Combining Philosophers

All the ideas for Oswald Veblen, Graeme Forbes and David Bostock

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154 ideas

1. Philosophy / E. Nature of Metaphysics / 2. Possibility of Metaphysics
12027 There must be a plausible epistemological theory alongside any metaphysical theory [Forbes,G]
     Full Idea: No metaphysical account which renders it impossible to give a plausible epistemological theory is to be countenanced.
     From: Graeme Forbes (The Metaphysics of Modality [1985], 9.1)
     A reaction: It is hard to object to this principle, though we certainly don't want to go verificationist, and thus rule out speculations about metaphysics which are beyond any possible knowledge. Some have tried to prove that something must exist (e.g. Jacquette).
2. Reason / D. Definition / 8. Impredicative Definition
18137 Impredicative definitions are wrong, because they change the set that is being defined? [Bostock]
     Full Idea: Poincaré suggested that what is wrong with an impredicative definition is that it allows the set defined to alter its composition as more sets are added to the theory.
     From: David Bostock (Philosophy of Mathematics [2009], 8.3)
4. Formal Logic / A. Syllogistic Logic / 2. Syllogistic Logic
13439 Venn Diagrams map three predicates into eight compartments, then look for the conclusion [Bostock]
     Full Idea: Venn Diagrams are a traditional method to test validity of syllogisms. There are three interlocking circles, one for each predicate, thus dividing the universe into eight possible basic elementary quantifications. Is the conclusion in a compartment?
     From: David Bostock (Intermediate Logic [1997], 3.8)
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / a. Symbols of PL
12005 The symbol 'ι' forms definite descriptions; (ιx)F(x) says 'the x which is such that F(x)' [Forbes,G]
     Full Idea: We use the symbol 'ι' (Greek 'iota') to form definite descriptions, reading (ιx)F(x) as 'the x which is such that F(x)', or simply as 'the F'.
     From: Graeme Forbes (The Metaphysics of Modality [1985], 4.1)
     A reaction: Compare the lambda operator in modal logic, which picks out predicates from similar formulae.
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL
13421 'Disjunctive Normal Form' is ensuring that no conjunction has a disjunction within its scope [Bostock]
     Full Idea: 'Disjunctive Normal Form' (DNF) is rearranging the occurrences of ∧ and ∨ so that no conjunction sign has any disjunction in its scope. This is achieved by applying two of the distribution laws.
     From: David Bostock (Intermediate Logic [1997], 2.6)
13422 'Conjunctive Normal Form' is ensuring that no disjunction has a conjunction within its scope [Bostock]
     Full Idea: 'Conjunctive Normal Form' (CNF) is rearranging the occurrences of ∧ and ∨ so that no disjunction sign has any conjunction in its scope. This is achieved by applying two of the distribution laws.
     From: David Bostock (Intermediate Logic [1997], 2.6)
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL
13355 'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,ψ|= [Bostock]
     Full Idea: The Principle of Disjunction says that Γ,φ∨ψ |= iff Γ,φ |= and Γ,ψ |=.
     From: David Bostock (Intermediate Logic [1997], 2.5.G)
     A reaction: That is, a disjunction leads to a contradiction if they each separately lead to contradictions.
13356 The 'conditional' is that Γ|=φ→ψ iff Γ,φ|=ψ [Bostock]
     Full Idea: The Conditional Principle says that Γ |= φ→ψ iff Γ,φ |= ψ. With the addition of negation, this implies φ,φ→ψ |= ψ, which is 'modus ponens'.
     From: David Bostock (Intermediate Logic [1997], 2.5.H)
     A reaction: [Second half is in Ex. 2.5.4]
13350 'Assumptions' says that a formula entails itself (φ|=φ) [Bostock]
     Full Idea: The Principle of Assumptions says that any formula entails itself, i.e. φ |= φ. The principle depends just upon the fact that no interpretation assigns both T and F to the same formula.
     From: David Bostock (Intermediate Logic [1997], 2.5.A)
     A reaction: Thus one can introduce φ |= φ into any proof, and then use it to build more complex sequents needed to attain a particular target formula. Bostock's principle is more general than anything in Lemmon.
13351 'Thinning' allows that if premisses entail a conclusion, then adding further premisses makes no difference [Bostock]
     Full Idea: The Principle of Thinning says that if a set of premisses entails a conclusion, then adding further premisses will still entail the conclusion. It is 'thinning' because it makes a weaker claim. If γ|=φ then γ,ψ|= φ.
     From: David Bostock (Intermediate Logic [1997], 2.5.B)
     A reaction: It is also called 'premise-packing'. It is the characteristic of a 'monotonic' logic - where once something is proved, it stays proved, whatever else is introduced.
13352 'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z [Bostock]
     Full Idea: The Principle of Cutting is the general point that entailment is transitive, extending this to cover entailments with more than one premiss. Thus if γ |= φ and φ,Δ |= ψ then γ,Δ |= ψ. Here φ has been 'cut out'.
     From: David Bostock (Intermediate Logic [1997], 2.5.C)
     A reaction: It might be called the Principle of Shortcutting, since you can get straight to the last conclusion, eliminating the intermediate step.
13353 'Negation' says that Γ,¬φ|= iff Γ|=φ [Bostock]
     Full Idea: The Principle of Negation says that Γ,¬φ |= iff Γ |= φ. We also say that φ,¬φ |=, and hence by 'thinning on the right' that φ,¬φ |= ψ, which is 'ex falso quodlibet'.
     From: David Bostock (Intermediate Logic [1997], 2.5.E)
     A reaction: That is, roughly, if the formula gives consistency, the negation gives contradiction. 'Ex falso' says that anything will follow from a contradiction.
13354 'Conjunction' says that Γ|=φ∧ψ iff Γ|=φ and Γ|=ψ [Bostock]
     Full Idea: The Principle of Conjunction says that Γ |= φ∧ψ iff Γ |= φ and Γ |= ψ. This implies φ,ψ |= φ∧ψ, which is ∧-introduction. It is also implies ∧-elimination.
     From: David Bostock (Intermediate Logic [1997], 2.5.F)
     A reaction: [Second half is Ex. 2.5.3] That is, if they are entailed separately, they are entailed as a unit. It is a moot point whether these principles are theorems of propositional logic, or derivation rules.
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL
13610 A logic with ¬ and → needs three axiom-schemas and one rule as foundation [Bostock]
     Full Idea: For ¬,→ Schemas: (A1) |-φ→(ψ→φ), (A2) |-(φ→(ψ→ξ)) → ((φ→ψ)→(φ→ξ)), (A3) |-(¬φ→¬ψ) → (ψ→φ), Rule:DET:|-φ,|-φ→ψ then |-ψ
     From: David Bostock (Intermediate Logic [1997], 5.2)
     A reaction: A1 says everything implies a truth, A2 is conditional proof, and A3 is contraposition. DET is modus ponens. This is Bostock's compact near-minimal axiom system for proposition logic. He adds two axioms and another rule for predicate logic.
4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
18122 Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism [Bostock]
     Full Idea: None of the classical ways of defining one logical constant in terms of others is available in intuitionist logic (and this includes the two quantifiers).
     From: David Bostock (Philosophy of Mathematics [2009], 7.2)
4. Formal Logic / E. Nonclassical Logics / 6. Free Logic
13846 A 'free' logic can have empty names, and a 'universally free' logic can have empty domains [Bostock]
     Full Idea: A 'free' logic is one in which names are permitted to be empty. A 'universally free' logic is one in which the domain of an interpretation may also be empty.
     From: David Bostock (Intermediate Logic [1997], 8.6)
4. Formal Logic / F. Set Theory ST / 1. Set Theory
18114 There is no single agreed structure for set theory [Bostock]
     Full Idea: There is so far no agreed set of axioms for set theory which is categorical, i.e. which does pick just one structure.
     From: David Bostock (Philosophy of Mathematics [2009], 6.4)
     A reaction: This contrasts with Peano Arithmetic, which is categorical in its second-order version.
4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set
18107 A 'proper class' cannot be a member of anything [Bostock]
     Full Idea: A 'proper class' cannot be a member of anything, neither of a set nor of another proper class.
     From: David Bostock (Philosophy of Mathematics [2009], 5.4)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
18115 We could add axioms to make sets either as small or as large as possible [Bostock]
     Full Idea: We could add the axiom that all sets are constructible (V = L), making the universe of sets as small as possible, or add the axiom that there is a supercompact cardinal (SC), making the universe as large as we no know how to.
     From: David Bostock (Philosophy of Mathematics [2009], 6.4)
     A reaction: Bostock says most mathematicians reject the first option, and are undecided about the second option.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
18139 The Axiom of Choice relies on reference to sets that we are unable to describe [Bostock]
     Full Idea: The usual accounts of ZF are not restricted to subsets that we can describe, and that is what justifies the axiom of choice.
