Combining Philosophers

All the ideas for Lynch,MP/Glasgow,JM, M Fitting/R Mendelsohn and Shaughan Lavine

unexpand these ideas     |    start again     |     specify just one area for these philosophers

---

81 ideas

4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables

9738	Each line of a truth table is a model [Fitting/Mendelsohn]
	Full Idea: Each line of a truth table is, in effect, a model.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.6)
	A reaction: I find this comment illuminating. It is being connected with the more complex models of modal logic. Each line of a truth table is a picture of how the world might be.

4. Formal Logic / D. Modal Logic ML / 2. Tools of Modal Logic / a. Symbols of ML

9727	Modal logic adds □ (necessarily) and ◊ (possibly) to classical logic [Fitting/Mendelsohn]
	Full Idea: For modal logic we add to the syntax of classical logic two new unary operators □ (necessarily) and ◊ (possibly).
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.3)

9726	We let 'R' be the accessibility relation: xRy is read 'y is accessible from x' [Fitting/Mendelsohn]
	Full Idea: We let 'R' be the accessibility relation: xRy is read 'y is accessible from x'.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.5)

9737	The symbol ||- is the 'forcing' relation; 'Γ ||- P' means that P is true in world Γ [Fitting/Mendelsohn]
	Full Idea: The symbol ||- is used for the 'forcing' relation, as in 'Γ ||- P', which means that P is true in world Γ.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.6)

13136	The prefix σ names a possible world, and σ.n names a world accessible from that one [Fitting/Mendelsohn]
	Full Idea: A 'prefix' is a finite sequence of positive integers. A 'prefixed formula' is an expression of the form σ X, where σ is a prefix and X is a formula. A prefix names a possible world, and σ.n names a world accessible from that one.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.2)

4. Formal Logic / D. Modal Logic ML / 2. Tools of Modal Logic / b. Terminology of ML

13727	A 'constant' domain is the same for all worlds; 'varying' domains can be entirely separate [Fitting/Mendelsohn]
	Full Idea: In 'constant domain' semantics, the domain of each possible world is the same as every other; in 'varying domain' semantics, the domains need not coincide, or even overlap.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 4.5)

9734	Modern modal logic introduces 'accessibility', saying xRy means 'y is accessible from x' [Fitting/Mendelsohn]
	Full Idea: Modern modal logic takes into consideration the way the modal relates the possible worlds, called the 'accessibility' relation. .. We let R be the accessibility relation, and xRy reads as 'y is accessible from x.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.5)
	A reaction: There are various types of accessibility, and these define the various modal logics.

9736	A 'model' is a frame plus specification of propositions true at worlds, written < G,R,||- > [Fitting/Mendelsohn]
	Full Idea: A 'model' is a frame plus a specification of which propositional letters are true at which worlds. It is written as , where ||- is a relation between possible worlds and propositional letters. So Γ ||- P means P is true at world Γ.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.6)

9735	A 'frame' is a set G of possible worlds, with an accessibility relation R, written < G,R > [Fitting/Mendelsohn]
	Full Idea: A 'frame' consists of a non-empty set G, whose members are generally called possible worlds, and a binary relation R, on G, generally called the accessibility relation. We say the frame is the pair so that a single object can be talked about.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.6)

9741	Accessibility relations can be 'reflexive' (self-referring), 'transitive' (carries over), or 'symmetric' (mutual) [Fitting/Mendelsohn]
	Full Idea: A relation R is 'reflexive' if every world is accessible from itself; 'transitive' if the first world is related to the third world (ΓRΔ and ΔRΩ → ΓRΩ); and 'symmetric' if the accessibility relation is mutual.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.7)
	A reaction: The different systems of modal logic largely depend on how these accessibility relations are specified. There is also the 'serial' relation, which just says that any world has another world accessible to it.

