16 ideas
| 15457 | Interdefinition is useless by itself, but if we grasp one separately, we have them both [Lewis] |
| Full Idea: All circles of interdefinition are useless by themselves. But if we reach one of the interdefined pair, then we have them both. | |
| From: David Lewis (Defining 'Intrinsic' (with Rae Langton) [1998], IV) |
| 13838 | A decent modern definition should always imply a semantics [Hacking] |
| Full Idea: Today we expect that anything worth calling a definition should imply a semantics. | |
| From: Ian Hacking (What is Logic? [1979], §10) | |
| A reaction: He compares this with Gentzen 1935, who was attempting purely syntactic definitions of the logical connectives. |
| 13833 | 'Thinning' ('dilution') is the key difference between deduction (which allows it) and induction [Hacking] |
| Full Idea: 'Dilution' (or 'Thinning') provides an essential contrast between deductive and inductive reasoning; for the introduction of new premises may spoil an inductive inference. | |
| From: Ian Hacking (What is Logic? [1979], §06.2) | |
| A reaction: That is, inductive logic (if there is such a thing) is clearly non-monotonic, whereas classical inductive logic is monotonic. |
| 13834 | Gentzen's Cut Rule (or transitivity of deduction) is 'If A |- B and B |- C, then A |- C' [Hacking] |
| Full Idea: If A |- B and B |- C, then A |- C. This generalises to: If Γ|-A,Θ and Γ,A |- Θ, then Γ |- Θ. Gentzen called this 'cut'. It is the transitivity of a deduction. | |
| From: Ian Hacking (What is Logic? [1979], §06.3) | |
| A reaction: I read the generalisation as 'If A can be either a premise or a conclusion, you can bypass it'. The first version is just transitivity (which by-passes the middle step). |
| 13835 | Only Cut reduces complexity, so logic is constructive without it, and it can be dispensed with [Hacking] |
| Full Idea: Only the cut rule can have a conclusion that is less complex than its premises. Hence when cut is not used, a derivation is quite literally constructive, building up from components. Any theorem obtained by cut can be obtained without it. | |
| From: Ian Hacking (What is Logic? [1979], §08) |
| 13845 | The various logics are abstractions made from terms like 'if...then' in English [Hacking] |
| Full Idea: I don't believe English is by nature classical or intuitionistic etc. These are abstractions made by logicians. Logicians attend to numerous different objects that might be served by 'If...then', like material conditional, strict or relevant implication. | |
| From: Ian Hacking (What is Logic? [1979], §15) | |
| A reaction: The idea that they are 'abstractions' is close to my heart. Abstractions from what? Surely 'if...then' has a standard character when employed in normal conversation? |
| 13840 | First-order logic is the strongest complete compact theory with Löwenheim-Skolem [Hacking] |
| Full Idea: First-order logic is the strongest complete compact theory with a Löwenheim-Skolem theorem. | |
| From: Ian Hacking (What is Logic? [1979], §13) |
| 13844 | A limitation of first-order logic is that it cannot handle branching quantifiers [Hacking] |
| Full Idea: Henkin proved that there is no first-order treatment of branching quantifiers, which do not seem to involve any idea that is fundamentally different from ordinary quantification. | |
| From: Ian Hacking (What is Logic? [1979], §13) | |
| A reaction: See Hacking for an example of branching quantifiers. Hacking is impressed by this as a real limitation of the first-order logic which he generally favours. |
| 13842 | Second-order completeness seems to need intensional entities and possible worlds [Hacking] |
| Full Idea: Second-order logic has no chance of a completeness theorem unless one ventures into intensional entities and possible worlds. | |
| From: Ian Hacking (What is Logic? [1979], §13) |
| 13837 | With a pure notion of truth and consequence, the meanings of connectives are fixed syntactically [Hacking] |
| Full Idea: My doctrine is that the peculiarity of the logical constants resides precisely in that given a certain pure notion of truth and consequence, all the desirable semantic properties of the constants are determined by their syntactic properties. | |
| From: Ian Hacking (What is Logic? [1979], §09) | |
| A reaction: He opposes this to Peacocke 1976, who claims that the logical connectives are essentially semantic in character, concerned with the preservation of truth. |
| 13839 | Perhaps variables could be dispensed with, by arrows joining places in the scope of quantifiers [Hacking] |
| Full Idea: For some purposes the variables of first-order logic can be regarded as prepositions and place-holders that could in principle be dispensed with, say by a system of arrows indicating what places fall in the scope of which quantifier. | |
| From: Ian Hacking (What is Logic? [1979], §11) | |
| A reaction: I tend to think of variables as either pronouns, or as definite descriptions, or as temporary names, but not as prepositions. Must address this new idea... |
| 13843 | If it is a logic, the Löwenheim-Skolem theorem holds for it [Hacking] |
| Full Idea: A Löwenheim-Skolem theorem holds for anything which, on my delineation, is a logic. | |
| From: Ian Hacking (What is Logic? [1979], §13) | |
| A reaction: I take this to be an unusually conservative view. Shapiro is the chap who can give you an alternative view of these things, or Boolos. |
| 9916 | Convention, yes! Arbitrary, no! [Poincaré, by Putnam] |
| Full Idea: Poincaré once exclaimed, 'Convention, yes! Arbitrary, no!'. | |
| From: report of Henri Poincaré (talk [1901]) by Hilary Putnam - Models and Reality | |
| A reaction: An interesting view. It mustn't be assumed that conventions are not rooted in something. Maybe a sort of pragmatism is implied. |
| 15400 | We must avoid circularity between what is intrinsic and what is natural [Lewis, by Cameron] |
| Full Idea: Lewis revised his analysis of duplication because he had assumed that as a matter of necessity perfectly natural properties are intrinsic, and that necessarily how a thing is intrinsically is determined completely by the natural properties it has. | |
| From: report of David Lewis (Defining 'Intrinsic' (with Rae Langton) [1998]) by Ross P. Cameron - Intrinsic and Extrinsic Properties 'Analysis' | |
| A reaction: [This compares Lewis 1986:61 with Langton and Lewis 1998] I am keen on both intrinsic and on natural properties, but I have not yet confronted this little problem. Time for a displacement activity, I think.... |
| 15458 | A property is 'intrinsic' iff it can never differ between duplicates [Lewis] |
| Full Idea: A property is 'intrinsic' iff it never can differ between duplicates; iff whenever two things (actual or possible) are duplicates, either both of them have the property or both of them lack it. | |
| From: David Lewis (Defining 'Intrinsic' (with Rae Langton) [1998], IV) | |
| A reaction: This leaves me wondering how one could arrive at a precise definition of 'duplicates'. Can it be done without mentioning that they have the same intrinsic properties? |
| 15459 | Ellipsoidal stars seem to have an intrinsic property which depends on other objects [Lewis] |
| Full Idea: The property of being an ellipsoidal star would seem offhand to be a basic intrinsic property, but it is incompatible (nomologically) with being an isolated object. | |
| From: David Lewis (Defining 'Intrinsic' (with Rae Langton) [1998], V) | |
| A reaction: Another nice example from Lewis. It makes you wonder whether the intrinsic/extrinsic distinction should go. Modern physics, with its 'entanglements', doesn't seem to suit the distinction. |