Logical Properties of Sentences, Sets of Sentences, and Arguments
Definition. A statement is logically indeterminate (or contingent) if and only if it's
possible for it to be true and it's possible for it to be false.
In the language of possible-world semantics, a statement is logically indeterminate (or
contingent) if and only if it's true in some possible worlds and false in others.
Definition. A statement is logically true (or a tautology) if and only if it's impossible for
it to be false.
In the language of possible-world semantics, a statement is logically true (or a
tautology) if and only if it's true in all possible worlds.
Definition. A statement is logically false (or contradictory) if and only if it's impossible
for it to be true.
In the language of possible-world semantics, a statement is logically false (or
contradictory) if and only if it's false in all possible worlds.
Definition (alternate). A statement is logically indeterminate (or contingent) if and
only if it's neither logically true nor logically false.
Definition. A set of statements is inconsistent (or contradictory or unsatisfiable) if and
only if it's impossible for all its members to be true at the same time.
In the language of possible-world semantics, a set of statements is inconsistent (or
contradictory or unsatisfiable) if and only if there’s no possible world in which all of
the statements in the set are true at the same time.
Definition. A set of statements is consistent (or satisfiable) if and only if it's possible for
all of its members to be true at the same time.
In the language of possible-world semantics, a set of statements is consistent (or
satisfiable) if and only if there’s at least one possible world in which all the members of
the set are true at the same time.
Definition. A statement P implies a statement Q if and only if it's impossible for P to be
true while Q is false.
In the language of possible-world semantics, a statement P implies a statement Q if and
only if every possible world in which P is true is a world in which Q is true as well.
Definition. P is logically equivalent to Q if, and only if, both P implies Q and Q implies
P.
Defintion (alternate) P is logically equivalent to Q if, and only if, P and Q must always
have the same truth value.
In the language of possible-world semantics, P is logically equivalent to Q if, and only if,
every world in which P is true is one in which Q is true, and vice versa..
Definition. An argument form is logically valid if and only if it’s impossible for all of
the premises to be true and the conclusion to be false at the same time.
Definition (alternate). An argument form is logically valid if and only if whenever the
premises are true, the conclusion is true as well.
In the language of possible-world semantics, an argument is valid if and only if every
possible world in which the premises are all true is a world in which the conclusion is
true as well. Or we can say that there is no possible world in which the premises are all
true and the conclusion is false.
Definition. An argument form is logically invalid if and only if it is possible for the
premises to be true and the conclusion to be false at the same time..
In the language of possible-world semantics, an argument form is invalid if and only if
there’s at least one possible world in which the premises are all true but the conclusion is
false.
Definition. An argument form is logically sound if and only if (a) it is logically valid,
and (b) all of its premises are true (in the actual world.)
Truth-functional properties of sentences and sets of sentences
Definition. A statement is truth-functionally indeterminate (or contingent) if and only
if some truth table assignments, under the main connective, are true and some are false.
Definition. A statement is truth-functionally true (or a tautology) if and only if all truth
table assignments, under the main connective, are true.
Definition. A statement is truth-functionally false (or contradictory) if and only if all
truth table assigments, under the main connective, are false.
Definition. A set of statements is truth-functionally inconsistent (or contradictory) if
and only if there is no truth table assignment, under the main connective of each formula,
for which all of the formulas are true.
Definition. A set of statements is truth-functionally consistent (or satisfiable) if and
only if there’s at least one truth table assignment, under the main connective of each
formula, for which all of the formulas are true.
Definition. A statement P truth-functionally implies a statement Q if and only if
whenever the truth table assignment under the main connective for P is true, then the truth
table assignment under the main connective of Q is true as well.
Definition. P is truth-functionally equivalent to Q if, and only if, all assignments of the
truth table under the main connective for each formula are the same .
Definition. An argument form is truth-functionally valid if and only if there’s no
assignment of truth table values under the main connective for each formula for which the
premises are all true and the conclusion is false.
Definition. An argument form is truth-functionally invalid if and only if there is at least
one assignment of truth table values under the main connective for each formula for
which the premises are true and the conclusion is false.