display all the ideas for this combination of texts
6 ideas
| 13688 | Good axioms should be indisputable logical truths [Sider] |
| Full Idea: Since they are the foundations on which a proof rests, the axioms in a good axiomatic system ought to represent indisputable logical truths. | |
| From: Theodore Sider (Logic for Philosophy [2010], 2.6) |
| 13687 | No assumptions in axiomatic proofs, so no conditional proof or reductio [Sider] |
| Full Idea: Axiomatic systems do not allow reasoning with assumptions, and therefore do not allow conditional proof or reductio ad absurdum. | |
| From: Theodore Sider (Logic for Philosophy [2010], 2.6) | |
| A reaction: Since these are two of the most basic techniques of proof which I have learned (in Lemmon), I shall avoid axiomatic proof systems at all costs, despites their foundational and Ockhamist appeal. |
| 13690 | Proof by induction 'on the length of the formula' deconstructs a formula into its accepted atoms [Sider] |
| Full Idea: The style of proof called 'induction on formula construction' (or 'on the number of connectives', or 'on the length of the formula') rest on the fact that all formulas are built up from atomic formulas according to strict rules. | |
| From: Theodore Sider (Logic for Philosophy [2010], 2.7) | |
| A reaction: Hence the proof deconstructs the formula, and takes it back to a set of atomic formulas have already been established. |
| 13691 | Induction has a 'base case', then an 'inductive hypothesis', and then the 'inductive step' [Sider] |
| Full Idea: A proof by induction starts with a 'base case', usually that an atomic formula has some property. It then assumes an 'inductive hypothesis', that the property is true up to a certain case. The 'inductive step' then says it will be true for the next case. | |
| From: Theodore Sider (Logic for Philosophy [2010], 2.7) | |
| A reaction: [compressed] |
| 13685 | Natural deduction helpfully allows reasoning with assumptions [Sider] |
| Full Idea: The method of natural deduction is popular in introductory textbooks since it allows reasoning with assumptions. | |
| From: Theodore Sider (Logic for Philosophy [2010], 2.5) | |
| A reaction: Reasoning with assumptions is generally easier, rather than being narrowly confined to a few tricky axioms, You gradually show that an inference holds whatever the assumption was, and so end up with the same result. |
| 13686 | We can build proofs just from conclusions, rather than from plain formulae [Sider] |
| Full Idea: We can construct proofs not out of well-formed formulae ('wffs'), but out of sequents, which are some premises followed by their logical consequence. We explicitly keep track of the assumptions upon which the conclusion depends. | |
| From: Theodore Sider (Logic for Philosophy [2010], 2.5.1) | |
| A reaction: He says the method of sequents was invented by Gerhard Gentzen (the great nazi logician) in 1935. The typical starting sequents are the introduction and elimination rules. E.J. Lemmon's book, used in this database, is an example. |