     From: David Bostock (Philosophy of Mathematics [2009], 8.4 n36)
     A reaction: This contrasts interestingly with predicativism, which says we can only discuss things which we can describe or define. Something like verificationism hovers in the background.
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
18105 Replacement enforces a 'limitation of size' test for the existence of sets [Bostock]
     Full Idea: The Axiom of Replacement (or the Axiom of Subsets, 'Aussonderung', Fraenkel 1922) in effect enforces the idea that 'limitation of size' is a crucial factor when deciding whether a proposed set or does not not exist.
     From: David Bostock (Philosophy of Mathematics [2009], 5.4)
5. Theory of Logic / A. Overview of Logic / 2. History of Logic
10247 We have no adequate logic at the moment, so mathematicians must create one [Veblen]
     Full Idea: Formal logic has to be taken over by mathematicians. The fact is that there does not exist an adequate logic at the present time, and unless the mathematicians create one, no one else is likely to do so.
     From: Oswald Veblen (Presidential Address of Am. Math. Soc [1924], 141), quoted by Stewart Shapiro - Philosophy of Mathematics
     A reaction: This remark was made well after Frege, but before the advent of Gödel and Tarski. That implies that he was really thinking of meta-logic.
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
18108 First-order logic is not decidable: there is no test of whether any formula is valid [Bostock]
     Full Idea: First-order logic is not decidable. That is, there is no test which can be applied to any arbitrary formula of that logic and which will tell one whether the formula is or is not valid (as proved by Church in 1936).
     From: David Bostock (Philosophy of Mathematics [2009], 5.5)
18109 The completeness of first-order logic implies its compactness [Bostock]
     Full Idea: From the fact that the usual rules for first-level logic are complete (as proved by Gödel 1930), it follows that this logic is 'compact'.
     From: David Bostock (Philosophy of Mathematics [2009], 5.5)
     A reaction: The point is that the completeness requires finite proofs.
5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
13545 Elementary logic cannot distinguish clearly between the finite and the infinite [Bostock]
     Full Idea: In very general terms, we cannot express the distinction between what is finite and what is infinite without moving essentially beyond the resources available in elementary logic.
     From: David Bostock (Intermediate Logic [1997], 4.8)
     A reaction: This observation concludes a discussion of Compactness in logic.
13346 Truth is the basic notion in classical logic [Bostock]
     Full Idea: The most fundamental notion in classical logic is that of truth.
     From: David Bostock (Intermediate Logic [1997], 1.1)
     A reaction: The opening sentence of his book. Hence the first half of the book is about semantics, and only the second half deals with proof. Compare Idea 10282. The thought seems to be that you could leave out truth, but that makes logic pointless.
13822 Fictional characters wreck elementary logic, as they have contradictions and no excluded middle [Bostock]
     Full Idea: Discourse about fictional characters leads to a breakdown of elementary logic. We accept P or ¬P if the relevant story says so, but P∨¬P will not be true if the relevant story says nothing either way, and P∧¬P is true if the story is inconsistent.
     From: David Bostock (Intermediate Logic [1997], 8.5)
     A reaction: I really like this. Does one need to invent a completely new logic for fictional characters? Or must their logic be intuitionist, or paraconsistent, or both?
5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-
13623 The syntactic turnstile |- φ means 'there is a proof of φ' or 'φ is a theorem' [Bostock]
     Full Idea: The syntactic turnstile |- φ means 'There is a proof of φ' (in the system currently being considered). Another way of saying the same thing is 'φ is a theorem'.
     From: David Bostock (Intermediate Logic [1997], 5.1)
5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
13349 Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid' [Bostock]
     Full Idea: If we write Γ |= φ, with one formula to the right, then the turnstile abbreviates 'entails'. For a sequent of the form Γ |= it can be read as 'is inconsistent'. For |= φ we read it as 'valid'.
     From: David Bostock (Intermediate Logic [1997], 1.3)
13347 Validity is a conclusion following for premises, even if there is no proof [Bostock]
     Full Idea: The classical definition of validity counts an argument as valid if and only if the conclusion does in fact follow from the premises, whether or not the argument contains any demonstration of this fact.
     From: David Bostock (Intermediate Logic [1997], 1.2)
     A reaction: Hence validity is given by |= rather than by |-. A common example is 'it is red so it is coloured', which seems true but beyond proof. In the absence of formal proof, you wonder whether validity is merely a psychological notion.
13348 It seems more natural to express |= as 'therefore', rather than 'entails' [Bostock]
     Full Idea: In practice we avoid quotation marks and explicitly set-theoretic notation that explaining |= as 'entails' appears to demand. Hence it seems more natural to explain |= as simply representing the word 'therefore'.
     From: David Bostock (Intermediate Logic [1997], 1.3)
     A reaction: Not sure I quite understand that, but I have trained myself to say 'therefore' for the generic use of |=. In other consequences it seems better to read it as 'semantic consequence', to distinguish it from |-.
5. Theory of Logic / B. Logical Consequence / 5. Modus Ponens
13614 MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment) [Bostock]
     Full Idea: The Rule of Detachment is a version of Modus Ponens, and says 'If |=φ and |=φ→ψ then |=ψ'. This has no assumptions. Modus Ponens is the more general rule that 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ'.
     From: David Bostock (Intermediate Logic [1997], 5.3)
     A reaction: Modus Ponens is actually designed for use in proof based on assumptions (which isn't always the case). In Detachment the formulae are just valid, without dependence on assumptions to support them.
13617 MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ [Bostock]
     Full Idea: Modus Ponens is equivalent to the converse of the Deduction Theorem, namely 'If Γ |- φ→ψ then Γ,φ|-ψ'.
     From: David Bostock (Intermediate Logic [1997], 5.3)
     A reaction: See 13615 for details of the Deduction Theorem. See 13614 for Modus Ponens.
5. Theory of Logic / D. Assumptions for Logic / 4. Identity in Logic
13799 The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b) [Bostock]
     Full Idea: We shall use 'a=b' as short for 'a is the same thing as b'. The sign '=' thus expresses a particular two-place predicate. Officially we will use 'I' as the identity predicate, so that 'Iab' is as formula, but we normally 'abbreviate' this to 'a=b'.
     From: David Bostock (Intermediate Logic [1997], 8.1)
13800 |= α=α and α=β |= φ(α/ξ ↔ φ(β/ξ) fix identity [Bostock]
     Full Idea: We usually take these two principles together as the basic principles of identity: |= α=α and α=β |= φ(α/ξ) ↔ φ(β/ξ). The second (with scant regard for history) is known as Leibniz's Law.
     From: David Bostock (Intermediate Logic [1997], 8.1)
13803 If we are to express that there at least two things, we need identity [Bostock]
     Full Idea: To say that there is at least one thing x such that Fx we need only use an existential quantifier, but to say that there are at least two things we need identity as well.
     From: David Bostock (Intermediate Logic [1997], 8.1)
     A reaction: The only clear account I've found of why logic may need to be 'with identity'. Without it, you can only reason about one thing or all things. Presumably plural quantification no longer requires '='?
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
13357 Truth-functors are usually held to be defined by their truth-tables [Bostock]
     Full Idea: The usual view of the meaning of truth-functors is that each is defined by its own truth-table, independently of any other truth-functor.
     From: David Bostock (Intermediate Logic [1997], 2.7)
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / d. and
12010 Is the meaning of 'and' given by its truth table, or by its introduction and elimination rules? [Forbes,G]
     Full Idea: The typical semantic account of validity for propositional connectives like 'and' presupposes that meaning is given by truth-tables. On the natural deduction view, the meaning of 'and' is given by its introduction and elimination rules.
     From: Graeme Forbes (The Metaphysics of Modality [1985], 4.4)
5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
13812 A 'zero-place' function just has a single value, so it is a name [Bostock]
     Full Idea: We can talk of a 'zero-place' function, which is a new-fangled name for a familiar item; it just has a single value, and so it has the same role as a name.
     From: David Bostock (Intermediate Logic [1997], 8.2)
13811 A 'total' function ranges over the whole domain, a 'partial' function over appropriate inputs [Bostock]
     Full Idea: Usually we allow that a function is defined for arguments of a suitable kind (a 'partial' function), but we can say that each function has one value for any object whatever, from the whole domain that our quantifiers range over (a 'total' function).
     From: David Bostock (Intermediate Logic [1997], 8.2)
     A reaction: He points out (p.338) that 'the father of..' is a functional expression, but it wouldn't normally take stones as input, so seems to be a partial function. But then it doesn't even take all male humans either. It only takes fathers!
5. Theory of Logic / F. Referring in Logic / 1. Naming / a. Names
13360 In logic, a name is just any expression which refers to a particular single object [Bostock]
     Full Idea: The important thing about a name, for logical purposes, is that it is used to make a singular reference to a particular object; ..we say that any expression too may be counted as a name, for our purposes, it it too performs the same job.
     From: David Bostock (Intermediate Logic [1997], 3.1)
     A reaction: He cites definite descriptions as the most notoriously difficult case, in deciding whether or not they function as names. I takes it as pretty obvious that sometimes they do and sometimes they don't (in ordinary usage).
5. Theory of Logic / F. Referring in Logic / 1. Naming / e. Empty names
13361 An expression is only a name if it succeeds in referring to a real object [Bostock]
     Full Idea: An expression is not counted as a name unless it succeeds in referring to an object, i.e. unless there really is an object to which it refers.
     From: David Bostock (Intermediate Logic [1997], 3.1)
     A reaction: His 'i.e.' makes the existence condition sound sufficient, but in ordinary language you don't succeed in referring to 'that man over there' just because he exists. In modal contexts we presumably refer to hypothetical objects (pace Lewis).
5. Theory of Logic / F. Referring in Logic / 2. Descriptions / b. Definite descriptions
13813 Definite descriptions don't always pick out one thing, as in denials of existence, or errors [Bostock]
     Full Idea: It is natural to suppose one only uses a definite description when one believes it describes only one thing, but exceptions are 'there is no such thing as the greatest prime number', or saying something false where the reference doesn't occur.
     From: David Bostock (Intermediate Logic [1997], 8.3)
13814 Definite desciptions resemble names, but can't actually be names, if they don't always refer [Bostock]
     Full Idea: Although a definite description looks like a complex name, and in many ways behaves like a name, still it cannot be a name if names must always refer to objects. Russell gave the first proposal for handling such expressions.
     From: David Bostock (Intermediate Logic [1997], 8.3)
     A reaction: I take the simple solution to be a pragmatic one, as roughly shown by Donnellan, that sometimes they are used exactly like names, and sometimes as something else. The same phrase can have both roles. Confusing for logicians. Tough.
13816 Because of scope problems, definite descriptions are best treated as quantifiers [Bostock]
     Full Idea: Because of the scope problem, it now seems better to 'parse' definition descriptions not as names but as quantifiers. 'The' is to be treated in the same category as acknowledged quantifiers like 'all' and 'some'. We write Ix - 'for the x such that..'.
     From: David Bostock (Intermediate Logic [1997], 8.3)
     A reaction: This seems intuitively rather good, since quantification in normal speech is much more sophisticated than the crude quantification of classical logic. But the fact is that they often function as names (but see Idea 13817).
13817 Definite descriptions are usually treated like names, and are just like them if they uniquely refer [Bostock]
     Full Idea: In practice, definite descriptions are for the most part treated as names, since this is by far the most convenient notation (even though they have scope). ..When a description is uniquely satisfied then it does behave like a name.
     From: David Bostock (Intermediate Logic [1997], 8.3)
     A reaction: Apparent names themselves have problems when they wander away from uniquely picking out one thing, as in 'John Doe'.
13848 We are only obliged to treat definite descriptions as non-names if only the former have scope [Bostock]
     Full Idea: If it is really true that definite descriptions have scopes whereas names do not, then Russell must be right to claim that definite descriptions are not names. If, however, this is not true, then it does no harm to treat descriptions as complex names.
     From: David Bostock (Intermediate Logic [1997], 8.8)
5. Theory of Logic / F. Referring in Logic / 2. Descriptions / c. Theory of definite descriptions
13815 Names do not have scope problems (e.g. in placing negation), but Russell's account does have that problem [Bostock]
     Full Idea: In orthodox logic names are not regarded as having scope (for example, in where a negation is placed), whereas on Russell's theory definite descriptions certainly do. Russell had his own way of dealing with this.
     From: David Bostock (Intermediate Logic [1997], 8.3)
5. Theory of Logic / G. Quantification / 1. Quantification
13438 'Prenex normal form' is all quantifiers at the beginning, out of the scope of truth-functors [Bostock]
     Full Idea: A formula is said to be in 'prenex normal form' (PNF) iff all its quantifiers occur in a block at the beginning, so that no quantifier is in the scope of any truth-functor.
     From: David Bostock (Intermediate Logic [1997], 3.7)
     A reaction: Bostock provides six equivalences which can be applied to manouevre any formula into prenex normal form. He proves that every formula can be arranged in PNF.
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
13818 If we allow empty domains, we must allow empty names [Bostock]
     Full Idea: We can show that if empty domains are permitted, then empty names must be permitted too.
     From: David Bostock (Intermediate Logic [1997], 8.4)
5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
18123 Substitutional quantification is just standard if all objects in the domain have a name [Bostock]
     Full Idea: Substitutional quantification and quantification understood in the usual 'ontological' way will coincide when every object in the (ontological) domain has a name.
     From: David Bostock (Philosophy of Mathematics [2009], 7.3 n23)
5. Theory of Logic / H. Proof Systems / 1. Proof Systems
13801 An 'informal proof' is in no particular system, and uses obvious steps and some ordinary English [Bostock]
     Full Idea: An 'informal proof' is not in any particular proof system. One may use any rule of proof that is 'sufficiently obvious', and there is quite a lot of ordinary English in the proof, explaining what is going on at each step.
     From: David Bostock (Intermediate Logic [1997], 8.1)
5. Theory of Logic / H. Proof Systems / 2. Axiomatic Proof
13619 Quantification adds two axiom-schemas and a new rule [Bostock]
     Full Idea: New axiom-schemas for quantifiers: (A4) |-∀ξφ → φ(α/ξ), (A5) |-∀ξ(ψ→φ) → (ψ→∀ξφ), plus the rule GEN: If |-φ the |-∀ξφ(ξ/α).
     From: David Bostock (Intermediate Logic [1997], 5.6)
     A reaction: This follows on from Idea 13610, where he laid out his three axioms and one rule for propositional (truth-functional) logic. This Idea plus 13610 make Bostock's proposed axiomatisation of first-order logic.
13622 Axiom systems from Frege, Russell, Church, Lukasiewicz, Tarski, Nicod, Kleene, Quine... [Bostock]
     Full Idea: Notably axiomatisations of first-order logic are by Frege (1879), Russell and Whitehead (1910), Church (1956), Lukasiewicz and Tarski (1930), Lukasiewicz (1936), Nicod (1917), Kleene (1952) and Quine (1951). Also Bostock (1997).
     From: David Bostock (Intermediate Logic [1997], 5.8)
     A reaction: My summary, from Bostock's appendix 5.8, which gives details of all of these nine systems. This nicely illustrates the status and nature of axiom systems, which have lost the absolute status they seemed to have in Euclid.
5. Theory of Logic / H. Proof Systems / 3. Proof from Assumptions
13615 'Conditonalised' inferences point to the Deduction Theorem: If Γ,φ|-ψ then Γ|-φ→ψ [Bostock]
     Full Idea: If a group of formulae prove a conclusion, we can 'conditionalize' this into a chain of separate inferences, which leads to the Deduction Theorem (or Conditional Proof), that 'If Γ,φ|-ψ then Γ|-φ→ψ'.
     From: David Bostock (Intermediate Logic [1997], 5.3)
     A reaction: This is the rule CP (Conditional Proof) which can be found in the rules for propositional logic I transcribed from Lemmon's book.
13616 The Deduction Theorem greatly simplifies the search for proof [Bostock]
     Full Idea: Use of the Deduction Theorem greatly simplifies the search for proof (or more strictly, the task of showing that there is a proof).
     From: David Bostock (Intermediate Logic [1997], 5.3)
     A reaction: See 13615 for details of the Deduction Theorem. Bostock is referring to axiomatic proof, where it can be quite hard to decide which axioms are relevant. The Deduction Theorem enables the making of assumptions.
13620 Proof by Assumptions can always be reduced to Proof by Axioms, using the Deduction Theorem [Bostock]
     Full Idea: By repeated transformations using the Deduction Theorem, any proof from assumptions can be transformed into a fully conditionalized proof, which is then an axiomatic proof.
     From: David Bostock (Intermediate Logic [1997], 5.6)
     A reaction: Since proof using assumptions is perhaps the most standard proof system (e.g. used in Lemmon, for many years the standard book at Oxford University), the Deduction Theorem is crucial for giving it solid foundations.
13621 The Deduction Theorem and Reductio can 'discharge' assumptions - they aren't needed for the new truth [Bostock]
     Full Idea: Like the Deduction Theorem, one form of Reductio ad Absurdum (If Γ,φ|-[absurdity] then Γ|-¬φ) 'discharges' an assumption. Assume φ and obtain a contradiction, then we know ¬&phi, without assuming φ.
     From: David Bostock (Intermediate Logic [1997], 5.7)
     A reaction: Thus proofs from assumption either arrive at conditional truths, or at truths that are true irrespective of what was initially assumed.
