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 (${\displaystyle P_{1}}$, ${\displaystyle P_{2}}$, ..., ${\displaystyle P_{n}}$) 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 a ∧ b and c

a	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.

Denotation and properties

Tautological consequence can also be defined as ${\displaystyle P_{1}}$ ∧ ${\displaystyle P_{2}}$ ∧ ... ∧ ${\displaystyle P_{n}}$ → ${\displaystyle Q}$ is a substitution instance of a tautology, with the same effect. ^[2]

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)
• Truth table