4. Formal Logic / D. Modal Logic ML / 2. Tools of Modal Logic / c. Derivation rules of ML

13149	S5: a) if n ◊X then kX b) if n ¬□X then k ¬X c) if n □X then k X d) if n ¬◊X then k ¬X [Fitting/Mendelsohn]
	Full Idea: Simplified S5 rules: a) if n ◊X then kX b) if n ¬□X then k ¬X c) if n □X then k X d) if n ¬◊X then k ¬X. 'n' picks any world; in a) and b) 'k' asserts a new world; in c) and d) 'k' refers to a known world
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.3)

9740	If a proposition is possibly true in a world, it is true in some world accessible from that world [Fitting/Mendelsohn]
	Full Idea: If a proposition is possibly true in a world, then it is also true in some world which is accessible from that world. That is: Γ ||- ◊X ↔ for some Δ ∈ G, ΓRΔ then Δ ||- X.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.6)

9739	If a proposition is necessarily true in a world, it is true in all worlds accessible from that world [Fitting/Mendelsohn]
	Full Idea: If a proposition is necessarily true in a world, then it is also true in all worlds which are accessible from that world. That is: Γ ||- □X ↔ for every Δ ∈ G, if ΓRΔ then Δ ||- X.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.6)

13137	Conj: a) if σ X∧Y then σ X and σ Y b) if σ ¬(X∧Y) then σ ¬X or σ ¬Y [Fitting/Mendelsohn]
	Full Idea: General tableau rules for conjunctions: a) if σ X ∧ Y then σ X and σ Y b) if σ ¬(X ∧ Y) then σ ¬X or σ ¬Y
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.2)

13140	Bicon: a)if σ(X↔Y) then σ(X→Y) and σ(Y→X) b) [not biconditional, one or other fails] [Fitting/Mendelsohn]
	Full Idea: General tableau rules for biconditionals: a) if σ (X ↔ Y) then σ (X → Y) and σ (Y → X) b) if σ ¬(X ↔ Y) then σ ¬(X → Y) or σ ¬(Y → X)
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.2)

13139	Implic: a) if σ ¬(X→Y) then σ X and σ ¬Y b) if σ X→Y then σ ¬X or σ Y [Fitting/Mendelsohn]
	Full Idea: General tableau rules for implications: a) if σ ¬(X → Y) then σ X and σ ¬Y b) if σ X → Y then σ ¬X or σ Y
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.2)

13143	Universal: a) if σ ¬◊X then σ.m ¬X b) if σ □X then σ.m X [m exists] [Fitting/Mendelsohn]
	Full Idea: General tableau rules for universal modality: a) if σ ¬◊ X then σ.m ¬X b) if σ □ X then σ.m X , where m refers to a world that can be seen (rather than introducing a new world).
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.2)
	A reaction: Note that the universal rule of □, usually read as 'necessary', only refers to worlds which can already be seen, whereas possibility (◊) asserts some thing about a new as yet unseen world.

13141	Negation: if σ ¬¬X then σ X [Fitting/Mendelsohn]
	Full Idea: General tableau rule for negation: if σ ¬¬X then σ X
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.2)

13138	Disj: a) if σ ¬(X∨Y) then σ ¬X and σ ¬Y b) if σ X∨Y then σ X or σ Y [Fitting/Mendelsohn]
	Full Idea: General tableau rules for disjunctions: a) if σ ¬(X ∨ Y) then σ ¬X and σ ¬Y b) if σ X ∨ Y then σ X or σ Y
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.2)

13142	Existential: a) if σ ◊X then σ.n X b) if σ ¬□X then σ.n ¬X [n is new] [Fitting/Mendelsohn]
	Full Idea: General tableau rules for existential modality: a) if σ ◊ X then σ.n X b) if σ ¬□ X then σ.n ¬X , where n introduces some new world (rather than referring to a world that can be seen).
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.2)
	A reaction: Note that the existential rule of ◊, usually read as 'possibly', asserts something about a new as yet unseen world, whereas □ only refers to worlds which can already be seen,

13144	T reflexive: a) if σ □X then σ X b) if σ ¬◊X then σ ¬X [Fitting/Mendelsohn]
	Full Idea: System T reflexive rules (also for B, S4, S5): a) if σ □X then σ X b) if σ ¬◊X then σ ¬X
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.3)

13145	D serial: a) if σ □X then σ ◊X b) if σ ¬◊X then σ ¬□X [Fitting/Mendelsohn]
	Full Idea: System D serial rules (also for T, B, S4, S5): a) if σ □X then σ ◊X b) if σ ¬◊X then σ ¬□X
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.3)

