27 ideas
| 6118 | Philosophy is logical analysis, followed by synthesis [Russell] |
| Full Idea: The business of philosophy, as I conceive it, is essentially that of logical analysis, followed by logical synthesis. | |
| From: Bertrand Russell (Logical Atomism [1924], p.162) | |
| A reaction: I am uneasy about Russell's hopes for the contribution that logic could make, but I totally agree that analysis is the route to wisdom, and I take Aristotle as my role model of an analytical philosopher, rather than the modern philosophers of logic. |
| 6116 | A logical language would show up the fallacy of inferring reality from ordinary language [Russell] |
| Full Idea: We are trying to create a perfectly logical language to prevent inferences from the nature of language to the nature of the world, which are fallacious because they depend upon the logical defects of language. | |
| From: Bertrand Russell (Logical Atomism [1924], p.159) | |
| A reaction: Wittgenstein seems to have rebelled against this idea, so that one strand of his later philosophy leads to 'ordinary language' philosophy, which is exactly what Russell is criticising. Wittgenstein seems to have seen 'logical language' as an oxymoron. |
| 6117 | Philosophy should be built on science, to reduce error [Russell] |
| Full Idea: We would be wise to build our philosophy upon science, because the risk of error in philosophy is pretty sure to be greater than in science. | |
| From: Bertrand Russell (Logical Atomism [1924], p.160) | |
| A reaction: If you do very little, it reduces the 'risk of error'. I agree that philosophers should start from the facts, and be responsive to new facts, and that science is excellent at discovering facts. But I don't think cognitive science is the new epistemology. |
| 10301 | The axiom of choice is controversial, but it could be replaced [Shapiro] |
| Full Idea: The axiom of choice has a troubled history, but is now standard in mathematics. It could be replaced with a principle of comprehension for functions), or one could omit the variables ranging over functions. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], n 3) |
| 6110 | Subject-predicate logic (and substance-attribute metaphysics) arise from Aryan languages [Russell] |
| Full Idea: It is doubtful whether the subject-predicate logic, with the substance-attribute metaphysic, would have been invented by people speaking a non-Aryan language. | |
| From: Bertrand Russell (Logical Atomism [1924], p.151) | |
| A reaction: This is not far off the Sapir-Whorf Hypothesis (e.g. Idea 3917), which Russell would never accept. I presume that Russell would see true logic as running deeper, and the 'Aryan' approach as just one possible way to describe it. |
| 6107 | It is logic, not metaphysics, that is fundamental to philosophy [Russell] |
| Full Idea: I hold that logic is what is fundamental in philosophy, and that schools should be characterised rather by their logic than by their metaphysics. | |
| From: Bertrand Russell (Logical Atomism [1924], p.143) | |
| A reaction: Personally I disagree. Russell seems to have been most interested in the logical form underlying language, but that seems to be because he was interested in the ontological implications of what we say, which is metaphysics. |
| 10588 | First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [Shapiro] |
| Full Idea: Early study of first-order logic revealed a number of important features. Gödel showed that there is a complete, sound and effective deductive system. It follows that it is Compact, and there are also the downward and upward Löwenheim-Skolem Theorems. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) |
| 10298 | Some say that second-order logic is mathematics, not logic [Shapiro] |
| Full Idea: Some authors argue that second-order logic (with standard semantics) is not logic at all, but is a rather obscure form of mathematics. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.4) |
| 10299 | If the aim of logic is to codify inferences, second-order logic is useless [Shapiro] |
| Full Idea: If the goal of logical study is to present a canon of inference, a calculus which codifies correct inference patterns, then second-order logic is a non-starter. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.4) | |
| A reaction: This seems to be because it is not 'complete'. However, moves like plural quantification seem aimed at capturing ordinary language inferences, so the difficulty is only that there isn't a precise 'calculus'. |
| 10300 | Logical consequence can be defined in terms of the logical terminology [Shapiro] |
| Full Idea: Informally, logical consequence is sometimes defined in terms of the meanings of a certain collection of terms, the so-called 'logical terminology'. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.4) | |
| A reaction: This seems to be a compositional account, where we build a full account from an account of the atomic bits, perhaps presented as truth-tables. |
| 6115 | Vagueness, and simples being beyond experience, are obstacles to a logical language [Russell] |
| Full Idea: The fact that we do not experience simples is one obstacle to the actual creation of a correct logical language, and vagueness is another. | |
| From: Bertrand Russell (Logical Atomism [1924], p.159) | |
| A reaction: The dream of creating a perfect logical language looks doomed from the start, but it is a very interesting project to try to pinpoint why it is unlikely to be possible. I say a perfect language cuts nature exactly at the joints, so find the joints. |
| 10290 | Second-order variables also range over properties, sets, relations or functions [Shapiro] |
| Full Idea: Second-order variables can range over properties, sets, or relations on the items in the domain-of-discourse, or over functions from the domain itself. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) |
| 10292 | Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro] |
| Full Idea: Downward Löwenheim-Skolem: a finite or denumerable set of first-order formulas that is satisfied by a model whose domain is infinite is satisfied in a model whose domain is the natural numbers | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) |
| 10590 | Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro] |
| Full Idea: Upward Löwenheim-Skolem: if a set of first-order formulas is satisfied by a domain of at least the natural numbers, then it is satisfied by a model of at least some infinite cardinal. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) |
| 10296 | The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro] |
