Notes
metavariable
An expression in the metalanguage that is used to
talk generally about expressions of the object language.
The language SL
'SL' stands for
'Sentential Logic',
the language
described in this chapter.
SL vocabulary
atomic sentence (SL)
Every capital letter with or without integer
subscripts is an SL atomic sentence.
sentence (SL)
All SL sentences are defined as follows:
-
An SL atomic sentence is an SL sentence.
-
$\mathscr{P}$ is an SL sentence
if and only if
$\sim\mathscr{P}$
is an SL sentence.
-
$\mathscr{P}$ and $\mathscr{Q}$ are SL sentences
if and only if
($\mathscr{P}\&\mathscr{Q})$
is an SL sentence.
-
$\mathscr{P}$ and $\mathscr{Q}$ are SL sentences
if and only if
$(\mathscr{P}\lor\mathscr{Q})$
is an SL sentence.
-
$\mathscr{P}$ and $\mathscr{Q}$ are SL sentences
if and only if
$(\mathscr{P}\supset\mathscr{Q})$
is an SL sentence.
-
$\mathscr{P}$ and $\mathscr{Q}$ are SL sentences
if and only if
$(\mathscr{P}\equiv\mathscr{Q}$)
is an SL sentence.
characteristic truth tables
main logical operator (SL)
immediate components (SL)
contrapositive
The contrapositive of a sentence of the form
$\mathscr{P}\supset\mathscr{Q}$ is the
sentence $\sim\mathscr{Q}\supset\sim\mathscr{P}.$
inverse
The inverse of a sentence of the form
$\mathscr{P}\supset\mathscr{Q}$ is
the sentence $\sim\mathscr{P}\supset\sim\mathscr{Q}.$
Note that the inverse of a statement is equivalent
to the converse of the statement.
Converse
The converse of a sentence of the form
$\mathscr{P}\supset\mathscr{Q}$ is
the sentence $\mathscr{Q}\supset{P}.$ Note
that the converse of a statement is equivalent
to the inverse of the statement.
Function
A function is a rule which relates to each element
in a set called the domain exactly one element from
another set called the range, and every element in
the range to at least one element in the domain.
(Elements not so related are not elements of either
the domain or the range.) (My definition)
truth function
A truth function is any function whose domain is
$\{\mathbf{T},\mathbf{F}\}$ and whose range is
either $\{\mathbf{T},\mathbf{F}\}$ or $\{T\}.$
(My definition)
truth function
A truth function is a mapping of each possible
combination of truth-values that $n$ sentences
have ($n$ sentences have $2^n$ possible combinations
of truth values) to a unique truth-value assignment
for some positive integer $n.$ The truth-values of
such a function are the functional values of the
function, and each of these values is a function
of one of the possible combinations of truth-values
of the arguments of the function. (The Logic Book’s
definition)
truth-functional connective
A sentential connective is used truth-functionally
if and only if it is used to generate a compound
sentence from one or more sentences in such a way
that the truth-value of the generated compound is
wholly determined by the truth-values of those one
or more sentences from which the compound is
generated, no matter what those truth-values may be.
Alex Notes