15 ideas
| 14249 | Boolos reinterprets second-order logic as plural logic [Boolos, by Oliver/Smiley] |
| Full Idea: Boolos's conception of plural logic is as a reinterpretation of second-order logic. | |
| From: report of George Boolos (On Second-Order Logic [1975]) by Oliver,A/Smiley,T - What are Sets and What are they For? n5 | |
| A reaction: Oliver and Smiley don't accept this view, and champion plural reference differently (as, I think, some kind of metalinguistic device?). |
| 10830 | Second-order logic metatheory is set-theoretic, and second-order validity has set-theoretic problems [Boolos] |
| Full Idea: The metatheory of second-order logic is hopelessly set-theoretic, and the notion of second-order validity possesses many if not all of the epistemic debilities of the notion of set-theoretic truth. | |
| From: George Boolos (On Second-Order Logic [1975], p.45) | |
| A reaction: Epistemic problems arise when a logic is incomplete, because some of the so-called truths cannot be proved, and hence may be unreachable. This idea indicates Boolos's motivation for developing a theory of plural quantification. |
| 10829 | A sentence can't be a truth of logic if it asserts the existence of certain sets [Boolos] |
| Full Idea: One may be of the opinion that no sentence ought to be considered as a truth of logic if, no matter how it is interpreted, it asserts that there are sets of certain sorts. | |
| From: George Boolos (On Second-Order Logic [1975], p.44) | |
| A reaction: My intuition is that in no way should any proper logic assert the existence of anything at all. Presumably interpretations can assert the existence of numbers or sets, but we should be able to identify something which is 'pure' logic. Natural deduction? |
| 10832 | '∀x x=x' only means 'everything is identical to itself' if the range of 'everything' is fixed [Boolos] |
| Full Idea: One may say that '∀x x=x' means 'everything is identical to itself', but one must realise that one's answer has a determinate sense only if the reference (range) of 'everything' is fixed. | |
| From: George Boolos (On Second-Order Logic [1975], p.46) | |
| A reaction: This is the problem now discussed in the recent book 'Absolute Generality', of whether one can quantify without specifying a fixed or limited domain. |
| 10834 | Weak completeness: if it is valid, it is provable. Strong: it is provable from a set of sentences [Boolos] |
| Full Idea: A weak completeness theorem shows that a sentence is provable whenever it is valid; a strong theorem, that a sentence is provable from a set of sentences whenever it is a logical consequence of the set. | |
| From: George Boolos (On Second-Order Logic [1975], p.52) | |
| A reaction: So the weak version says |- φ → |= φ, and the strong versions says Γ |- φ → Γ |= φ. Presumably it is stronger if it can specify the source of the inference. |
| 13841 | Why should compactness be definitive of logic? [Boolos, by Hacking] |
| Full Idea: Boolos asks why on earth compactness, whatever its virtues, should be definitive of logic itself. | |
| From: report of George Boolos (On Second-Order Logic [1975]) by Ian Hacking - What is Logic? §13 |
| 10833 | Many concepts can only be expressed by second-order logic [Boolos] |
| Full Idea: The notions of infinity and countability can be characterized by second-order sentences, though not by first-order sentences (as compactness and Skolem-Löwenheim theorems show), .. as well as well-ordering, progression, ancestral and identity. | |
| From: George Boolos (On Second-Order Logic [1975], p.48) |
| 23651 | Universals are not objects of sense and cannot be imagined - but can be conceived [Reid] |
| Full Idea: A universal is not an object of any sense, and therefore cannot be imagined; but it may be distinctly conceived. | |
| From: Thomas Reid (Essays on Intellectual Powers 5: Abstraction [1785], 6) | |
| A reaction: If you try to imagine whiteness, what size is it, and what substance embodies it? Neither are needed to think of whiteness, so Reid is right. A nice observation. |
| 23650 | Only individuals exist [Reid] |
| Full Idea: Everything that really exists is an individual. | |
| From: Thomas Reid (Essays on Intellectual Powers 5: Abstraction [1785], 6) | |
| A reaction: Locke is the probable inspiration for this nominalist affirmation. Not sure how high temperature plasma, or the oceans of the world, fit into this. On the whole I agree with him. He is mainly rejecting abstract universals. |
| 23649 | No one thinks two sheets possess a single whiteness, but all agree they are both white [Reid] |
| Full Idea: If we say that the whiteness of this sheet is the whiteness of another sheet, every man perceives this to be absurd; but when he says both sheets are white, this is true and perfectly understood. | |
| From: Thomas Reid (Essays on Intellectual Powers 5: Abstraction [1785], 3) | |
| A reaction: Well said. Only a philosopher could think the whiteness of one sheet is exactly the same entity as the whiteness of a different sheet. We seem to have brilliantly and correctly labelled them both as white, and then thought that one word implies one thing. |
| 11874 | Real identity admits of no degrees [Reid] |
| Full Idea: Wherever identity is real, it admits of no degrees. | |
| From: Thomas Reid (Essays on Intellectual Powers 5: Abstraction [1785]), quoted by David Wiggins - Sameness and Substance Renewed 6 epig | |
| A reaction: Wiggins quotes this with strong approval. Personally I am inclined to think that identity may admit of no degrees in human thought, because that is the only way we can do it, but the world is full of uncertain identities, at every level. |
| 23652 | We must first conceive things before we can consider them [Reid] |
| Full Idea: No man can consider a thing which he does not conceive. | |
| From: Thomas Reid (Essays on Intellectual Powers 5: Abstraction [1785], 6) | |
| A reaction: This seems to imply concepts, but we should not take this to be linguistic, since animals obviously consider things and make judgements. |
| 23648 | First we notice and name attributes ('abstracting'); then we notice that subjects share them ('generalising') [Reid] |
| Full Idea: First we resolve or analyse a subject into its known attributes, and give a name to each attribute. Then we observe one or more attributes to be common to many subjects. The first philosophers call 'abstraction', and the second is 'generalising'. | |
| From: Thomas Reid (Essays on Intellectual Powers 5: Abstraction [1785], 3) | |
| A reaction: It is very unfashionable in analytic philosophy to view universals in this way, but it strikes me as obviously correct. There are not weird abstract entities awaiting a priori intuition. There are just features of the world to be observed and picked out. |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |