Notes
truth-value assignment
truth on a truth value assignment.
A sentence is true on a truth-value assignment if
and only if it has the truth-value $\mathbf{T}$ (true) on the
truth-value assignment.
false on a truth value assignment.
A sentence is false on a truth-value assignment if
and only if it has the truth-value $\mathbf{F}$ (false) on
the truth-value assignment.
truth-functionally true.
A sentence $\mathscr{P}$ is truth-functionally true
if and only if $\mathscr{P}$ is true on every truth-value
assignment. (Hence, such a sentence is logically true as well.)
truth-functionally false.
A sentence $\mathscr{P}$ is truth-functionally false
if and only if $\mathscr{P}$ is false on every truth-value
assignment. (Hence, such a sentence is logically false as well.)
truth-functionally indeterminate.
A sentence $\mathscr{P}$ is truth-functionally
indeterminate if and only if $\mathscr{P}$ is neither
truth-functionally true nor truth-functionally false.
(Hence, such a sentence is logically indeterminate as
well.) Equivalently, a truth-functionally indeterminate
sentence is true on at least one truth-value assignment
and false on at least one truth-value assignment.
truth-functionally consistent.
A set of sentences of SL is truth-functionally consistent
if and only if there is at least one truth-value
assignment on which all the members of the set are
true. (Hence, such sentences are always logically
consistent as well.)
truth-functionally inconsistent.
A set of sentences of SL is truth-functionally
inconsistent if and only if it is not truth-functionally
consistent. (Hence, such sentences are always
logically inconsistent as well.)
truth-functionally equivalent.
Sentences $\mathscr{P}$ and $\mathscr{Q}$ are
truth-functionally equivalent if and only if there
is no truth-value assignment on which $\mathscr{P}$
and $\mathscr{Q}$ have different truth-values.
(Hence, such sentences are always logically
equivalent as well.)
truth-functional entailment.
A set $\Gamma$ of sentences of SL truth-functionally entails
a sentence $\mathscr{P}$ if and only if there is no
truth-value assignment on which every member of $\Gamma$
is true and $\mathscr{P}$ is false. This is symbolized
as
'$\Gamma\models\mathscr{P}$'
and is read,
"$\Gamma$
truth-functionally
entails
$\mathscr{P}$."
If $\Gamma$ does not truth-functionally entail
$\mathscr{P},$ we write
'$\Gamma\not\models\mathscr{P}$',
which reads,
"$\Gamma$
doesn’t truth-functionally entail
$\mathscr{P}$."
truth-functional validity
An argument is truth-functionally valid if and only
if there is no truth-value assignment on which all
the premises are true and the conclusion is false.
(Hence, such an argument is deductively valid as
well.) This means that an argument is truth-functionally
valid if and only if the set consisting of the premises
of the argument truth-functionally entails the conclusion.
truth-functional invalidity.
An argument of SL is truth-functionally invalid if
and only if it is not truth-functionally valid.
(Hence, such an argument is deductively invalid as
well.) This means that an argument is truth-functionally
invalid if and only if the set consisting of the
premises of the argument doesn’t truth-functionally
entail the conclusion.
Alex Notes