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All the ideas for 'fragments/reports', 'Principia Mathematica' and 'Logical Necessity'

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39 ideas

4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL
The best known axiomatization of PL is Whitehead/Russell, with four axioms and two rules [Russell/Whitehead, by Hughes/Cresswell]
     Full Idea: The best known axiomatization of PL is Whitehead/Russell. There are four axioms: (p∨p)→p, q→(p∨q), (p→q)→(q∨p), and (q→r)→((p∨q)→(p∨r)), plus Substitution and Modus Ponens rules.
     From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by GE Hughes/M Cresswell - An Introduction to Modal Logic Ch.1
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / h. System S5
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.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
Russell saw Reducibility as legitimate for reducing classes to logic [Linsky,B on Russell/Whitehead]
     Full Idea: The axiom of Reducibility ...is crucial in the reduction of classes to logic, ...and seems to be a quite legitimate logical notion for Russell.
     From: comment on B Russell/AN Whitehead (Principia Mathematica [1913]) by Bernard Linsky - Russell's Metaphysical Logic 6.4
     A reaction: This is an unusual defence of the axiom, which is usually presumed to have been kicked into the long grass by Quine. If one could reduce classes to logic, that would destroy the opposition to logicism in a single neat coup.
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
Russell denies extensional sets, because the null can't be a collection, and the singleton is just its element [Russell/Whitehead, by Shapiro]
     Full Idea: Russell adduces two reasons against the extensional view of classes, namely the existence of the null class (which cannot very well be a collection), and the unit classes (which would have to be identical with their single elements).
     From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Stewart Shapiro - Structure and Ontology p.459
     A reaction: Gödel believes in the reality of classes. I have great sympathy with Russell, when people start to claim that sets are not just conveniences to help us think about things, but actual abstract entities. Is the singleton of my pencil is on this table?
We regard classes as mere symbolic or linguistic conveniences [Russell/Whitehead]
     Full Idea: Classes, so far as we introduce them, are merely symbolic or linguistic conveniences, not genuine objects.
     From: B Russell/AN Whitehead (Principia Mathematica [1913], p.72), quoted by Penelope Maddy - Naturalism in Mathematics III.2
5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
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.
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)
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)
5. Theory of Logic / B. Logical Consequence / 7. Strict Implication
Lewis's 'strict implication' preserved Russell's confusion of 'if...then' with implication [Quine on Russell/Whitehead]
     Full Idea: Russell call 'if...then' implication, when the material conditional is a much better account; C.I.Lewis (in founding modern modal logic) preserved Russell's confusion by creating 'strict implication', and called that implication.
     From: comment on B Russell/AN Whitehead (Principia Mathematica [1913]) by Willard Quine - Reply to Professor Marcus p.177
     A reaction: [A compession of Quine's paragraph]. All of this assumes that logicians can give an accurate account of what if...then means, when ordinary usage is broad and vague. Strict implication seems to drain all the normal meaning out of 'if...then'.
Russell's implication means that random sentences imply one another [Lewis,CI on Russell/Whitehead]
     Full Idea: In Mr Russell's idea of implication, if twenty random sentences from a newspaper were put in a hat, and two of them drawn at random, one will certainly imply the other, and it is an even bet the implication will be mutual.
     From: comment on B Russell/AN Whitehead (Principia Mathematica [1913]) by C.I. Lewis - A Pragmatic Conception of the A Priori p.366
     A reaction: This sort of lament leads modern logicians to suggest 'relevance' as an important criterion. It certainly seems odd that so-called 'classical logic' should contain a principle so at variance with everyday reasoning.
5. Theory of Logic / C. Ontology of Logic / 1. Ontology of Logic
Russell unusually saw logic as 'interpreted' (though very general, and neutral) [Russell/Whitehead, by Linsky,B]
     Full Idea: Russell did not view logic as an uninterpreted calculus awaiting interpretations [the modern view]. Rather, logic is a single 'interpreted' body of a priori truths, of propositions rather than sentence forms - but maximally general and topic neutral.
     From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Bernard Linsky - Russell's Metaphysical Logic 1
     A reaction: This is the view which Wittgenstein challenged, saying logic is just conventional. Linsky claims that Russell's logicism is much more plausible, once you understand his view of logic.
5. Theory of Logic / D. Assumptions for Logic / 3. Contradiction
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)
5. Theory of Logic / E. Structures of Logic / 6. Relations in Logic
In 'Principia' a new abstract theory of relations appeared, and was applied [Russell/Whitehead, by Gödel]
     Full Idea: In 'Principia' a young science was enriched with a new abstract theory of relations, ..and not only Cantor's set theory but also ordinary arithmetic and the theory of measurement are treated from this abstract relational standpoint.
     From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Kurt Gödel - Russell's Mathematical Logic p.448
     A reaction: I presume this is accounting for relations in terms of ordered sets.