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
13753 Natural deduction takes proof from assumptions (with its rules) as basic, and axioms play no part [Bostock]
     Full Idea: Natural deduction takes the notion of proof from assumptions as a basic notion, ...so it will use rules for use in proofs from assumptions, and axioms (as traditionally understood) will have no role to play.
     From: David Bostock (Intermediate Logic [1997], 6.1)
     A reaction: The main rules are those for introduction and elimination of truth functors.
13755 Excluded middle is an introduction rule for negation, and ex falso quodlibet will eliminate it [Bostock]
     Full Idea: Many books take RAA (reductio) and DNE (double neg) as the natural deduction introduction- and elimination-rules for negation, but RAA is not a natural introduction rule. I prefer TND (tertium) and EFQ (ex falso) for ¬-introduction and -elimination.
     From: David Bostock (Intermediate Logic [1997], 6.2)
13758 In natural deduction we work from the premisses and the conclusion, hoping to meet in the middle [Bostock]
     Full Idea: When looking for a proof of a sequent, the best we can do in natural deduction is to work simultaneously in both directions, forward from the premisses, and back from the conclusion, and hope they will meet in the middle.
     From: David Bostock (Intermediate Logic [1997], 6.5)
13754 Natural deduction rules for → are the Deduction Theorem (→I) and Modus Ponens (→E) [Bostock]
     Full Idea: Natural deduction adopts for → as rules the Deduction Theorem and Modus Ponens, here called →I and →E. If ψ follows φ in the proof, we can write φ→ψ (→I). φ and φ→ψ permit ψ (→E).
     From: David Bostock (Intermediate Logic [1997], 6.2)
     A reaction: Natural deduction has this neat and appealing way of formally introducing or eliminating each connective, so that you know where you are, and you know what each one means.
18120 The Deduction Theorem is what licenses a system of natural deduction [Bostock]
     Full Idea: The Deduction Theorem is what licenses a system of 'natural deduction' in the first place.
     From: David Bostock (Philosophy of Mathematics [2009], 7.2)
5. Theory of Logic / H. Proof Systems / 5. Tableau Proof
13756 A tree proof becomes too broad if its only rule is Modus Ponens [Bostock]
     Full Idea: When the only rule of inference is Modus Ponens, the branches of a tree proof soon spread too wide for comfort.
     From: David Bostock (Intermediate Logic [1997], 6.4)
13611 Tableau proofs use reduction - seeking an impossible consequence from an assumption [Bostock]
     Full Idea: A tableau proof is a proof by reduction ad absurdum. One begins with an assumption, and one develops the consequences of that assumption, seeking to derive an impossible consequence.
     From: David Bostock (Intermediate Logic [1997], 4.1)
13612 Non-branching rules add lines, and branching rules need a split; a branch with a contradiction is 'closed' [Bostock]
     Full Idea: Rules for semantic tableaus are of two kinds - non-branching rules and branching rules. The first allow the addition of further lines, and the second requires splitting the branch. A branch which assigns contradictory values to a formula is 'closed'.
     From: David Bostock (Intermediate Logic [1997], 4.1)
     A reaction: [compressed] Thus 'and' stays on one branch, asserting both formulae, but 'or' splits, checking first one and then the other. A proof succeeds when all the branches are closed, showing that the initial assumption leads only to contradictions.
13613 A completed open branch gives an interpretation which verifies those formulae [Bostock]
     Full Idea: An open branch in a completed tableau will always yield an interpretation that verifies every formula on the branch.
     From: David Bostock (Intermediate Logic [1997], 4.7)
     A reaction: In other words the open branch shows a model which seems to work (on the available information). Similarly a closed branch gives a model which won't work - a counterexample.
13761 In a tableau proof no sequence is established until the final branch is closed; hypotheses are explored [Bostock]
     Full Idea: In a tableau system no sequent is established until the final step of the proof, when the last branch closes, and until then we are simply exploring a hypothesis.
     From: David Bostock (Intermediate Logic [1997], 7.3)
     A reaction: This compares sharply with a sequence calculus, where every single step is a conclusive proof of something. So use tableaux for exploring proofs, and then sequence calculi for writing them up?
13757 Unlike natural deduction, semantic tableaux have recipes for proving things [Bostock]
     Full Idea: With semantic tableaux there are recipes for proof-construction that we can operate, whereas with natural deduction there are not.
     From: David Bostock (Intermediate Logic [1997], 6.5)
13762 Tableau rules are all elimination rules, gradually shortening formulae [Bostock]
     Full Idea: In their original setting, all the tableau rules are elimination rules, allowing us to replace a longer formula by its shorter components.
     From: David Bostock (Intermediate Logic [1997], 7.3)
5. Theory of Logic / H. Proof Systems / 6. Sequent Calculi
13760 A sequent calculus is good for comparing proof systems [Bostock]
     Full Idea: A sequent calculus is a useful tool for comparing two systems that at first look utterly different (such as natural deduction and semantic tableaux).
     From: David Bostock (Intermediate Logic [1997], 7.2)
13759 Each line of a sequent calculus is a conclusion of previous lines, each one explicitly recorded [Bostock]
     Full Idea: A sequent calculus keeps an explicit record of just what sequent is established at each point in a proof. Every line is itself the sequent proved at that point. It is not a linear sequence or array of formulae, but a matching array of whole sequents.
     From: David Bostock (Intermediate Logic [1997], 7.1)
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
13364 Interpretation by assigning objects to names, or assigning them to variables first [Bostock, by PG]
     Full Idea: There are two approaches to an 'interpretation' of a logic: the first method assigns objects to names, and then defines connectives and quantifiers, focusing on truth; the second assigns objects to variables, then variables to names, using satisfaction.
     From: report of David Bostock (Intermediate Logic [1997], 3.4) by PG - Db (lexicon)
     A reaction: [a summary of nine elusive pages in Bostock] He says he prefers the first method, but the second method is more popular because it handles open formulas, by treating free variables as if they were names.
5. Theory of Logic / I. Semantics of Logic / 5. Extensionalism
13821 Extensionality is built into ordinary logic semantics; names have objects, predicates have sets of objects [Bostock]
     Full Idea: Extensionality is built into the semantics of ordinary logic. When a name-letter is interpreted as denoting something, we just provide the object denoted. All that we provide for a one-place predicate-letter is the set of objects that it is true of..
     From: David Bostock (Intermediate Logic [1997])
     A reaction: Could we keep the syntax of ordinary logic, and provide a wildly different semantics, much closer to real life? We could give up these dreadful 'objects' that Frege lumbered us with. Logic for processes, etc.
13362 If an object has two names, truth is undisturbed if the names are swapped; this is Extensionality [Bostock]
     Full Idea: If two names refer to the same object, then in any proposition which contains either of them the other may be substituted in its place, and the truth-value of the proposition of the proposition will be unaltered. This is the Principle of Extensionality.
     From: David Bostock (Intermediate Logic [1997], 3.1)
     A reaction: He acknowledges that ordinary language is full of counterexamples, such as 'he doesn't know the Morning Star and the Evening Star are the same body' (when he presumably knows that the Morning Star is the Morning Star). This is logic. Like maths.
5. Theory of Logic / K. Features of Logics / 2. Consistency
13541 For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ [Bostock]
     Full Idea: Any system of proof S is said to be 'negation-consistent' iff there is no formula such that |-(S)φ and |-(S)¬φ.
     From: David Bostock (Intermediate Logic [1997], 4.5)
     A reaction: Compare Idea 13542. This version seems to be a 'strong' version, as it demands a higher standard than 'absolute consistency'. Both halves of the condition would have to be established.
13540 A set of formulae is 'inconsistent' when there is no interpretation which can make them all true [Bostock]
     Full Idea: 'Γ |=' means 'Γ is a set of closed formulae, and there is no (standard) interpretation in which all of the formulae in Γ are true'. We abbreviate this last to 'Γ is inconsistent'.
     From: David Bostock (Intermediate Logic [1997], 4.5)
     A reaction: This is a semantic approach to inconsistency, in terms of truth, as opposed to saying that we cannot prove both p and ¬p. I take this to be closer to the true concept, since you need never have heard of 'proof' to understand 'inconsistent'.
13542 A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula [Bostock]
     Full Idea: Any system of proof S is said to be 'absolutely consistent' iff it is not the case that for every formula we have |-(S)φ.
     From: David Bostock (Intermediate Logic [1997], 4.5)
     A reaction: Bostock notes that a sound system will be both 'negation-consistent' (Idea 13541) and absolutely consistent. 'Tonk' systems can be shown to be unsound because the two come apart.
5. Theory of Logic / K. Features of Logics / 6. Compactness
13544 Inconsistency or entailment just from functors and quantifiers is finitely based, if compact [Bostock]
     Full Idea: Being 'compact' means that if we have an inconsistency or an entailment which holds just because of the truth-functors and quantifiers involved, then it is always due to a finite number of the propositions in question.
