Take a set
with two elements. The elements have to be distinct and that’s all.
Usually these two elements are designated by T and F, sometimes by 0 and
1, or by any other marks you prefer (provided that the marks are of
distinct types). We shall designate the very set by
How many functions are there from
Let us call them
Tv (A) will be either true (T )
or false (F). Now take a sentential connective say
Tv
(A Intuitively,
what all this means is that the truth value of the whole expression
“A The
beauty of truth-functions (or, if you prefer, of “binary logic”)
is that with only few truth-functions you can define any other
truth-functions of any arity. For example
*R
For example
say R
. . . . ... . .
*R
As *R
&”
F (
p_{1}, p_{2} ,... p_{n })
is defined only in terms of “
”,
“̴&”,
and “∨”
. Thus any truth
functional connective can be defined in terms of “
”,
“̴&”,
and “∨”
. Now again “&”can
be defined in terms of of “
”
and “̴∨”
or “∨”can
be defined in terms of of “
”
and “̴&”
. So again any truth functional sentence can be defined either by
“
”
and “̴∨
”
or by
“
”
and “̴&”
. Interestingly any of these three expressions can be defined only by
the connectives
“
& q =_{def }(_{ }p↓p)↓(q↓q
) _{ }=_{def } (p↑q)↑(p↑q)
Thus any truth functional compound can have a monolithic definition either exclusively in terms of “↓” (down stroke) or exclusively in terms of “↑” (up stroke).
1
At the very outset we can take the variables
2
Note we are not using the little subscript 3
Generally, if p_{1},p_{2}
,..., p_{n
}) is an n-ary
sentential connective and if we get an n-ary truth function F_{2
}
so that
Tv
( p_{1},p_{2}
,... p_{n }) ) = F_{2
}(Tv (
p_{1}), Tv (p_{2}),
...,(Tv (p_{n })
) then F is truth-functional. Following our text we can call
F_{2
}as the characteristic truth function of .
Thus F&_{2
}is the characteristic truth function of &.
Such subtle distinction between a truth-functional connective and
its characteristic truth function is often ignored in the
literature.4 We are now back to the usual practice, ignoring the difference between a connective and its characteristic truth-function and the difference between a sentence and its truth-value. 5 We are ignoring the strict well-formed-formula rules , which requires that there should be proper grouping in terms of parentheses 6
The two expressions |