20 ideas
| 9978 | Analytic philosophy focuses too much on forms of expression, instead of what is actually said [Tait] |
| Full Idea: The tendency to attack forms of expression rather than attempting to appreciate what is actually being said is one of the more unfortunate habits that analytic philosophy inherited from Frege. | |
| From: William W. Tait (Frege versus Cantor and Dedekind [1996], IV) | |
| A reaction: The key to this, I say, is to acknowledge the existence of propositions (in brains). For example, this belief will make teachers more sympathetic to pupils who are struggling to express an idea, and verbal nit-picking becomes totally irrelevant. |
| 12204 | The logic of metaphysical necessity is S5 [Rumfitt] |
| Full Idea: It is a widely accepted thesis that the logic of metaphysical necessity is S5. | |
| From: Ian Rumfitt (Logical Necessity [2010], §5) | |
| A reaction: Rumfitt goes on to defend this standard view (against Dummett's defence of S4). The point, I take it, is that one can only assert that something is 'true in all possible worlds' only when the worlds are all accessible to one another. |
| 9986 | The null set was doubted, because numbering seemed to require 'units' [Tait] |
| Full Idea: The conception that what can be numbered is some object (including flocks of sheep) relative to a partition - a choice of unit - survived even in the late nineteenth century in the form of the rejection of the null set (and difficulties with unit sets). | |
| From: William W. Tait (Frege versus Cantor and Dedekind [1996], IX) | |
| A reaction: This old view can't be entirely wrong! Frege makes the point that if asked to count a pack of cards, you must decide whether to count cards, or suits, or pips. You may not need a 'unit', but you need a concept. 'Units' name concept-extensions nicely! |
| 9984 | We can have a series with identical members [Tait] |
| Full Idea: Why can't we have a series (as opposed to a linearly ordered set) all of whose members are identical, such as (a, a, a...,a)? | |
| From: William W. Tait (Frege versus Cantor and Dedekind [1996], VII) | |
| A reaction: The question is whether the items order themselves, which presumably the natural numbers are supposed to do, or whether we impose the order (and length) of the series. What decides how many a's there are? Do we order, or does nature? |
| 12195 | Soundness in argument varies with context, and may be achieved very informally indeed [Rumfitt] |
| Full Idea: Our ordinary standards for deeming arguments to be sound vary greatly from context to context. Even the package tourist's syllogism ('It's Tuesday, so this is Belgium') may meet the operative standards for soundness. | |
| From: Ian Rumfitt (Logical Necessity [2010], Intro) | |
| A reaction: No doubt one could spell out the preconceptions of package tourist reasoning, and arrive at the logical form of the implication which is being offered. |
| 12199 | There is a modal element in consequence, in assessing reasoning from suppositions [Rumfitt] |
| Full Idea: There is a modal element in consequence, in its applicability to assessing reasoning from suppositions. | |
| From: Ian Rumfitt (Logical Necessity [2010], §2) |
| 12201 | We reject deductions by bad consequence, so logical consequence can't be deduction [Rumfitt] |
| Full Idea: A rule is to be rejected if it enables us to deduce from some premisses a purported conclusion that does not follow from them in the broad sense. The idea that deductions answer to consequence is incomprehensible if consequence consists in deducibility. | |
| From: Ian Rumfitt (Logical Necessity [2010], §2) |
| 12194 | Contradictions include 'This is red and not coloured', as well as the formal 'B and not-B' [Rumfitt] |
| Full Idea: Overt contradictions include formal contradictions of form 'B and not B', but I also take them to include 'This is red all over and green all over' and 'This is red and not coloured'. | |
| From: Ian Rumfitt (Logical Necessity [2010], Intro) |
| 12198 | Geometrical axioms in logic are nowadays replaced by inference rules (which imply the logical truths) [Rumfitt] |
| Full Idea: The geometrical style of formalization of logic is now little more than a quaint anachronism, largely because it fails to show logical truths for what they are: simply by-products of rules of inference that are applicable to suppositions. | |
| From: Ian Rumfitt (Logical Necessity [2010], §1) | |
| A reaction: This is the rejection of Russell-style axiom systems in favour of Gentzen-style natural deduction systems (starting from rules). Rumfitt quotes Dummett in support. |
| 14532 | A distinctive type of necessity is found in logical consequence [Rumfitt, by Hale/Hoffmann,A] |
| Full Idea: Rumfitt argues that there is a distinctive notion of necessity implicated in the notion of logical consequence. | |
| From: report of Ian Rumfitt (Logical Necessity [2010]) by Bob Hale/ Aviv Hoffmann - Introduction to 'Modality' 2 |
| 12200 | A logically necessary statement need not be a priori, as it could be unknowable [Rumfitt] |
| Full Idea: There is no reason to suppose that any statement that is logically necessary (in the present sense) is knowable a priori. ..If a statement is logically necessary, its negation will yield a contradiction, but that does not imply that someone could know it. | |
