other binary truth-functions? We have connectives for four,
and we know how to define a fifth, the truth-function
>>
⇒
⊥
>⊥
⇒
⊥
⊥>
⇒
⊥
⊥⊥
⇒
>
which is expressed by ‘neither…nor…’ (to repeat,
[
neither
p
nor
q
\
is true just
in case both
p
and
q
are false, so we express it with
[
~
p
&
~
q
\
). But there are