13146	B symmetric: a) if σ.n □X then σ X b) if σ.n ¬◊X then σ ¬X [n occurs] [Fitting/Mendelsohn]
	Full Idea: System B symmetric rules (also for S5): a) if σ.n □X then σ X b) if σ.n ¬◊X then σ ¬X [where n is a world which already occurs]
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.3)

13147	4 transitive: a) if σ □X then σ.n □X b) if σ ¬◊X then σ.n ¬◊X [n occurs] [Fitting/Mendelsohn]
	Full Idea: System 4 transitive rules (also for K4, S4, S5): a) if σ □X then σ.n □X b) if σ ¬◊X then σ.n ¬◊X [where n is a world which already occurs]
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.3)

13148	4r rev-trans: a) if σ.n □X then σ □X b) if σ.n ¬◊X then σ ¬◊X [n occurs] [Fitting/Mendelsohn]
	Full Idea: System 4r reversed-transitive rules (also for S5): a) if σ.n □X then σ □X b) if σ.n ¬◊X then σ ¬◊X [where n is a world which already occurs]
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.3)

4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / b. System K

9742	The system K has no accessibility conditions [Fitting/Mendelsohn]
	Full Idea: The system K has no frame conditions imposed on its accessibility relation.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.8)
	A reaction: The system is named K in honour of Saul Kripke.

4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / c. System D

13114	□P → P is not valid in D (Deontic Logic), since an obligatory action may be not performed [Fitting/Mendelsohn]
	Full Idea: System D is usually thought of as Deontic Logic, concerning obligations and permissions. □P → P is not valid in D, since just because an action is obligatory, it does not follow that it is performed.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.12.2 Ex)

9743	The system D has the 'serial' conditon imposed on its accessibility relation [Fitting/Mendelsohn]
	Full Idea: The system D has the 'serial' condition imposed on its accessibility relation - that is, every world must have some world which is accessible to it.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.8)

4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / d. System T

9744	The system T has the 'reflexive' conditon imposed on its accessibility relation [Fitting/Mendelsohn]
	Full Idea: The system T has the 'reflexive' condition imposed on its accessibility relation - that is, every world must be accessible to itself.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.8)

4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / e. System K4

9746	The system K4 has the 'transitive' condition on its accessibility relation [Fitting/Mendelsohn]
	Full Idea: The system K4 has the 'transitive' condition imposed on its accessibility relation - that is, if a relation holds between worlds 1 and 2 and worlds 2 and 3, it must hold between worlds 1 and 3. The relation carries over.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.8)

4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / f. System B

9745	The system B has the 'reflexive' and 'symmetric' conditions on its accessibility relation [Fitting/Mendelsohn]
	Full Idea: The system B has the 'reflexive' and 'symmetric' conditions imposed on its accessibility relation - that is, every world must be accessible to itself, and any relation between worlds must be mutual.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.8)

4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / g. System S4

9747	The system S4 has the 'reflexive' and 'transitive' conditions on its accessibility relation [Fitting/Mendelsohn]
	Full Idea: The system S4 has the 'reflexive' and 'transitive' conditions imposed on its accessibility relation - that is, every world is accessible to itself, and accessibility carries over a series of worlds.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.8)

4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / h. System S5

9748	System S5 has the 'reflexive', 'symmetric' and 'transitive' conditions on its accessibility relation [Fitting/Mendelsohn]
	Full Idea: The system S5 has the 'reflexive', 'symmetric' and 'transitive' conditions imposed on its accessibility relation - that is, every world is self-accessible, and accessibility is mutual, and it carries over a series of worlds.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.8)
	A reaction: S5 has total accessibility, and hence is the most powerful system (though it might be too powerful).

4. Formal Logic / D. Modal Logic ML / 4. Alethic Modal Logic

9404	Modality affects content, because P→◊P is valid, but ◊P→P isn't [Fitting/Mendelsohn]
	Full Idea: P→◊P is usually considered to be valid, but its converse, ◊P→P is not, so (by Frege's own criterion) P and possibly-P differ in conceptual content, and there is no reason why logic should not be widened to accommodate this.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.2)
	A reaction: Frege had denied that modality affected the content of a proposition (1879:p.4). The observation here is the foundation for the need for a modal logic.

4. Formal Logic / D. Modal Logic ML / 5. Epistemic Logic

13112	In epistemic logic knowers are logically omniscient, so they know that they know [Fitting/Mendelsohn]
	Full Idea: In epistemic logic the knower is treated as logically omniscient. This is puzzling because one then cannot know something and yet fail to know that one knows it (the Principle of Positive Introspection).
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.11)
	A reaction: This is nowadays known as the K-K Problem - to know, must you know that you know. Broadly, we find that externalists say you don't need to know that you know (so animals know things), but internalists say you do need to know that you know.

13111	Read epistemic box as 'a knows/believes P' and diamond as 'for all a knows/believes, P' [Fitting/Mendelsohn]
	Full Idea: In epistemic logic we read Υ as 'KaP: a knows that P', and ◊ as 'PaP: it is possible, for all a knows, that P' (a is an individual). For belief we read them as 'BaP: a believes that P' and 'CaP: compatible with everything a believes that P'.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.11)
	A reaction: [scripted capitals and subscripts are involved] Hintikka 1962 is the source of this. Fitting and Mendelsohn prefer □ to read 'a is entitled to know P', rather than 'a knows that P'.

4. Formal Logic / D. Modal Logic ML / 6. Temporal Logic

13113	F: will sometime, P: was sometime, G: will always, H: was always [Fitting/Mendelsohn]
	Full Idea: We introduce four future and past tense operators: FP: it will sometime be the case that P. PP: it was sometime the case that P. GP: it will always be the case that P. HP: it has always been the case that P. (P itself is untensed).
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.10)
	A reaction: Temporal logic begins with A.N. Prior, and starts with □ as 'always', and ◊ as 'sometimes', but then adds these past and future divisions. Two different logics emerge, taking □ and ◊ as either past or as future.

4. Formal Logic / D. Modal Logic ML / 7. Barcan Formula

13728	The Barcan says nothing comes into existence; the Converse says nothing ceases; the pair imply stability [Fitting/Mendelsohn]
	Full Idea: The Converse Barcan says nothing passes out of existence in alternative situations. The Barcan says that nothing comes into existence. The two together say the same things exist no matter what the situation.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 4.9)
	A reaction: I take the big problem to be that these reflect what it is you want to say, and that does not keep stable across a conversation, so ordinary rational discussion sometimes asserts these formulas, and 30 seconds later denies them.

13729	The Barcan corresponds to anti-monotonicity, and the Converse to monotonicity [Fitting/Mendelsohn]
	Full Idea: The Barcan formula corresponds to anti-monotonicity, and the Converse Barcan formula corresponds to monotonicity.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 6.3)

4. Formal Logic / F. Set Theory ST / 1. Set Theory

15945	Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine]
	Full Idea: Second-order set theory is just like first-order set-theory, except that we use the version of Replacement with a universal second-order quantifier over functions from set to sets.
	From: Shaughan Lavine (Understanding the Infinite [1994], VII.4)

4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST

15914	An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine]
	Full Idea: A member m of M is an 'upper bound' of a subset N of M if m is not less than any member of N. A member m of M is a 'least upper bound' of N if m is an upper bound of N such that if l is any other upper bound of N, then m is less than l.
	From: Shaughan Lavine (Understanding the Infinite [1994], III.4)
	A reaction: [if you don't follow that, you'll have to keep rereading it till you do]

4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set

15921	Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine]
	Full Idea: Since combinatorial collections are enumerated, some multiplicities may be too large to be gathered into combinatorial collections. But the size of a multiplicity seems quite irrelevant to whether it forms a logical connection.
	From: Shaughan Lavine (Understanding the Infinite [1994], IV.2)

4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets

15937	Those who reject infinite collections also want to reject the Axiom of Choice [Lavine]
	Full Idea: Many of those who are skeptical about the existence of infinite combinatorial collections would want to doubt or deny the Axiom of Choice.
	From: Shaughan Lavine (Understanding the Infinite [1994], VI.2)

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI

15936	The Power Set is just the collection of functions from one collection to another [Lavine]
	Full Idea: The Power Set is just he codification of the fact that the collection of functions from a mathematical collection to a mathematical collection is itself a mathematical collection that can serve as a domain of mathematical study.
	From: Shaughan Lavine (Understanding the Infinite [1994], VI.1)