     From: David Bostock (Intermediate Logic [1997], 4.8)
     A reaction: Bostock says this is surprising, given the examples 'a is not a parent of a parent of b...' etc, where an infinity seems to establish 'a is not an ancestor of b'. The point, though, is that this truth doesn't just depend on truth-functors and quantifiers.
13618 Compactness means an infinity of sequents on the left will add nothing new [Bostock]
     Full Idea: The logic of truth-functions is compact, which means that sequents with infinitely many formulae on the left introduce nothing new. Hence we can confine our attention to finite sequents.
     From: David Bostock (Intermediate Logic [1997], 5.5)
     A reaction: This makes it clear why compactness is a limitation in logic. If you want the logic to be unlimited in scope, it isn't; it only proves things from finite numbers of sequents. This makes it easier to prove completeness for the system.
5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / c. Berry's paradox
18125 Berry's Paradox considers the meaning of 'The least number not named by this name' [Bostock]
     Full Idea: Berry's Paradox can be put in this form, by considering the alleged name 'The least number not named by this name'.
     From: David Bostock (Philosophy of Mathematics [2009], 8.1)
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
18101 Each addition changes the ordinality but not the cardinality, prior to aleph-1 [Bostock]
     Full Idea: If you add to the ordinals you produce many different ordinals, each measuring the length of the sequence of ordinals less than it. They each have cardinality aleph-0. The cardinality eventually increases, but we can't say where this break comes.
     From: David Bostock (Philosophy of Mathematics [2009], 4.5)
18100 ω + 1 is a new ordinal, but its cardinality is unchanged [Bostock]
     Full Idea: If we add ω onto the end of 0,1,2,3,4..., it then has a different length, of ω+1. It has a different ordinal (since it can't be matched with its first part), but the same cardinal (since adding 1 makes no difference).
     From: David Bostock (Philosophy of Mathematics [2009], 4.5)
     A reaction: [compressed] The ordinals and cardinals coincide up to ω, but this is the point at which they come apart.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
18102 A cardinal is the earliest ordinal that has that number of predecessors [Bostock]
     Full Idea: It is the usual procedure these days to identify a cardinal number with the earliest ordinal number that has that number of predecessors.
     From: David Bostock (Philosophy of Mathematics [2009], 4.5)
     A reaction: This sounds circular, since you need to know the cardinal in order to decide which ordinal is the one you want, but, hey, what do I know?
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
18106 Aleph-1 is the first ordinal that exceeds aleph-0 [Bostock]
     Full Idea: The cardinal aleph-1 is identified with the first ordinal to have more than aleph-0 members, and so on.
     From: David Bostock (Philosophy of Mathematics [2009], 5.4)
     A reaction: That is, the succeeding infinite ordinals all have the same cardinal number of members (aleph-0), until the new total is triggered (at the number of the reals). This is Continuum Hypothesis territory.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
18095 Instead of by cuts or series convergence, real numbers could be defined by axioms [Bostock]
     Full Idea: In addition to cuts, or converging series, Cantor suggests we can simply lay down a set of axioms for the real numbers, and this can be done without any explicit mention of the rational numbers [note: the axioms are those for a complete ordered field].
     From: David Bostock (Philosophy of Mathematics [2009], 4.4)
     A reaction: It is interesting when axioms are best, and when not. Set theory depends entirely on axioms. Horsten and Halbach are now exploring treating truth as axiomatic. You don't give the 'nature' of the thing - just rules for its operation.
18099 The number of reals is the number of subsets of the natural numbers [Bostock]
     Full Idea: It is not difficult to show that the number of the real numbers is the same as the number of all the subsets of the natural numbers.
     From: David Bostock (Philosophy of Mathematics [2009], 4.5)
     A reaction: The Continuum Hypothesis is that this is the next infinite number after the number of natural numbers. Why can't there be a number which is 'most' of the subsets of the natural numbers?
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
18093 For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock]
     Full Idea: As Eudoxus claimed, two distinct real numbers cannot both make the same cut in the rationals, for any two real numbers must be separated by a rational number. He did not say, though, that for every such cut there is a real number that makes it.
     From: David Bostock (Philosophy of Mathematics [2009], 4.4)
     A reaction: This is in Bostock's discussion of Dedekind's cuts. It seems that every cut is guaranteed to produce a real. Fine challenges the later assumption.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals
18110 Infinitesimals are not actually contradictory, because they can be non-standard real numbers [Bostock]
     Full Idea: Non-standard natural numbers will yield non-standard rational and real numbers. These will include reciprocals which will be closer to 0 than any standard real number. These are like 'infinitesimals', so that notion is not actually a contradiction.
     From: David Bostock (Philosophy of Mathematics [2009], 5.5)
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
18156 Modern axioms of geometry do not need the real numbers [Bostock]
     Full Idea: A modern axiomatisation of geometry, such as Hilbert's (1899), does not need to claim the existence of real numbers anywhere in its axioms.
     From: David Bostock (Philosophy of Mathematics [2009], 9.B.5.ii)
     A reaction: This is despite the fact that geometry is reduced to algebra, and the real numbers are the equivalent of continuous lines. Bostock votes for a Greek theory of proportion in this role.
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
18097 The Peano Axioms describe a unique structure [Bostock]
     Full Idea: The Peano Axioms are categorical, meaning that they describe a unique structure.
     From: David Bostock (Philosophy of Mathematics [2009], 4.4 n20)
     A reaction: So if you think there is nothing more to the natural numbers than their structure, then the Peano Axioms give the essence of arithmetic. If you think that 'objects' must exist to generate a structure, there must be more to the numbers.
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
13358 Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all [Bostock]
     Full Idea: The principle of mathematical (or ordinary) induction says suppose the first number, 0, has a property; suppose that if any number has that property, then so does the next; then it follows that all numbers have the property.
     From: David Bostock (Intermediate Logic [1997], 2.8)
     A reaction: Ordinary induction is also known as 'weak' induction. Compare Idea 13359 for 'strong' or complete induction. The number sequence must have a first element, so this doesn't work for the integers.
13359 Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers [Bostock]
     Full Idea: The principle of complete induction says suppose that for every number, if all the numbers less than it have a property, then so does it; it then follows that every number has the property.
     From: David Bostock (Intermediate Logic [1997], 2.8)
     A reaction: Complete induction is also known as 'strong' induction. Compare Idea 13358 for 'weak' or mathematical induction. The number sequence need have no first element.
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
18148 Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock]
     Full Idea: Hume's Principle will not do as an implicit definition because it makes a positive claim about the size of the universe (which no mere definition can do), and because it does not by itself explain what the numbers are.
     From: David Bostock (Philosophy of Mathematics [2009], 9.A.2)
18145 Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock]
     Full Idea: Hume's Principle gives a criterion of identity for numbers, but it is obvious that many other things satisfy that criterion. The simplest example is probably the numerals (in any notation, decimal, binary etc.), giving many different interpretations.
     From: David Bostock (Philosophy of Mathematics [2009], 9.A.2)
18149 There are many criteria for the identity of numbers [Bostock]
     Full Idea: There is not just one way of giving a criterion of identity for numbers.
     From: David Bostock (Philosophy of Mathematics [2009], 9.A.2)
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
18143 Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock]
     Full Idea: The Julius Caesar problem was one reason that led Frege to give an explicit definition of numbers as special sets. He does not appear to notice that the same problem affects his Axiom V for introducing sets (whether Caesar is or is not a set).
     From: David Bostock (Philosophy of Mathematics [2009], 9.A.2)
     A reaction: The Julius Caesar problem is a sceptical acid that eats into everything in philosophy of mathematics. You give all sorts of wonderful accounts of numbers, but at what point do you know that you now have a number, and not something else?
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
18116 Numbers can't be positions, if nothing decides what position a given number has [Bostock]
     Full Idea: There is no ground for saying that a number IS a position, if the truth is that there is nothing to determine which number is which position.
     From: David Bostock (Philosophy of Mathematics [2009], 6.4)
     A reaction: If numbers lose touch with the empirical ability to count physical objects, they drift off into a mad world where they crumble away.
18117 Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock]
     Full Idea: Structuralism begins from a false premise, namely that numbers have no properties other than their relations to other numbers.
     From: David Bostock (Philosophy of Mathematics [2009], 6.5)
     A reaction: Well said. Describing anything purely relationally strikes me as doomed, because you have to say why those things relate in those ways.
6. Mathematics / C. Sources of Mathematics / 3. Mathematical Nominalism
18141 Nominalism about mathematics is either reductionist, or fictionalist [Bostock]
     Full Idea: Nominalism has two main versions, one which tries to 'reduce' the objects of mathematics to something simpler (Russell and Wittgenstein), and another which claims that such objects are mere 'fictions' which have no reality (Field).
     From: David Bostock (Philosophy of Mathematics [2009], 9)
18157 Nominalism as based on application of numbers is no good, because there are too many applications [Bostock]
     Full Idea: The style of nominalism which aims to reduce statements about numbers to statements about their applications does not work for the natural numbers, because they have many applications, and it is arbitrary to choose just one of them.
     From: David Bostock (Philosophy of Mathematics [2009], 9.B.5.iii)
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
18150 Actual measurement could never require the precision of the real numbers [Bostock]
     Full Idea: We all know that in practice no physical measurement can be 100 per cent accurate, and so it cannot require the existence of a genuinely irrational number, rather than some of the rational numbers close to it.
     From: David Bostock (Philosophy of Mathematics [2009], 9.A.3)
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
18158 Ordinals are mainly used adjectively, as in 'the first', 'the second'... [Bostock]
     Full Idea: The basic use of the ordinal numbers is their use as ordinal adjectives, in phrases such as 'the first', 'the second' and so on.
     From: David Bostock (Philosophy of Mathematics [2009], 9.5.iii)
     A reaction: That is because ordinals seem to attach to particulars, whereas cardinals seem to attach to groups. Then you say 'three is greater than four', it is not clear which type you are talking about.
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
18127 Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock]
     Full Idea: The simple theory of types distinguishes sets into different 'levels', but this is quite different from the distinction into 'orders' which is imposed by the ramified theory.
     From: David Bostock (Philosophy of Mathematics [2009], 8.1)
     A reaction: The ramified theory has both levels and orders (p.235). Russell's terminology is, apparently, inconsistent.
6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
18147 Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock]
     Full Idea: The response of neo-logicists to the Julius Caesar problem is to strengthen Hume's Principle in the hope of ensuring that only numbers will satisfy it. They say the criterion of identity provided by HP is essential to number, and not to anything else.
     From: David Bostock (Philosophy of Mathematics [2009], 9.A.2)
18144 Neo-logicists agree that HP introduces number, but also claim that it suffices for the job [Bostock]
     Full Idea: The neo-logicists take up Frege's claim that Hume's Principle introduces a new concept (of a number), but unlike Frege they go on to claim that it by itself gives a complete account of that concept.
     From: David Bostock (Philosophy of Mathematics [2009], 9.A.2)
     A reaction: So the big difference between Frege and neo-logicists is the Julius Caesar problem.
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
18111 Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock]
     Full Idea: If logic is neutral on the number of objects there are, then logicists can't construe numbers as objects, for arithmetic is certainly not neutral on the number of numbers there are. They must be treated in some other way, perhaps as numerical quantifiers.
     From: David Bostock (Philosophy of Mathematics [2009], 5.5)
18146 If Hume's Principle is the whole story, that implies structuralism [Bostock]
     Full Idea: If Hume's Principle is all we are given, by way of explanation of what the numbers are, the only conclusion to draw would seem to be the structuralists' conclusion, ...studying all systems that satisfy that principle.
     From: David Bostock (Philosophy of Mathematics [2009], 9.A.2)
     A reaction: Any approach that implies a set of matching interpretations will always imply structuralism. To avoid it, you need to pin the target down uniquely.
18129 Many crucial logicist definitions are in fact impredicative [Bostock]
     Full Idea: Many of the crucial definitions in the logicist programme are in fact impredicative.
     From: David Bostock (Philosophy of Mathematics [2009], 8.2)
6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
18159 Higher cardinalities in sets are just fairy stories [Bostock]
     Full Idea: In its higher reaches, which posit sets of huge cardinalities, set theory is just a fairy story.
     From: David Bostock (Philosophy of Mathematics [2009], 9.5.iii)
     A reaction: You can't say the higher reaches are fairy stories but the lower reaches aren't, if the higher is directly derived from the lower. The empty set and the singleton are fairy stories too. Bostock says the axiom of infinity triggers the fairy stories.
18155 A fairy tale may give predictions, but only a true theory can give explanations [Bostock]
     Full Idea: A common view is that although a fairy tale may provide very useful predictions, it cannot provide explanations for why things happen as they do. In order to do that a theory must also be true (or, at least, an approximation to the truth).
     From: David Bostock (Philosophy of Mathematics [2009], 9.B.5)
     A reaction: Of course, fictionalism offers an explanation of mathematics as a whole, but not of the details (except as the implications of the initial fictional assumptions).
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
18140 The best version of conceptualism is predicativism [Bostock]
     Full Idea: In my personal opinion, predicativism is the best version of conceptualism that we have yet discovered.
     From: David Bostock (Philosophy of Mathematics [2009], 8.4)
     A reaction: Since conceptualism is a major player in the field, this makes predicativism a very important view. I won't vote Predicativist quite yet, but I'm tempted.
18138 Conceptualism fails to grasp mathematical properties, infinity, and objective truth values [Bostock]
     Full Idea: Three simple objections to conceptualism in mathematics are that we do not ascribe mathematical properties to our ideas, that our ideas are presumably finite, and we don't think mathematics lacks truthvalue before we thought of it.
     From: David Bostock (Philosophy of Mathematics [2009], 8.4)
     A reaction: [compressed; Bostock refers back to his Ch 2] Plus Idea 18134. On the whole I sympathise with conceptualism, so I will not allow myself to be impressed by any of these objections. (So, what's actually wrong with them.....?).
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
18131 If abstracta only exist if they are expressible, there can only be denumerably many of them [Bostock]
     Full Idea: If an abstract object exists only when there is some suitable way of expressing it, then there are at most denumerably many abstract objects.
     From: David Bostock (Philosophy of Mathematics [2009], 8.2)
     A reaction: Fine by me. What an odd view, to think there are uncountably many abstract objects in existence, only a countable portion of which will ever be expressed! [ah! most people agree with me, p.243-4]
18134 Predicativism makes theories of huge cardinals impossible [Bostock]
     Full Idea: Classical mathematicians say predicative mathematics omits areas of great interest, all concerning non-denumerable real numbers, such as claims about huge cardinals. There cannot be a predicative version of this theory.
     From: David Bostock (Philosophy of Mathematics [2009], 8.3)
     A reaction: I'm not sure that anyone will really miss huge cardinals if they are prohibited, though cryptography seems to flirt with such things. Are we ever allowed to say that some entity conjured up by mathematicians is actually impossible?
18135 If mathematics rests on science, predicativism may be the best approach [Bostock]
     Full Idea: It has been claimed that only predicative mathematics has a justification through its usefulness to science (an empiricist approach).
     From: David Bostock (Philosophy of Mathematics [2009], 8.3)
     A reaction: [compressed. Quine is the obvious candidate] I suppose predicativism gives your theory roots, whereas impredicativism is playing an abstract game.
18136 If we can only think of what we can describe, predicativism may be implied [Bostock]
     Full Idea: If we accept the initial idea that we can think only of what we ourselves can describe, then something like the theory of predicativism quite naturally results
     From: David Bostock (Philosophy of Mathematics [2009], 8.3)
     A reaction: I hate the idea that we can only talk of what falls under a sortal, but 'what we can describe' is much more plausible. Whether or not you agree with this approach (I'm pondering it), this makes predicativism important.
18133 The usual definitions of identity and of natural numbers are impredicative [Bostock]
     Full Idea: The predicative approach cannot accept either the usual definition of identity or the usual definition of the natural numbers, for both of these definitions are impredicative.
     From: David Bostock (Philosophy of Mathematics [2009], 8.3)
     A reaction: [Bostock 237-8 gives details]
18132 The predicativity restriction makes a difference with the real numbers [Bostock]
     Full Idea: It is with the real numbers that the restrictions imposed by predicativity begin to make a real difference.
     From: David Bostock (Philosophy of Mathematics [2009], 8.3)
7. Existence / D. Theories of Reality / 10. Vagueness / d. Vagueness as linguistic
12023 Vagueness problems arise from applying sharp semantics to vague languages [Forbes,G]
     Full Idea: It is very plausible that the sorites paradoxes arose from the application of a semantic apparatus appropriate only for sharp predicates to languages containing vague predicates (rather than from deficiency of meaning, or from incoherence).
     From: Graeme Forbes (The Metaphysics of Modality [1985], 7.3)
     A reaction: Sounds wrong. Of course, logic has been designed for sharp predicates, and natural languages are awash with vagueness. But the problems of vagueness bothered lawyers long before logicians like Russell began to worry about it.
8. Modes of Existence / A. Relations / 4. Formal Relations / a. Types of relation
13543 A relation is not reflexive, just because it is transitive and symmetrical [Bostock]
     Full Idea: It is easy to fall into the error of supposing that a relation which is both transitive and symmetrical must also be reflexive.
     From: David Bostock (Intermediate Logic [1997], 4.7)
     A reaction: Compare Idea 14430! Transivity will take you there, and symmetricality will get you back, but that doesn't entitle you to take the shortcut?