5. Theory of Logic / H. Proof Systems / 2. Axiomatic Proof
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.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
A real number is the class of rationals less than the number [Russell/Whitehead, by Shapiro]
     Full Idea: For Russell the real number 2 is the class of rationals less than 2 (i.e. 2/1). ...Notice that on this definition, real numbers are classes of rational numbers.
     From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Stewart Shapiro - Thinking About Mathematics 5.2
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / a. Defining numbers
Russell takes numbers to be classes, but then reduces the classes to numerical quantifiers [Russell/Whitehead, by Bostock]
     Full Idea: Although Russell takes numbers to be certain classes, his 'no-class' theory then eliminates all mention of classes in favour of the 'propositional functions' that define them; and in the case of the numbers these just are the numerical quantifiers.
     From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by David Bostock - Philosophy of Mathematics 9.B.4
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
Russell and Whitehead took arithmetic to be higher-order logic [Russell/Whitehead, by Hodes]
     Full Idea: Russell and Whitehead took arithmetic to be higher-order logic, ..and came close to identifying numbers with numerical quantifiers.
     From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Harold Hodes - Logicism and Ontological Commits. of Arithmetic p.148
     A reaction: The point here is 'higher-order'.
Russell and Whitehead were not realists, but embraced nearly all of maths in logic [Russell/Whitehead, by Friend]
     Full Idea: Unlike Frege, Russell and Whitehead were not realists about mathematical objects, and whereas Frege thought that only arithmetic and analysis are branches of logic, they think the vast majority of mathematics (including geometry) is essentially logical.
     From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Michčle Friend - Introducing the Philosophy of Mathematics 3.1
     A reaction: If, in essence, Descartes reduced geometry to algebra (by inventing co-ordinates), then geometry ought to be included. It is characteristic of Russell's hubris to want to embrace everything.
'Principia' lacks a precise statement of the syntax [Gödel on Russell/Whitehead]
     Full Idea: What is missing, above all, in 'Principia', is a precise statement of the syntax of the formalism.
     From: comment on B Russell/AN Whitehead (Principia Mathematica [1913]) by Kurt Gödel - Russell's Mathematical Logic p.448
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
The ramified theory of types used propositional functions, and covered bound variables [Russell/Whitehead, by George/Velleman]
     Full Idea: Russell and Whitehead's ramified theory of types worked not with sets, but with propositional functions (similar to Frege's concepts), with a more restrictive assignment of variables, insisting that bound, as well as free, variables be of lower type.
     From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.3
     A reaction: I don't fully understand this (and no one seems much interested any more), but I think variables are a key notion, and there is something interesting going on here. I am intrigued by ordinary language which behaves like variables.
The Russell/Whitehead type theory was limited, and was not really logic [Friend on Russell/Whitehead]
     Full Idea: The Russell/Whitehead type theory reduces mathematics to a consistent founding discipline, but is criticised for not really being logic. They could not prove the existence of infinite sets, and introduced a non-logical 'axiom of reducibility'.
     From: comment on B Russell/AN Whitehead (Principia Mathematica [1913]) by Michčle Friend - Introducing the Philosophy of Mathematics 3.6
     A reaction: To have reduced most of mathematics to a founding discipline sounds like quite an achievement, and its failure to be based in pure logic doesn't sound too bad. However, it seems to reduce some maths to just other maths.
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
In 'Principia Mathematica', logic is exceeded in the axioms of infinity and reducibility, and in the domains [Bernays on Russell/Whitehead]
     Full Idea: In the system of 'Principia Mathematica', it is not only the axioms of infinity and reducibility which go beyond pure logic, but also the initial conception of a universal domain of individuals and of a domain of predicates.
     From: comment on B Russell/AN Whitehead (Principia Mathematica [1913], p.267) by Paul Bernays - On Platonism in Mathematics p.267
     A reaction: This sort of criticism seems to be the real collapse of the logicist programme, rather than Russell's paradox, or Gödel's Incompleteness Theorems. It just became impossible to stick strictly to logic in the reduction of arithmetic.
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
Russell and Whitehead consider the paradoxes to indicate that we create mathematical reality [Russell/Whitehead, by Friend]
     Full Idea: Russell and Whitehead are particularly careful to avoid paradox, and consider the paradoxes to indicate that we create mathematical reality.
     From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Michčle Friend - Introducing the Philosophy of Mathematics 3.1
     A reaction: This strikes me as quite a good argument. It is certainly counterintuitive that reality, and abstractions from reality, would contain contradictions. The realist view would be that we have paradoxes because we have misdescribed the facts.
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
To avoid vicious circularity Russell produced ramified type theory, but Ramsey simplified it [Russell/Whitehead, by Shapiro]
     Full Idea: Russell insisted on the vicious circle principle, and thus rejected impredicative definitions, which resulted in an unwieldy ramified type theory, with the ad hoc axiom of reducibility. Ramsey's simpler theory was impredicative and avoided the axiom.
     From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Stewart Shapiro - Thinking About Mathematics 5.2
     A reaction: Nowadays the theory of types seems to have been given up, possibly because it has no real attraction if it lacks the strict character which Russell aspired to.
