Rule of Addition/Sequent Form/Formulation 2/Form 1/Proof 2
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Theorem
- $\vdash p \implies \paren {p \lor q}$
Proof
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): $\S 4.7$: The Derivation of Formulae: $D \, 4$