| From: Ian Rumfitt (Logical Necessity [2010], §2) | |
| A reaction: This remark is aimed at Dorothy Edgington, who holds the opposite view. Rumfitt largely defends McFetridge's view (q.v.). |
| 12193 | Logical necessity is when 'necessarily A' implies 'not-A is contradictory' [Rumfitt] |
| Full Idea: By the notion of 'logical necessity' I mean that there is a sense of 'necessary' for which 'It is necessary that A' implies and is implied by 'It is logically contradictory that not A'. ...From this, logical necessity is implicated in logical consequence. | |
| From: Ian Rumfitt (Logical Necessity [2010], Intro) | |
| A reaction: Rumfitt expresses a commitment to classical logic at this point. We will need to be quite sure what we mean by 'contradiction', which will need a clear notion of 'truth'.... |
| 12202 | Narrow non-modal logical necessity may be metaphysical, but real logical necessity is not [Rumfitt] |
| Full Idea: While Fine suggests defining a narrow notion of logical necessity in terms of metaphysical necessity by 'restriction' (to logical truths that can be defined in non-modal terms), this seems unpromising for broad logical necessity, which is modal. | |
| From: Ian Rumfitt (Logical Necessity [2010], §2) | |
| A reaction: [compressed] He cites Kit Fine 2002. Rumfitt glosses the non-modal definitions as purely formal. The metaphysics lurks somewhere in the proof. |
| 12203 | If a world is a fully determinate way things could have been, can anyone consider such a thing? [Rumfitt] |
| Full Idea: A world is usually taken to be a fully determinate way that things could have been; but then one might seriously wonder whether anyone is capable of 'considering' such a thing at all. | |
| From: Ian Rumfitt (Logical Necessity [2010], §4) | |
| A reaction: This has always worried me. If I say 'maybe my coat is in the car', I would hate to think that I had to be contemplating some entire possible world (including all the implications of my coat not being on the hat stand). |
| 9981 | Abstraction is 'logical' if the sense and truth of the abstraction depend on the concrete [Tait] |
| Full Idea: If the sense of a proposition about the abstract domain is given in terms of the corresponding proposition about the (relatively) concrete domain, ..and the truth of the former is founded upon the truth of the latter, then this is 'logical abstraction'. | |
| From: William W. Tait (Frege versus Cantor and Dedekind [1996], V) | |
| A reaction: The 'relatively' in parentheses allows us to apply his idea to levels of abstraction, and not just to the simple jump up from the concrete. I think Tait's proposal is excellent, rather than purloining 'abstraction' for an internal concept within logic. |
| 9982 | Cantor and Dedekind use abstraction to fix grammar and objects, not to carry out proofs [Tait] |
| Full Idea: Although (in Cantor and Dedekind) abstraction does not (as has often been observed) play any role in their proofs, but it does play a role, in that it fixes the grammar, the domain of meaningful propositions, and so determining the objects in the proofs. | |
| From: William W. Tait (Frege versus Cantor and Dedekind [1996], V) | |
| A reaction: [compressed] This is part of a defence of abstractionism in Cantor and Dedekind (see K.Fine also on the subject). To know the members of a set, or size of a domain, you need to know the process or function which created the set. |
| 9985 | Abstraction may concern the individuation of the set itself, not its elements [Tait] |
| Full Idea: A different reading of abstraction is that it concerns, not the individuating properties of the elements relative to one another, but rather the individuating properties of the set itself, for example the concept of what is its extension. | |
| From: William W. Tait (Frege versus Cantor and Dedekind [1996], VIII) | |
| A reaction: If the set was 'objects in the room next door', we would not be able to abstract from the objects, but we might get to the idea of things being contain in things, or the concept of an object, or a room. Wrong. That's because they are objects... Hm. |
| 9972 | Why should abstraction from two equipollent sets lead to the same set of 'pure units'? [Tait] |
| Full Idea: Why should abstraction from two equipollent sets lead to the same set of 'pure units'? | |
| From: William W. Tait (Frege versus Cantor and Dedekind [1996]) | |
| A reaction: [Tait is criticising Cantor] This expresses rather better than Frege or Dummett the central problem with the abstractionist view of how numbers are derived from matching groups of objects. |
| 9980 | If abstraction produces power sets, their identity should imply identity of the originals [Tait] |
| Full Idea: If the power |A| is obtained by abstraction from set A, then if A is equipollent to set B, then |A| = |B|. But this does not imply that A = B. So |A| cannot just be A, taken in abstraction, unless that can identify distinct sets, ..or create new objects. | |
| From: William W. Tait (Frege versus Cantor and Dedekind [1996], V) | |
| A reaction: An elegant piece of argument, which shows rather crucial facts about abstraction. We are then obliged to ask how abstraction can create an object or a set, if the central activity of abstraction is just ignoring certain features. |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |