11 ideas
13639 | Quine says higher-order items are intensional, and lack a clearly defined identity relation [Quine, by Shapiro] |
Full Idea: Quine (in 1941) attacked 'Principia Mathematica' because the items in the range of higher-order variables (attributes etc) are intensional and thus do not have a clearly defined identity relation. | |
From: report of Willard Quine (Whitehead and the Rise of Modern Logic [1941]) by Stewart Shapiro - Foundations without Foundationalism 1.3 |
18755 | Validity is explained as truth in all models, because that relies on the logical terms [McGee] |
Full Idea: A model of a language assigns values to non-logical terms. If a sentence is true in every model, its truth doesn't depend on those non-logical terms. Hence the validity of an argument comes from its logical form. Thus models explain logical validity. | |
From: Vann McGee (Logical Consequence [2014], 4) | |
A reaction: [compressed] Thus you get a rigorous account of logical validity by only allowing the rigorous input of model theory. This is the modern strategy of analytic philosophy. But is 'it's red so it's coloured' logically valid? |
18751 | Natural language includes connectives like 'because' which are not truth-functional [McGee] |
Full Idea: Natural language includes connectives that are not truth-functional. In order for 'p because q' to be true, both p and q have to be true, but knowing the simpler sentences are true doesn't determine whether the larger sentence is true. | |
From: Vann McGee (Logical Consequence [2014], 2) |
18761 | Second-order variables need to range over more than collections of first-order objects [McGee] |
Full Idea: To get any advantage from moving to second-order logic, we need to assign to second-order variables a role different from merely ranging over collections made up of things the first-order variables range over. | |
From: Vann McGee (Logical Consequence [2014], 7) | |
A reaction: Thus it is exciting if they range over genuine properties, but not so exciting if you merely characterise those properties as sets of first-order objects. This idea leads into a discussion of plural quantification. |
18753 | An ontologically secure semantics for predicate calculus relies on sets [McGee] |
Full Idea: We can get a less ontologically perilous presentation of the semantics of the predicate calculus by using sets instead of concepts. | |
From: Vann McGee (Logical Consequence [2014], 4) | |
A reaction: The perilous versions rely on Fregean concepts, and notably Russell's 'concept that does not fall under itself'. The sets, of course, have to be ontologically secure, and so will involve the iterative conception, rather than naive set theory. |
18754 | Logically valid sentences are analytic truths which are just true because of their logical words [McGee] |
Full Idea: Logically valid sentences are a species of analytic sentence, being true not just in virtue of the meanings of their words, but true in virtue of the meanings of their logical words. | |
From: Vann McGee (Logical Consequence [2014], 4) | |
A reaction: A helpful link between logical truths and analytic truths, which had not struck me before. |
18757 | Soundness theorems are uninformative, because they rely on soundness in their proofs [McGee] |
Full Idea: Soundness theorems are seldom very informative, since typically we use informally, in proving the theorem, the very same rules whose soundness we are attempting to establish. | |
From: Vann McGee (Logical Consequence [2014], 5) | |
A reaction: [He cites Quine 1935] |
18760 | The culmination of Euclidean geometry was axioms that made all models isomorphic [McGee] |
Full Idea: One of the culminating achievements of Euclidean geometry was categorical axiomatisations, that describe the geometric structure so completely that any two models of the axioms are isomorphic. The axioms are second-order. | |
From: Vann McGee (Logical Consequence [2014], 7) | |
A reaction: [He cites Veblen 1904 and Hilbert 1903] For most mathematicians, categorical axiomatisation is the best you can ever dream of (rather than a single true axiomatisation). |
21557 | Russell confused use and mention, and reduced classes to properties, not to language [Quine, by Lackey] |
Full Idea: Quine (1941) said that Russell had confused use and mention, and thus thought he had reduced classes to linguistic entities, while in fact he reduced them only to Platonic properties. | |
From: report of Willard Quine (Whitehead and the Rise of Modern Logic [1941]) by Douglas Lackey - Intros to Russell's 'Essays in Analysis' p.133 | |
A reaction: This is cited as the 'orthodox critical interpretation' of Russell and Whitehead. Confusion of use and mention was a favourite charge of Quine's. |
1556 | By nature people are close to one another, but culture drives them apart [Hippias] |
Full Idea: I regard you all as relatives - by nature, not by convention. By nature like is akin to like, but convention is a tyrant over humankind and often constrains people to act contrary to nature. | |
From: Hippias (fragments/reports [c.430 BCE]), quoted by Plato - Protagoras 337c8 |
18762 | A maxim claims that if we are allowed to assert a sentence, that means it must be true [McGee] |
Full Idea: If our linguistic conventions entitle us to assert a sentence, they thereby make it true, because of the maxim that 'truth is the norm of assertion'. | |
From: Vann McGee (Logical Consequence [2014], 8) | |
A reaction: You could only really deny that maxim if you had no belief at all in truth, but then you can assert anything you like (with full entitlement). Maybe you can assert anything you like as long as it doesn't upset anyone? Etc. |