10 ideas
| 11022 | Gentzen introduced a natural deduction calculus (NK) in 1934 [Gentzen, by Read] |
| Full Idea: Gentzen introduced a natural deduction calculus (NK) in 1934. | |
| From: report of Gerhard Gentzen (works [1938]) by Stephen Read - Thinking About Logic Ch.8 |
| 11065 | The inferential role of a logical constant constitutes its meaning [Gentzen, by Hanna] |
| Full Idea: Gentzen argued that the inferential role of a logical constant constitutes its meaning. | |
| From: report of Gerhard Gentzen (works [1938]) by Robert Hanna - Rationality and Logic 5.3 | |
| A reaction: Possibly inspired by Wittgenstein's theory of meaning as use? This idea was the target of Prior's famous connective 'tonk', which has the role of implying anything you like, proving sentences which are not logical consequences. |
| 11023 | The logical connectives are 'defined' by their introduction rules [Gentzen] |
| Full Idea: The introduction rules represent, as it were, the 'definitions' of the symbols concerned, and the elimination rules are no more, in the final analysis, than the consequences of these definitions. | |
| From: Gerhard Gentzen (works [1938]), quoted by Stephen Read - Thinking About Logic Ch.8 | |
| A reaction: If an introduction-rule (or a truth table) were taken as fixed and beyond dispute, then it would have the status of a definition, since there would be nothing else to appeal to. So is there anything else to appeal to here? |
| 11213 | Each logical symbol has an 'introduction' rule to define it, and hence an 'elimination' rule [Gentzen] |
| Full Idea: To every logical symbol there belongs precisely one inference figure which 'introduces' the symbol ..and one which 'eliminates' it. The introductions represent the 'definitions' of the symbols concerned, and eliminations are consequences of these. | |
| From: Gerhard Gentzen (works [1938], II.5.13), quoted by Ian Rumfitt - "Yes" and "No" III | |
| A reaction: [1935 paper] This passage is famous, in laying down the basics of natural deduction systems of logic (ones using only rules, and avoiding axioms). Rumfitt questions whether Gentzen's account gives the sense of the connectives. |
| 10067 | Gentzen proved the consistency of arithmetic from assumptions beyond arithmetic [Gentzen, by Musgrave] |
| Full Idea: Gentzen proved the consistency of arithmetic from assumptions which transcend arithmetic. | |
| From: report of Gerhard Gentzen (works [1938]) by Alan Musgrave - Logicism Revisited §5 | |
| A reaction: This does not contradict Gödel's famous result, but reinforces it. The interesting question is what assumptions Gentzen felt he had to make. |
| 14590 | If we accept scattered objects such as archipelagos, why not think of cars that way? [Hawthorne] |
| Full Idea: In being willing to countenance archipelagos, one embraces scattered objects. Why not then embrace the 'archipelago' of my car and the Eiffel Tower? | |
| From: John Hawthorne (Three-Dimensionalism v Four-Dimensionalism [2008], 2.1) | |
| A reaction: This is a beautifully simple and striking point. Language is full of embracing terms like 'the furniture', but that doesn't mean we assume the furniture is unified. The archipelago is less of an 'object' if you live on one of the islands. |
| 14591 | Four-dimensionalists say instantaneous objects are more fundamental than long-lived ones [Hawthorne] |
| Full Idea: Self-proclaimed four-dimensionalists typically adopt a picture that reckons instantaneous objects (and facts about them) to be more fundamental than long-lived ones. | |
| From: John Hawthorne (Three-Dimensionalism v Four-Dimensionalism [2008], 2.2) | |
| A reaction: A nice elucidation. As in Idea 14588, this seems motivated by a desire for some sort of foundationalism or atomism. Why shouldn't a metaphysic treat the middle-sized or temporally extended as foundational, and derive the rest that way? |
| 14589 | A modal can reverse meaning if the context is seen differently, so maybe context is all? [Hawthorne] |
| Full Idea: One person says 'He can't dig a hole; he hasn't got a spade', and another says 'He can dig a hole; just give him a spade', and both uses of the modal 'can' will be true. So some philosophers say that all modal predications are thus context-dependent. | |
| From: John Hawthorne (Three-Dimensionalism v Four-Dimensionalism [2008], 1.2) | |
| A reaction: Quine is the guru for this view of modality. Hawthorne's example seems to me to rely too much on the linguistic feature of contrasting 'can' and 'can't'. The underlying assertion in the propositions says something real about the possibilities. |
| 9425 | Lewis later proposed the axioms at the intersection of the best theories (which may be few) [Mumford on Lewis] |
| Full Idea: Later Lewis said we must choose between the intersection of the axioms of the tied best systems. He chose for laws the axioms that are in all the tied systems (but then there may be few or no axioms in the intersection). | |
| From: comment on David Lewis (Subjectivist's Guide to Objective Chance [1980], p.124) by Stephen Mumford - Laws in Nature |
| 14588 | Modern metaphysicians tend to think space-time points are more fundamental than space-time regions [Hawthorne] |
| Full Idea: Nowadays it is common for metaphysicians to hold both that space-time regions are less fundamental than the space-time points that compose them, and that facts about the regions are less fundamental than facts about the points and their arrangements. | |
| From: John Hawthorne (Three-Dimensionalism v Four-Dimensionalism [2008], 1) | |
| A reaction: I'm not quite sure what a physicist would make of this. It seems to be motivated by some a priori preference for atomism, and for system-building from minimal foundations. |