In
propositional logic
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
, tautological consequence is a strict form of
logical consequence[Barwise and Etchemendy 1999, p. 110] in which the
tautologousness of a
proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
is preserved from one line of a proof to the next. Not all logical consequences are tautological consequences. A
proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
is said to be a tautological consequence of one or more other propositions (
,
, ...,
) in a
proof with respect to some
logical system
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system.
A form ...
if one is
validly able to introduce the proposition onto a line of the proof within the
rules
Rule or ruling may refer to:
Education
* Royal University of Law and Economics (RULE), a university in Cambodia
Human activity
* The exercise of political or personal control by someone with authority or power
* Business rule, a rule pert ...
of the system and in all cases when each of those one or more other propositions (
,
, ...,
) are true, the proposition
also is true.
Another way to express this preservation of tautologousness is by using
truth table
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional argumen ...
s. A proposition
is said to be a tautological consequence of one or more other propositions (
,
, ...,
) if and only if in every row of a joint truth table that assigns "T" to all propositions (
,
, ...,
) the truth table also assigns "T" to
.
Example
= "Socrates is a man."
= "All men are mortal."
= "Socrates is mortal."
:
:
:
The conclusion of this argument is a logical consequence of the premises because it is impossible for all the premises to be true while the conclusion false.
Reviewing the truth table, it turns out the conclusion of the argument is ''not'' a tautological consequence of the premise. Not every row that assigns T to the premise also assigns T to the conclusion. In particular, it is the second row that assigns T to ''a'' ∧ ''b'', but does not assign T to ''c''.
Denotation and properties
It follows from the definition that if a proposition ''p'' is a contradiction then ''p'' tautologically implies every proposition, because there is no truth valuation that causes ''p'' to be true and so the definition of tautological implication is trivially satisfied. Similarly, if ''p'' is a tautology then ''p'' is tautologically implied by every proposition.
See also
*
Logical consequence
*
Tautology (logic)
In mathematical logic, a tautology (from el, ταυτολογία) is a formula or assertion that is true in every possible interpretation. An example is "x=y or x≠y". Similarly, "either the ball is green, or the ball is not green" is always ...
*
Truth table
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional argumen ...
Notes
References
*
Barwise, Jon, and
John Etchemendy
John W. Etchemendy (born 1952 in Reno, Nevada) is an American logician and philosopher who served as Stanford University's twelfth Provost. He succeeded John L. Hennessy to the post on September 1, 2000 and stepped down on January 31, 2017.
E ...
. ''Language, Proof and Logic''. Stanford: CSLI (Center for the Study of Language and Information) Publications, 1999. Print.
*
Kleene, S. C. (1967) ''Mathematical Logic'', reprinted 2002,
Dover Publications
Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, book ...
, {{ISBN, 0-486-42533-9.
Logical consequence