Tautological Consequent
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Theorem
A conditional with a tautology as consequent:
- $p \implies \top \dashv \vdash \top$
Proof by Natural Deduction
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $p \implies \top$ | Premise | (None) | ||
2 | $\top$ | Rule of Top-Introduction: $\top \II$ | (None) |
$\Box$
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $p$ | Assumption | (None) | ||
2 | 2 | $\top$ | Premise | (None) | ||
3 | 1 | $p \implies \top$ | Rule of Implication: $\implies \II$ | 1 – 2 | Assumption 1 has been discharged |
$\blacksquare$
Proof by Truth Table
We apply the Method of Truth Tables to the proposition.
As can be seen by inspection, in each case, the truth values in the appropriate columns match for all boolean interpretations.
$\begin{array}{|c|ccc||c|ccc|} \hline p & p & \implies & \top & \top \\ \hline F & F & T & T & T \\ T & T & T & T & T \\ \hline \end{array}$
$\blacksquare$
Also see
Sources
- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.3.3$