20 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) |
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) |
15510 | Classes are a host of ethereal, platonic, pseudo entities [Goodman] |
Full Idea: I will not willingly use apparatus that peoples the world with a host of ethereal, platonic, pseudo entities. | |
From: Nelson Goodman (The Structure of Appearance [1951], II.2), quoted by David Lewis - Parts of Classes 2.1 | |
A reaction: This represents the big gap that opened up with Goodman's former comrade in arms, Quine. Lewis quotes it in order to ask whether he means ethereal or platonic, as they are very different. I sympathise with Goodman. |
9920 | Two objects can apparently make up quite distinct arrangements in sets [Goodman, by Burgess/Rosen] |
Full Idea: Goodman argues that the set or class {{a}},{a,b}} is supposed to be distinct from the set or class {{b},{a,b}}, even though both are ultimately constituted from the same a and b. | |
From: report of Nelson Goodman (The Structure of Appearance [1951]) by JP Burgess / G Rosen - A Subject with No Object I.A.2.a | |
A reaction: I'm with Goodman all the way here, even though it is deeply unfashionable, particularly in the circles I move in. If there are trillion grains of sand on a beach, how many sets are we supposed to be committed to? |
10657 | The counties of Utah, and the state, and its acres, are in no way different [Goodman] |
Full Idea: A class (counties of Utah) is different neither from the individual (state of Utah) that contains its members, nor from any other class (acres of Utah) whose members exhaust the whole. For nominalists, distinction of entity means distinction of content. | |
From: Nelson Goodman (The Structure of Appearance [1951], p.26), quoted by Achille Varzi - Mereology 3.1 | |
A reaction: This is a nice credo for the nominalist version of mereology. You can still have a mereology that commits you to the wholes as well as the parts. Cf. Lewis in Idea 10660. |
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) |
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] |
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) |
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) |
7956 | If all and only red things were round things, we would need to specify the 'respect' of the resemblance [Goodman, by Macdonald,C] |
Full Idea: According to Goodman's 'companionship difficulty', resemblance nominalism has a problem if, say, all and only the red things were the round things, because we cannot distinguish the two different respects in which the things resemble one another. | |
From: report of Nelson Goodman (The Structure of Appearance [1951]) by Cynthia Macdonald - Varieties of Things Ch.6 | |
A reaction: Goodman opts for extreme linguististic nominalism in response to this (Idea 7952), whereas Russell opts for a sort of Platonism (4441). The current idea gives Russell a further problem, of needing a universal of the respect of the resemblance. |
7957 | Without respects of resemblance, we would collect blue book, blue pen, red pen, red clock together [Goodman, by Macdonald,C] |
Full Idea: Goodman's 'imperfect community' problem for Resemblance Nominalism says that without mention of respects in which things resemble, we end up with a heterogeneous collection with nothing wholly in common (blue book, blue pen, red pen, red clock). | |
From: report of Nelson Goodman (The Structure of Appearance [1951]) by Cynthia Macdonald - Varieties of Things Ch.6 | |
A reaction: This suggests Wittgenstein's 'family' resemblance as a way out (Idea 4141), but a blue book and a red clock seem totally unrelated. Nice objection! At this point we start to think that the tropes resemble, rather than the objects. |
7952 | If we apply the same word to different things, it is only because we are willing to do so [Goodman, by Macdonald,C] |
Full Idea: Predicate nominalism is the view that what all things to which the same word applies have in common is simply our willingness to apply the same word to them. | |
From: report of Nelson Goodman (The Structure of Appearance [1951], Ch.6) by Cynthia Macdonald - Varieties of Things | |
A reaction: This is Goodman's 'extreme nominalist' position. This seems also to be an anti-realist position, as it denies any 'joints' to nature (Idea 7953). It strikes me as daft. WHY are we willing to apply words in certain ways? |