Q
P1
P2
Pn
P1
P2
Pn
Q
Another way to express this preservation of tautologousness is by using truth tables. A proposition
Q
P1
P2
Pn
P1
P2
Pn
Q
= "Socrates is a man." = "All men are mortal." = "Socrates is mortal."
{\thereforec}
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.
| b | c | a ∧ b | 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 a ∧ b, but does not assign T to c.
Tautological consequence can also be defined as
P1
P2
Pn
Q
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.