14 ideas
| 10688 | 'Equivocation' is when terms do not mean the same thing in premises and conclusion [Beall/Restall] |
| Full Idea: 'Equivocation' is when the terms do not mean the same thing in the premises and in the conclusion. | |
| From: JC Beall / G Restall (Logical Consequence [2005], Intro) |
| 10304 | Very few things in set theory remain valid in intuitionist mathematics [Bernays] |
| Full Idea: Very few things in set theory remain valid in intuitionist mathematics. | |
| From: Paul Bernays (On Platonism in Mathematics [1934]) |
| 10690 | Formal logic is invariant under permutations, or devoid of content, or gives the norms for thought [Beall/Restall] |
| Full Idea: Logic is purely formal either when it is invariant under permutation of object (Tarski), or when it has totally abstracted away from all contents, or it is the constitutive norms for thought. | |
| From: JC Beall / G Restall (Logical Consequence [2005], 2) | |
| A reaction: [compressed] The third account sounds rather woolly, and the second one sounds like a tricky operation, but the first one sounds clear and decisive, so I vote for Tarski. |
| 10691 | Logical consequence needs either proofs, or absence of counterexamples [Beall/Restall] |
| Full Idea: Technical work on logical consequence has either focused on proofs, where validity is the existence of a proof of the conclusions from the premises, or on models, which focus on the absence of counterexamples. | |
| From: JC Beall / G Restall (Logical Consequence [2005], 3) |
| 10695 | Logical consequence is either necessary truth preservation, or preservation based on interpretation [Beall/Restall] |
| Full Idea: Two different views of logical consequence are necessary truth-preservation (based on modelling possible worlds; favoured by Realists), or truth-preservation based on the meanings of the logical vocabulary (differing in various models; for Anti-Realists). | |
| From: JC Beall / G Restall (Logical Consequence [2005], 2) | |
| A reaction: Thus Dummett prefers the second view, because the law of excluded middle is optional. My instincts are with the first one. |
| 10689 | A step is a 'material consequence' if we need contents as well as form [Beall/Restall] |
| Full Idea: A logical step is a 'material consequence' and not a formal one, if we need the contents as well as the structure or form. | |
| From: JC Beall / G Restall (Logical Consequence [2005], 2) |
| 10696 | A 'logical truth' (or 'tautology', or 'theorem') follows from empty premises [Beall/Restall] |
| Full Idea: If a conclusion follows from an empty collection of premises, it is true by logic alone, and is a 'logical truth' (sometimes a 'tautology'), or, in the proof-centred approach, 'theorems'. | |
| From: JC Beall / G Restall (Logical Consequence [2005], 4) | |
| A reaction: These truths are written as following from the empty set Φ. They are just implications derived from the axioms and the rules. |
| 10693 | Models are mathematical structures which interpret the non-logical primitives [Beall/Restall] |
| Full Idea: Models are abstract mathematical structures that provide possible interpretations for each of the non-logical primitives in a formal language. | |
| From: JC Beall / G Restall (Logical Consequence [2005], 3) |
| 10692 | Hilbert proofs have simple rules and complex axioms, and natural deduction is the opposite [Beall/Restall] |
| Full Idea: There are many proof-systems, the main being Hilbert proofs (with simple rules and complex axioms), or natural deduction systems (with few axioms and many rules, and the rules constitute the meaning of the connectives). | |
| From: JC Beall / G Restall (Logical Consequence [2005], 3) |
| 10303 | Restricted Platonism is just an ideal projection of a domain of thought [Bernays] |
| Full Idea: A restricted Platonism does not claim to be more than, so to speak, an ideal projection of a domain of thought. | |
| From: Paul Bernays (On Platonism in Mathematics [1934], p.261) | |
| A reaction: I have always found Platonism to be congenial when it talks of 'ideals', and ridiculous when it talks of a special form of 'existence'. Ideals only 'exist' because we idealise things. I may declare myself, after all, to be a Restricted Platonist. |
| 10306 | Mathematical abstraction just goes in a different direction from logic [Bernays] |
| Full Idea: Mathematical abstraction does not have a lesser degree than logical abstraction, but rather another direction. | |
| From: Paul Bernays (On Platonism in Mathematics [1934], p.268) | |
| A reaction: His point is that the logicists seem to think that if you increasingly abstract from mathematics, you end up with pure logic. |
| 21354 | It may be that internal relations like proportion exist, because we directly perceive it [MacBride] |
| Full Idea: Some philosophers maintain that we literally perceive proportions and other internal relations. These relations must exist, otherwise we couldn't perceive them. | |
| From: Fraser MacBride (Relations [2016], 3) | |
| A reaction: [He cites Mulligan 1991, and Hochberg 2013:232] This seems a rather good point. You can't perceive the differing heights of two people, yet fail to perceive that one is taller. You also perceive 'below', which is external. |
| 21353 | Internal relations are fixed by existences, or characters, or supervenience on characters [MacBride] |
| Full Idea: Internal relations are determined either by the mere existence of the things they relate, or by their intrinsic characters, or they supervene on the intrinsic characters of the things they relate. | |
| From: Fraser MacBride (Relations [2016], 3) | |
| A reaction: Suggesting that they 'supervene' doesn't explain anything (and supervenience never explains anything). I vote for the middle one - the intrinsic character. It has to be something about the existence, and not the mere fact of existence. |
| 21352 | 'Multigrade' relations are those lacking a fixed number of relata [MacBride] |
| Full Idea: A 'unigrade' relation R has a definite degree or adicity: R is binary, or ternary....or n-ary (for some unique n). By contrast a relation is 'multigrade' if it fails to be unigrade. Causation appears to be multigrade. | |
| From: Fraser MacBride (Relations [2016], 1) | |
| A reaction: He also cites entailment, which may have any number of premises. |