9. Objects / F. Identity among Objects / 7. Indiscernible Objects
An object is identical with itself, and no different indiscernible object can share that [Russell/Whitehead, by Adams,RM]
     Full Idea: Trivially, the Identity of Indiscernibles says that two individuals, Castor and Pollux, cannot have all properties in common. For Castor must have the properties of being identical with Castor and not being identical with Pollux, which Pollux can't share.
     From: report of B Russell/AN Whitehead (Principia Mathematica [1913], I p.57) by Robert Merrihew Adams - Primitive Thisness and Primitive Identity 2
     A reaction: I suspect that either the property of being identical with itself is quite vacuous, or it is parasytic on primitive identity, or it is the criterion which is actually used to define identity. Either way, I don't find this claim very illuminating.
10. Modality / A. Necessity / 3. Types of Necessity
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
10. Modality / A. Necessity / 6. Logical Necessity
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.).
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'....
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.
10. Modality / E. Possible worlds / 1. Possible Worlds / e. Against possible worlds
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).
12. Knowledge Sources / E. Direct Knowledge / 2. Intuition
Russell showed, through the paradoxes, that our basic logical intuitions are self-contradictory [Russell/Whitehead, by Gödel]
     Full Idea: By analyzing the paradoxes to which Cantor's set theory had led, ..Russell brought to light the amazing fact that our logical intuitions (concerning such notions as truth, concept, being, class) are self-contradictory.
     From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Kurt Gödel - Russell's Mathematical Logic p.452
     A reaction: The main intuition that failed was, I take it, that every concept has an extension, that is, there are always objects which will or could fall under the concept.
18. Thought / A. Modes of Thought / 6. Judgement / a. Nature of Judgement
The multiple relations theory says assertions about propositions are about their ingredients [Russell/Whitehead, by Linsky,B]
     Full Idea: The multiple relations theory of judgement proposes that assertions about propositions are dependent upon genuine facts involving belief and other attitude relations, subjects of those attitudes, and the constituents of the belief.
     From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Bernard Linsky - Russell's Metaphysical Logic 7.2
     A reaction: This seems to require a commitment to universals (especially relations) with which we can be directly acquainted. I prefer propositions, but as mental entities, not platonic entities.
A judgement is a complex entity, of mind and various objects [Russell/Whitehead]
     Full Idea: When a judgement occurs, there is a certain complex entity, composed of the mind and the various objects of the judgement.
     From: B Russell/AN Whitehead (Principia Mathematica [1913], p.44)
     A reaction: This is Russell's multiple-relation theory of judgement, which replaced his earlier belief in unified propositions (now 'false abstractions'). He seems to have accepted Locke's view, that the act of judgement produces the unity.
The meaning of 'Socrates is human' is completed by a judgement [Russell/Whitehead]
     Full Idea: When I judge 'Socrates is human', the meaning is completed by the act of judging.
     From: B Russell/AN Whitehead (Principia Mathematica [1913], p.44), quoted by Michael Morris - Guidebook to Wittgenstein's Tractatus
     A reaction: Morris says this is Russell's multiple-relations theory of judgement. The theory accompanies the rejection of the concept of the unified proposition. When I hear 'Socrates had a mole on his shoulder' I get the meaning without judging.
The multiple relation theory of judgement couldn't explain the unity of sentences [Morris,M on Russell/Whitehead]
     Full Idea: When Russell moved to his multiple relation theory of judgement …he then faced difficulties making sense of the unity of sentences.
     From: comment on B Russell/AN Whitehead (Principia Mathematica [1913], p.44) by Michael Morris - Guidebook to Wittgenstein's Tractatus 3A
     A reaction: Roughly, he seems committed to saying that there is only unity if you think there is unity; there is no unity in a sentence prior to the act of judgement.
Only the act of judging completes the meaning of a statement [Russell/Whitehead]
     Full Idea: When I judge 'Socrates is human', the meaning is completed by the act of judging, and we no longer have an incomplete symbol.
     From: B Russell/AN Whitehead (Principia Mathematica [1913], p.44), quoted by J. Alberto Coffa - The Semantic Tradition from Kant to Carnap
     A reaction: Personally I would have thought that you needed to know the meaning properly before you could make the judgement, but then he is Bertrand Russell and I'm not.
19. Language / D. Propositions / 3. Concrete Propositions
Propositions as objects of judgement don't exist, because we judge several objects, not one [Russell/Whitehead]
     Full Idea: A 'proposition', in the sense in which a proposition is supposed to be the object of a judgement, is a false abstraction, because a judgement has several objects, not one.
     From: B Russell/AN Whitehead (Principia Mathematica [1913], p.44), quoted by Michael Morris - Guidebook to Wittgenstein's Tractatus 2E
     A reaction: This is the rejection of the 'Russellian' theory of propositions, in favour of his multiple-relations theory of judgement. But why don't the related objects add up to a proposition about a state of affairs?
26. Natural Theory / A. Speculations on Nature / 5. Infinite in Nature
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
27. Natural Reality / G. Biology / 3. Evolution
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.