14 ideas
| 10163 | Propositional modal logic has been proved to be complete [Kripke, by Feferman/Feferman] |
| Full Idea: At the age of 19 Saul Kripke published a completeness proof of propositional modal logic. | |
| From: report of Saul A. Kripke (A Completeness Theorem in Modal Logic [1959]) by Feferman / Feferman - Alfred Tarski: life and logic Int V |
| 10760 | With possible worlds, S4 and S5 are sound and complete, but S1-S3 are not even sound [Kripke, by Rossberg] |
| Full Idea: Kripke gave a possible worlds semantics to a whole range of modal logics, and S4 and S5 turned out to be both sound and complete with this semantics. Hence more systems could be designed. S1-S3 failed in soundness, leading to 'impossible worlds'. | |
| From: report of Saul A. Kripke (A Completeness Theorem in Modal Logic [1959]) by Marcus Rossberg - First-order Logic, 2nd-order, Completeness §4 |
| 16189 | The variable domain approach to quantified modal logic invalidates the Barcan Formula [Kripke, by Simchen] |
| Full Idea: Kripke's variable domain approach to quantified modal logic famously invalidates the Barcan Formula. | |
| From: report of Saul A. Kripke (A Completeness Theorem in Modal Logic [1959]) by Ori Simchen - The Barcan Formula and Metaphysics §3 | |
| A reaction: [p.9 and p.16] In a single combined domain all the possibilia must be present, but with variable domains objects in remote domains may not exist in your local domain. BF is committed to those possible objects. |
| 15132 | The Barcan formulas fail in models with varying domains [Kripke, by Williamson] |
| Full Idea: Kripke showed that the Barcan formula ∀x□A⊃□∀xA and its converse fail in models which require varying domains. | |
| From: report of Saul A. Kripke (A Completeness Theorem in Modal Logic [1959]) by Timothy Williamson - Truthmakers and Converse Barcan Formula §1 | |
| A reaction: I think this is why I reject the Barcan formulas for metaphysics - because the domain of metaphysics should be seen as varying, since some objects are possible in some contexts and not in others. Hmm… |
| 17884 | Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner] |
| Full Idea: There are many coherent stopping points in the hierarchy of increasingly strong mathematical systems, starting with strict finitism, and moving up through predicativism to the higher reaches of set theory. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], Intro) |
| 17893 | 'Reflection principles' say the whole truth about sets can't be captured [Koellner] |
| Full Idea: Roughly speaking, 'reflection principles' assert that anything true in V [the set hierarchy] falls short of characterising V in that it is true within some earlier level. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 2.1) |
| 17894 | We have no argument to show a statement is absolutely undecidable [Koellner] |
| Full Idea: There is at present no solid argument to the effect that a given statement is absolutely undecidable. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 5.3) |
| 17890 | There are at least eleven types of large cardinal, of increasing logical strength [Koellner] |
| Full Idea: Some of the standard large cardinals (in order of increasing (logical) strength) are: inaccessible, Mahlo, weakly compact, indescribable, Erdös, measurable, strong, Wodin, supercompact, huge etc. (...and ineffable). | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) | |
| A reaction: [I don't understand how cardinals can have 'logical strength', but I pass it on anyway] |
| 17887 | PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner] |
| Full Idea: To the extent that we are justified in accepting Peano Arithmetic we are justified in accepting its consistency, and so we know how to expand the axiom system so as to overcome the limitation [of Gödel's Second Theorem]. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.1) | |
| A reaction: Each expansion brings a limitation, but then you can expand again. |
| 17891 | Arithmetical undecidability is always settled at the next stage up [Koellner] |
| Full Idea: The arithmetical instances of undecidability that arise at one stage of the hierarchy are settled at the next. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) |
| 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. |
| 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. |