13802 Relations can be one-many (at most one on the left) or many-one (at most one on the right) [Bostock]
     Full Idea: A relation is 'one-many' if for anything on the right there is at most one on the left (∀xyz(Rxz∧Ryz→x=y), and is 'many-one' if for anything on the left there is at most one on the right (∀xyz(Rzx∧Rzy→x=y).
     From: David Bostock (Intermediate Logic [1997], 8.1)
9. Objects / A. Existence of Objects / 5. Individuation / a. Individuation
12017 In all instances of identity, there must be some facts to ensure the identity [Forbes,G]
     Full Idea: For each instance of identity or failure of identity, there must be facts in virtue of which that instance obtains. ..Enough has been said to lend this doctrine some plausibility.
     From: Graeme Forbes (The Metaphysics of Modality [1985], 5.5)
     A reaction: Penelope Mackie picks this out from Forbes as a key principle. It sounds to be in danger of circularity, unless the 'facts' can be cited without referring to, or implicitly making use of, identities - which seems unlikely.
9. Objects / B. Unity of Objects / 3. Unity Problems / d. Coincident objects
12024 If we combined two clocks, it seems that two clocks may have become one clock. [Forbes,G]
     Full Idea: If we imagine a possible world in which two clocks in a room make one clock from half the parts of each, the judgement 'these two actual clocks could have been a single clock' does not seem wholly false.
     From: Graeme Forbes (The Metaphysics of Modality [1985], 7.4)
     A reaction: You would, of course, have sufficient parts left over to make a second clock, so they look like a destroyed clock, so I don't think I find Forbes's intuition on this one very persuasive.
9. Objects / D. Essence of Objects / 3. Individual Essences
11885 Only individual essences will ground identities across worlds in other properties [Forbes,G, by Mackie,P]
     Full Idea: Forbes argues that, unless we posit individual essences, we cannot guarantee that identities across possible worlds will be appropriately grounded in other properties.
     From: report of Graeme Forbes (The Metaphysics of Modality [1985]) by Penelope Mackie - How Things Might Have Been 2.4
     A reaction: There is a confrontation between Wiggins, who says identity is primitive, and Forbes, who says identity must be grounded in other properties. I think I side with Forbes.
12014 An individual essence is a set of essential properties which only that object can have [Forbes,G]
     Full Idea: An individual essence of an object x is a set of properties I which satisfies the following conditions: i. every property P in I is an essential property of x; ii. it is not possible that some object y distinct from x has every member of I.
     From: Graeme Forbes (The Metaphysics of Modality [1985], 5.1)
     A reaction: I am coming to the view that stable natural kinds (like electrons or gold) do not have individual essences, but complex kinds (like tigers or tables) do. The view is based on the idea that explanatory power is what individuates an essence.
12015 Non-trivial individual essence is properties other than de dicto, or universal, or relational [Forbes,G]
     Full Idea: A non-trivial individual essence is properties other than a) those following from a de dicto truth, b) properties of existence and self-identity (or their cognates), c) properties derived from necessities in some other category.
     From: Graeme Forbes (The Metaphysics of Modality [1985], 5.1)
     A reaction: [I have compressed Forbes] Rather than adding all these qualificational clauses to our concept, we could just tighten up on the notion of a property, saying it is something which is causally efficacious, and hence explanatory.
9. Objects / D. Essence of Objects / 5. Essence as Kind
12013 Essential properties depend on a category, and perhaps also on particular facts [Forbes,G]
     Full Idea: The essential properties of a thing will typically depend upon what category of thing it is, and perhaps also on some more particular facts about the thing itself.
     From: Graeme Forbes (The Metaphysics of Modality [1985], 5.1)
     A reaction: I see no way of dispensing with the second requirement, in the cases of complex entities like animals. If all samples are the same, then of course we can define a sample's essence through its kind, but not if samples differ in any way.
9. Objects / D. Essence of Objects / 6. Essence as Unifier
13804 A property is essential iff the object would not exist if it lacked that property [Forbes,G]
     Full Idea: A property P is an essential property of an object x iff x could not exist and lack P, that is, as they say, iff x has P at every world at which x exists.
     From: Graeme Forbes (In Defense of Absolute Essentialism [1986], 1)
     A reaction: This immediately places the existence of x outside the normal range of its properties, so presumably 'existence is not a predicate', but that dictum may be doubted. As it stands this definition will include trivial and vacuous properties.
13805 Properties are trivially essential if they are not grounded in a thing's specific nature [Forbes,G]
     Full Idea: Essential properties may be trivial or nontrivial. It is characteristic of P's being trivially essential to x that x's possession of P is not grounded in the specific nature of x.
     From: Graeme Forbes (In Defense of Absolute Essentialism [1986], 2)
     A reaction: This is where my objection to the modal view of essence arises. How is he going to explain 'grounded' and 'specific nature' without supplying an entirely different account of essence?
9. Objects / D. Essence of Objects / 7. Essence and Necessity / a. Essence as necessary properties
12012 Essential properties are those without which an object could not exist [Forbes,G]
     Full Idea: An essential property of an object x is a property without possessing which x could not exist.
     From: Graeme Forbes (The Metaphysics of Modality [1985], 5.1)
     A reaction: This is certainly open to question. See Joan Kung's account of Aristotle on essence. I am necessarily more than eight years old (now), and couldn't exist without that property, but is the property part of my essence?
13808 A relation is essential to two items if it holds in every world where they exist [Forbes,G]
     Full Idea: A relation R is essential to x and y (in that order) iff Rxy holds at every world where x and y both exist.
     From: Graeme Forbes (In Defense of Absolute Essentialism [1986], 2)
     A reaction: I find this bizarre. Not only does this seem to me to have nothing whatever to do with essence, but also the relation might hold even though it is a purely contingent matter. All rabbits are a reasonable distance from the local star. Essence of rabbit?
9. Objects / D. Essence of Objects / 7. Essence and Necessity / c. Essentials are necessary
13806 Trivially essential properties are existence, self-identity, and de dicto necessities [Forbes,G]
     Full Idea: The main groups of trivially essential properties are (a) existence, self-identity, or their consequences in S5; and (b) properties possessed in virtue of some de dicto necessary truth.
     From: Graeme Forbes (In Defense of Absolute Essentialism [1986], 2)
     A reaction: He adds 'extraneously essential' properties, which also strike me as being trivial, involving relations. 'Is such that 2+2=4' or 'is such that something exists' might be necessary, but they don't, I would say, have anything to do with essence.
9. Objects / D. Essence of Objects / 9. Essence and Properties
13807 A property is 'extraneously essential' if it is had only because of the properties of other objects [Forbes,G]
     Full Idea: P is 'extraneously essential' to x iff it is possessed by x at any world w only in virtue of the possession at w of certain properties by other objects.
     From: Graeme Forbes (In Defense of Absolute Essentialism [1986], 2)
     A reaction: I would say that these are the sorts of properties which have nothing to do with being essential, even if they are deemed to be necessary.
9. Objects / D. Essence of Objects / 11. Essence of Artefacts
12025 Artefacts have fuzzy essences [Forbes,G]
     Full Idea: Artefacts can be ascribed fuzzy essences. ...We might say that it is essential to an artefact to have 'most' of its parts.
     From: Graeme Forbes (The Metaphysics of Modality [1985], 7.6)
     A reaction: I think I prefer to accept the idea that essences are unstable things, in all cases. For all we know, electrons might subtly change their general character, or cease to be uniform, tomorrow. Essences explain, and what needs explaining changes.
12022 Same parts does not ensure same artefact, if those parts could constitute a different artefact [Forbes,G]
     Full Idea: Sameness of parts is not sufficient for identity of artefacts at a world, since the very same parts may turn up at different times as the parts of artefacts with different designs and functions.
     From: Graeme Forbes (The Metaphysics of Modality [1985], 7.2)
     A reaction: Thus the Ship of Theseus could be dismantled and turned into a barn (as happened with the 'Mayflower'). They could then be reconstituted as the ship, which would then have two beginnings (as Chris Hughes has pointed out).
13809 One might be essentialist about the original bronze from which a statue was made [Forbes,G]
     Full Idea: In the case of artefacts, there is an essentialism about original matter; for instance, it would be said of any particular bronze statue that it could not have been cast from a totally different quantity of bronze.
     From: Graeme Forbes (In Defense of Absolute Essentialism [1986], 3)
     A reaction: Forbes isn't endorsing this, and it doesn't sound convincing. He quotes the thought 'I wish I had made this pot from a different piece of clay'. We might corrupt a statue by switching bronze, but I don't think the sculptor could do so.
9. Objects / E. Objects over Time / 12. Origin as Essential
12020 An individual might change their sex in a world, but couldn't have differed in sex at origin [Forbes,G]
     Full Idea: In the time of a single world, the same individual can undergo a change of sex, but it is less clear that an individual of one sex could have been, from the outset, an individual of another.
