16 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) |
| 5953 | For the Cyrenaics experience was not enough to give certainty about reality [Aristippus young, by Plutarch] |
| Full Idea: The Cyrenaics, placing all experience within themselves, thought such evidence was insufficient warrant for certainty about reality, and withdrew as in a siege from the world, admitting that objects 'appear', but refusing to pronounce the word 'are'. | |
| From: report of Aristippus the younger (fragments/reports [c.335 BCE]) by Plutarch - 74: Reply to Colotes §1120 | |
| A reaction: This seems to be the most extreme position found in ancient thought. It accompanies their extreme hedonism, based on the reality of experience and lack of interest in anything external. A bit daft, really. |
| 19347 | Substance needs independence, unity, and stability (for individuation); also it is a subject, for predicates [Perkins] |
| Full Idea: For individuation, substance needs three properties: independence, to separate it from other things; unity, to call it one thing, rather than an aggregate; and permanence or stability over time. Its other role is as subject for predicates. | |
| From: Franklin Perkins (Leibniz: Guide for the Perplexed [2007], 3.1) | |
| A reaction: Perkins is describing the Aristotelian view, which is taken up by Leibniz. 'Substance' is not a controversial idea, if we see that it only means that the world is full of 'things'. It is an unusual philosopher wholly totally denies that. |
| 3023 | Even the foolish may have some virtues [Aristippus young, by Diog. Laertius] |
| Full Idea: The Cyrenaics say that some of the virtues may exist even in the foolish. | |
| From: report of Aristippus the younger (fragments/reports [c.335 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.7.8 |
| 3026 | Actions are influenced by circumstances, so Cyrenaics say felons should be reformed, not hated [Aristippus young, by Diog. Laertius] |
| Full Idea: Cyrenaics say errors should be pardoned, because men do not err intentionally but are influenced by circumstances; one should not hate a person, but only teach him better. | |
| From: report of Aristippus the younger (fragments/reports [c.335 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.7.9 | |
| A reaction: A very appealing suggestion, and rather wonderful for its time. There is still implied agreement about what is 'error', and what counts as 'better'. |
| 3024 | Cyrenaics teach that honour, justice and shame are all based on custom and fashion [Aristippus young, by Diog. Laertius] |
| Full Idea: The Cyrenaics taught that there was nothing naturally and intrinsically just, or honourable, or disgraceful; but that things were considered so because of law and fashion. | |
| From: report of Aristippus the younger (fragments/reports [c.335 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.7.8 | |
| A reaction: As we would say now, values and virtues are 'cultural constructs'. This obviously contains a lot of truth, but I don't think our opposition of genocide is just 'fashion'. |
| 3025 | For a Cyrenaic no one is of equal importance to himself [Aristippus young, by Diog. Laertius] |
| Full Idea: A Cyrenaic will not consider anyone else of equal importance with himself. | |
| From: report of Aristippus the younger (fragments/reports [c.335 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.7.9 |
| 3019 | No one pleasure is different from or more pleasant than another [Aristippus young, by Diog. Laertius] |
| Full Idea: No one pleasure is different from or more pleasant than another. | |
| From: report of Aristippus the younger (fragments/reports [c.335 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.7.8 |
| 3021 | The Cyrenaics asserted that corporeal pleasures were superior to mental ones [Aristippus young, by Diog. Laertius] |
| Full Idea: The Cyrenaics asserted that corporeal pleasures were superior to mental ones. | |
| From: report of Aristippus the younger (fragments/reports [c.335 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.7.8 |
| 3027 | Cyrenaics say wise men are self-sufficient, needing no friends [Aristippus young, by Diog. Laertius] |
| Full Idea: Cyrenaics say wise men are sufficient to themselves, and so have no need of friends. | |
| From: report of Aristippus the younger (fragments/reports [c.335 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.7.13 |