Rule of Addition/Sequent Form/Formulation 2/Form 1
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Theorem
- $\vdash p \implies \paren {p \lor q}$
Proof 1
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $p$ | Premise | (None) | ||
2 | 1 | $p \lor q$ | Rule of Addition: $\lor \II_1$ | 1 | ||
3 | $p \implies \paren {p \lor q}$ | Rule of Implication: $\implies \II$ | 1 – 3 | Assumption 1 has been discharged |
$\blacksquare$
Proof 2
This proof is derived in the context of the following proof system: Instance 2 of the Hilbert-style systems.
By the tableau method:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | $q \implies \paren {p \lor q}$ | Axiom $\text A 2$ | ||||
2 | $p \implies \paren {q \lor p}$ | Rule $\text {RST} 1$ | 1 | $p \,/\, q, q \,/\, p$ | ||
3 | $\paren {p \lor q} \implies \paren {q \lor p}$ | Axiom $\text A 3$ | ||||
4 | $\paren {q \lor p} \implies \paren {p \lor q}$ | Rule $\text {RST} 1$ | 3 | $p \,/\, q, q \,/\, p$ | ||
5 | $p \implies \paren {p \lor q}$ | Hypothetical Syllogism | 2, 4 |
$\blacksquare$
Sources
- 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 3.6$: Reference Formulae: $RF \, 6$
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $2$: The Propositional Calculus $2$: $2$: Theorems and Derived Rules: Exercise $3 \ \text{(a)}$