# Tautological consequence

In propositional logic, tautological consequence is a strict form of logical consequence[1] in which the tautologousness of a proposition is preserved from one line of a proof to the next. Not all logical consequences are tautological consequences. A proposition ${\displaystyle Q}$ is said to be a tautological consequence of one or more other propositions (${\displaystyle P_{1}}$, ${\displaystyle P_{2}}$, ..., ${\displaystyle P_{n}}$) in a proof with respect to some logical system if one is validly able to introduce the proposition onto a line of the proof within the rules of the system and in all cases when each of those one or more other propositions (${\displaystyle P_{1}}$, ${\displaystyle P_{2}}$, ..., ${\displaystyle P_{n}}$) are true, the proposition ${\displaystyle Q}$ also is true.

Another way to express this preservation of tautologousness is by using truth tables. A proposition ${\displaystyle Q}$ is said to be a tautological consequence of one or more other propositions (${\displaystyle P_{1}}$, ${\displaystyle P_{2}}$, ..., ${\displaystyle P_{n}}$) if and only if in every row of a joint truth table that assigns "T" to all propositions (${\displaystyle P_{1}}$, ${\displaystyle P_{2}}$, ..., ${\displaystyle P_{n}}$) the truth table also assigns "T" to ${\displaystyle Q}$.

## Example

a = "Socrates is a man." b = "All men are mortal." c = "Socrates is mortal."

a
b
${\displaystyle {\therefore c}}$

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.

Joint Truth Table for ab and c
a b c ab c
T T T T T
T T F T F
T F T F T
T F F F F
F T T F T
F T F F F
F F T F T
F F F F F

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 ab, 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.