| Full Idea: Both of the Löwenheim-Skolem Theorems fail for second-order languages with a standard semantics | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.3.2) |
| 10297 | The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro] |
| Full Idea: The Löwenheim-Skolem theorem is usually taken as a sort of defect (often thought to be inevitable) of the first-order logic. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.4) | |
| A reaction: [He is quoting Wang 1974 p.154] |
| 6109 | Some axioms may only become accepted when they lead to obvious conclusions [Russell] |
| Full Idea: Some of the premisses (of my logicist theory) are much less obvious than some of their consequences, and are believed chiefly because of their consequences. This will be found to be always the case when a science is arranged as a deductive system. | |
| From: Bertrand Russell (Logical Atomism [1924], p.145) | |
| A reaction: We shouldn't assume the model of self-evident axioms leading to surprising conclusions, which is something like the standard model for rationalist foundationalists. Russell nicely points out that the situation could be just the opposite |
| 10294 | Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro] |
| Full Idea: Full second-order logic has all the expressive power needed to do mathematics, but has an unworkable model theory. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) | |
| A reaction: [he credits Cowles for this remark] Having an unworkable model theory sounds pretty serious to me, as I'm not inclined to be interested in languages which don't produce models of some sort. Surely models are the whole point? |
| 6108 | Maths can be deduced from logical axioms and the logic of relations [Russell] |
| Full Idea: I think that no one will dispute that from certain ideas and axioms of formal logic, but with the help of the logic of relations, all pure mathematics can be deduced. | |
| From: Bertrand Russell (Logical Atomism [1924], p.145) | |
| A reaction: It has been said for a long time that Gödel's Incompleteness Theorems of 1930 disproved this claim, though recently there have been defenders of logicism. Beginning with 'certain ideas' sounds like begging the question. |
| 10968 | Russell gave up logical atomism because of negative, general and belief propositions [Russell, by Read] |
| Full Idea: Russell preceded Wittgenstein in deciding that the reduction of all propositions to atomic propositions could not be achieved. The problem cases were negative propositions, general propositions, and belief propositions. | |
| From: report of Bertrand Russell (Logical Atomism [1924]) by Stephen Read - Thinking About Logic Ch.1 |
| 6113 | To mean facts we assert them; to mean simples we name them [Russell] |
| Full Idea: The way to mean a fact is to assert it; the way to mean a simple is to name it. | |
| From: Bertrand Russell (Logical Atomism [1924], p.156) | |
| A reaction: Thus logical atomism is a linguistic programme, of reducing our language to a foundation of pure names. The recent thought of McDowell and others is aimed at undermining any possibility of a 'simple' in perception. The myth of 'The Given'. |
| 6114 | 'Simples' are not experienced, but are inferred at the limits of analysis [Russell] |
| Full Idea: When I speak of 'simples' I am speaking of something not experienced as such, but known only inferentially as the limits of analysis. | |
| From: Bertrand Russell (Logical Atomism [1924], p.158) | |
| A reaction: He claims that the simples are 'known', so he does not mean purely theoretical entities. They have something like the status of quarks in physics, whose existence is inferred from experience. |
| 21722 | Better to construct from what is known, than to infer what is unknown [Russell] |
| Full Idea: Whenever possible, substitute constructions out of known entities for inferences to unknown entities. | |
| From: Bertrand Russell (Logical Atomism [1924], p.161), quoted by Bernard Linsky - Russell's Metaphysical Logic 7 | |
| A reaction: In 1919 he said that the alternative, of 'postulating' new entities, has 'all the advantages of theft over honest toil' [IMP p.71]. This is Russell's commitment to 'constructing' everything, even his concept of matter. Arithmetic as PA is postulation. |
| 6111 | As propositions can be put in subject-predicate form, we wrongly infer that facts have substance-quality form [Russell] |
| Full Idea: Since any proposition can be put into a form with a subject and a predicate, united by a copula, it is natural to infer that every fact consists in the possession of a quality by a substance, which seems to me a mistake. | |
| From: Bertrand Russell (Logical Atomism [1924], p.152) | |
| A reaction: This disagrees with McGinn on facts (Idea 6075). I approve of this warning from Russell, which is a recognition that we can't just infer our metaphysics from our language. I think of this as the 'Frege Fallacy', which ensnared Quine and others. |
| 10591 | Logicians use 'property' and 'set' interchangeably, with little hanging on it [Shapiro] |
| Full Idea: In studying second-order logic one can think of relations and functions as extensional or intensional, or one can leave it open. Little turns on this here, and so words like 'property', 'class', and 'set' are used interchangeably. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.2.1) | |
| A reaction: Important. Students of the metaphysics of properties, who arrive with limited experience of logic, are bewildered by this attitude. Note that the metaphysics is left wide open, so never let logicians hijack the metaphysical problem of properties. |
| 22200 | If you eliminate the impossible, the truth will remain, even if it is weird [Conan Doyle] |
| Full Idea: When you have eliminated the impossible, whatever remains, however improbable, must be the truth. | |
| From: Arthur Conan Doyle (The Sign of Four [1890], Ch. 6) | |
| A reaction: A beautiful statement, by Sherlock Holmes, of Eliminative Induction. It is obviously not true, of course. Many options may still face you after you have eliminated what is actually impossible. |
| 6112 | Meaning takes many different forms, depending on different logical types [Russell] |
| Full Idea: There is not one relation of meaning between words and what they stand for, but as many relations of meaning, each of a different logical type, as there are logical types among the objects for which there are words. | |
| From: Bertrand Russell (Logical Atomism [1924], p.153) | |
| A reaction: This might be a good warning for those engaged in the externalist/internalist debate over the meaning of concepts such as natural kind terms like 'water'. I could have an external meaning for 'elms', but an internal meaning for 'ferns'. |