     From: Graeme Forbes (The Metaphysics of Modality [1985], 6.5)
     A reaction: I don't find this support for essentiality of origin very persuasive. I struggle with these ideas. Given my sex yesterday, then presumably I couldn't have had a different sex yesterday. Given that pigs can fly, pigs can fly. What am I missing?
9. Objects / F. Identity among Objects / 1. Concept of Identity
11888 Identities must hold because of other facts, which must be instrinsic [Forbes,G, by Mackie,P]
     Full Idea: Forbes has two principles of identity, which we can call the No Bare Identities Principle (identities hold in virtue of other facts), and the No Extrinsic Determination Principle (that only intrinsic facts of a thing establish identity).
     From: report of Graeme Forbes (The Metaphysics of Modality [1985], 127-8) by Penelope Mackie - How Things Might Have Been 2.7
     A reaction: The job of the philosopher is to prise apart the real identities of things from the way in which we conceive of identities. I take these principles to apply to real identities, not conceptual identities.
9. Objects / F. Identity among Objects / 5. Self-Identity
13847 If non-existent things are self-identical, they are just one thing - so call it the 'null object' [Bostock]
     Full Idea: If even non-existent things are still counted as self-identical, then all non-existent things must be counted as identical with one another, so there is at most one non-existent thing. We might arbitrarily choose zero, or invent 'the null object'.
     From: David Bostock (Intermediate Logic [1997], 8.6)
10. Modality / A. Necessity / 4. De re / De dicto modality
12003 De re modal formulae, unlike de dicto, are sensitive to transworld identities [Forbes,G]
     Full Idea: The difference between de re and de dicto formulae is a difference between formulae which are, and formulae which are not, sensitive to the identities of objects at various worlds.
     From: Graeme Forbes (The Metaphysics of Modality [1985], 3.1)
10. Modality / A. Necessity / 6. Logical Necessity
13820 The idea that anything which can be proved is necessary has a problem with empty names [Bostock]
     Full Idea: The common Rule of Necessitation says that what can be proved is necessary, but this is incorrect if we do not permit empty names. The most straightforward answer is to modify elementary logic so that only necessary truths can be proved.
     From: David Bostock (Intermediate Logic [1997], 8.4)
10. Modality / C. Sources of Modality / 4. Necessity from Concepts
12028 De re necessity is a form of conceptual necessity, just as de dicto necessity is [Forbes,G]
     Full Idea: De re necessity does not differ from de dicto necessity in respect of how it arises: it is still a form of conceptual necessity.
     From: Graeme Forbes (The Metaphysics of Modality [1985], 9.4)
     A reaction: [Forbes proceeds to argue for this claim] Forbes defends a form of essentialism, but takes the necessity to arise from a posteriori truths because of the a priori involvement of other concepts (rather as Kripke argues).
13810 The source of de dicto necessity is not concepts, but the actual properties of the thing [Forbes,G]
     Full Idea: It is widely held that the source of de dicto necessity is in concepts, ..but I deny this... even with simple de dicto necessities, the source of the necessity is to be found in the properties to which the predicates of the de dicto truth refer.
     From: Graeme Forbes (In Defense of Absolute Essentialism [1986], 3)
     A reaction: It is normal nowadays to say this about de re necessities, but this is more unusual.
10. Modality / E. Possible worlds / 1. Possible Worlds / c. Possible worlds realism
12008 Unlike places and times, we cannot separate possible worlds from what is true at them [Forbes,G]
     Full Idea: There is no means by which we might distinguish a possible world from what is true at it. ...Whereas our ability to separate a place, or a time, from its occupier is crucial to realism about places and times, as is a distance relation.
     From: Graeme Forbes (The Metaphysics of Modality [1985], 4.2)
     A reaction: He is objecting to Lewis's modal realism. I'm not fully convinced. It depends whether we are discussing real ontology or conceptual space. In the latter I see no difference between times and possible worlds. In ontology, a 'time' is weird.
12009 The problem with possible worlds realism is epistemological; we can't know properties of possible objects [Forbes,G]
     Full Idea: The main objection to realism about worlds is from epistemology. Knowledge of properties of objects requires experience of these objects, which must be within the range of our sensory faculties, but only concrete actual objects achieve that.
     From: Graeme Forbes (The Metaphysics of Modality [1985], 4.2)
     A reaction: This pinpoints my dislike of the whole possible worlds framework, ontologically speaking. I seem to be an actualist. I take possibilities to be inferences to the best explanation from the powers we know of in the actual world. We experience potentiality.
10. Modality / E. Possible worlds / 2. Nature of Possible Worlds / a. Nature of possible worlds
12007 Possible worlds are points of logical space, rather like other times than our own [Forbes,G]
     Full Idea: Someone impressed by the parallel between tense and modal operators ...might suggest that just as we can speak of places and times forming their own manifolds or spaces, so we can say that worlds are the points of logical space.
     From: Graeme Forbes (The Metaphysics of Modality [1985], 4.2)
     A reaction: I particularly like the notion of worlds being "points of logical space", and am inclined to remove it from this context and embrace it as the correct way to understand possible worlds. We must understand logical or conceptual space.
10. Modality / E. Possible worlds / 3. Transworld Objects / a. Transworld identity
12011 Transworld identity concerns the limits of possibility for ordinary things [Forbes,G]
     Full Idea: An elucidation of transworld identity can be regarded as an elucidation of the boundaries of possibility for ordinary things.
     From: Graeme Forbes (The Metaphysics of Modality [1985], 5.1)
     A reaction: I presume that if we don't search for some such criterion, we just have to face the possibility that Aristotle could have been a poached egg in some possible world. To know the bounds of possibility, study the powers of actual objects.
12016 The problem of transworld identity can be solved by individual essences [Forbes,G]
     Full Idea: The motivation for investigating individual essences should be obvious, since if every object has such an essence, the problem of elucidating transworld identity can be solved.
     From: Graeme Forbes (The Metaphysics of Modality [1985], 5.1)
     A reaction: It is important that, if necessary, the identities be 'individual', and not just generic, by sortal, or natural kind. We want to reason about (and explain) truths at the fine-grained level of the individual, not just at the broad level of generalisation.
10. Modality / E. Possible worlds / 3. Transworld Objects / c. Counterparts
12004 Counterpart theory is not good at handling the logic of identity [Forbes,G]
     Full Idea: The outstanding technical objection to counterpart-theoretic semantics concerns its handling of the logic of identity. In quantified S5 (the orthodox semantics) a = b → □(a = b) is valid, but 'a' must not attach to two objects.
     From: Graeme Forbes (The Metaphysics of Modality [1985], 3.5)
10. Modality / E. Possible worlds / 3. Transworld Objects / d. Haecceitism
12021 Haecceitism attributes to each individual a primitive identity or thisness [Forbes,G]
     Full Idea: Haecceitism attributes to each individual a primitive identity or thisness, as opposed to the sort of essentialism that gives non-trivial conditions sufficient for transworld identity.
     From: Graeme Forbes (The Metaphysics of Modality [1985], 6.6)
     A reaction: 'Haecceitism' is the doctrine that things have primitive identity. A 'haecceity' is a postulated property which actually does the job. The key point of the view is that whatever it is is 'primitive', and not complex, or analysable. I don't believe it.
12029 We believe in thisnesses, because we reject bizarre possibilities as not being about that individual [Forbes,G]
     Full Idea: The natural response to an unreasonable hypothesis of possibility for an object x, that in such a state of affairs it would not be x which satisfies the conditions, is evidence that we do possess concepts of thisness for individuals.
     From: Graeme Forbes (The Metaphysics of Modality [1985], 9.4)
     A reaction: We may have a 'concept' of thisness, but we needn't be committed to the 'existence' of a thisness. There is a fairly universal intuition that cessation of existence of an entity when it starts to change can be a very vague matter.
19. Language / C. Assigning Meanings / 3. Predicates
13363 A (modern) predicate is the result of leaving a gap for the name in a sentence [Bostock]
     Full Idea: A simple way of approaching the modern notion of a predicate is this: given any sentence which contains a name, the result of dropping that name and leaving a gap in its place is a predicate. Very different from predicates in Aristotle and Kant.
     From: David Bostock (Intermediate Logic [1997], 3.2)
     A reaction: This concept derives from Frege. To get to grips with contemporary philosophy you have to relearn all sorts of basic words like 'predicate' and 'object'.
19. Language / F. Communication / 2. Assertion
18121 In logic a proposition means the same when it is and when it is not asserted [Bostock]
     Full Idea: In Modus Ponens where the first premise is 'P' and the second 'P→Q', in the first premise P is asserted but in the second it is not. Yet it must mean the same in both premises, or it would be guilty of the fallacy of equivocation.
     From: David Bostock (Philosophy of Mathematics [2009], 7.2)
     A reaction: This is Geach's thought (leading to an objection to expressivism in ethics, that P means the same even if it is not expressed).