display all the ideas for this combination of texts
2 ideas
| 13698 | In a complete logic you can avoid axiomatic proofs, by using models to show consequences [Sider] |
| Full Idea: You can establish facts of the form Γ|-φ while avoiding the agonies of axiomatic proofs by reasoning directly about models to conclusions about semantic consequence, and then citing completeness. | |
| From: Theodore Sider (Logic for Philosophy [2010], 4.5) | |
| A reaction: You cite completeness by saying that anything which you have shown to be a semantic consequence must therefore be provable (in some way). |
| 13699 | Compactness surprisingly says that no contradictions can emerge when the set goes infinite [Sider] |
| Full Idea: Compactness is intuitively surprising, ..because one might have thought there could be some contradiction latent within some infinite set, preventing it from being satisfiable, only discovered when you consider the whole set. But this can't happen. | |
| From: Theodore Sider (Logic for Philosophy [2010], 4.5) |