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII

15899	Replacement was immediately accepted, despite having very few implications [Lavine]
	Full Idea: The Axiom of Replacement (of Skolem and Fraenkel) was remarkable for its universal acceptance, though it seemed to have no consequences except for the properties of the higher reaches of the Cantorian infinite.
	From: Shaughan Lavine (Understanding the Infinite [1994], I)

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII

15930	Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine]
	Full Idea: The Axiom of Foundation (Zermelo 1930) says 'Every (descending) chain in which each element is a member of the previous one is of finite length'. ..This forbids circles of membership, or ungrounded sets. ..The iterative conception gives this centre stage.
	From: Shaughan Lavine (Understanding the Infinite [1994], V.4)

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX

15920	Pure collections of things obey Choice, but collections defined by a rule may not [Lavine]
	Full Idea: Combinatorial collections (defined just by the members) obviously obey the Axiom of Choice, while it is at best dubious whether logical connections (defined by a rule) do.
	From: Shaughan Lavine (Understanding the Infinite [1994], IV.2)

15898	The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine]
	Full Idea: The controversy was not about Choice per se, but about the correct notion of function - between advocates of taking mathematics to be about arbitrary functions and advocates of taking it to be about functions given by rules.
	From: Shaughan Lavine (Understanding the Infinite [1994], I)

4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / c. Logical sets

15919	The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine]
	Full Idea: The Peano-Russell notion of class is the 'logical' notion, where each collection is associated with some kind of definition or rule that characterises the members of the collection.
	From: Shaughan Lavine (Understanding the Infinite [1994], IV.1)

4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets

15900	The iterative conception of set wasn't suggested until 1947 [Lavine]
	Full Idea: The iterative conception of set was not so much as suggested, let alone advocated by anyone, until 1947.
	From: Shaughan Lavine (Understanding the Infinite [1994], I)

15931	The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine]
	Full Idea: The iterative conception of sets does not tell us how far to iterate, and so we must start with an Axiom of Infinity. It also presupposes the notion of 'transfinite iteration'.
	From: Shaughan Lavine (Understanding the Infinite [1994], V.5)

15932	The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine]
	Full Idea: The iterative conception does not provide a conception that unifies the axioms of set theory, ...and it has had very little impact on what theorems can be proved.
	From: Shaughan Lavine (Understanding the Infinite [1994], V.5)
	A reaction: He says he would like to reject the iterative conception, but it may turn out that Foundation enables new proofs in mathematics (though it hasn't so far).

4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size

15933	Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine]
	Full Idea: Limitation of Size has it that if a collection is the same size as a set, then it is a set. The Axiom of Replacement is characteristic of limitation of size.
	From: Shaughan Lavine (Understanding the Infinite [1994], V.5)

4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets

15913	A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine]
	Full Idea: A collection M is 'well-ordered' by a relation < if < linearly orders M with a least element, and every subset of M that has an upper bound not in it has an immediate successor.
	From: Shaughan Lavine (Understanding the Infinite [1994], III.4)

5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic

15926	Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine]
	Full Idea: The distinctive feature of second-order logic is that it presupposes that, given a domain, there is a fact of the matter about what the relations on it are, so that the range of the second-order quantifiers is fixed as soon as the domain is fixed.
	From: Shaughan Lavine (Understanding the Infinite [1994], V.3)
	A reaction: This sounds like a rather large assumption, which is open to challenge. I am not sure whether it was the basis of Quine's challenge to second-order logic. He seems to have disliked its vagueness, because it didn't stick with 'objects'.

5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle

15934	Mathematical proof by contradiction needs the law of excluded middle [Lavine]
	Full Idea: The Law of Excluded Middle is (part of) the foundation of the mathematical practice of employing proofs by contradiction.
	From: Shaughan Lavine (Understanding the Infinite [1994], VI.1)
	A reaction: This applies in a lot of logic, as well as in mathematics. Come to think of it, it applies in Sudoku.

5. Theory of Logic / F. Referring in Logic / 3. Property (λ-) Abstraction

9725	'Predicate abstraction' abstracts predicates from formulae, giving scope for constants and functions [Fitting/Mendelsohn]
	Full Idea: 'Predicate abstraction' is a key idea. It is a syntactic mechanism for abstracting a predicate from a formula, providing a scoping mechanism for constants and function symbols similar to that provided for variables by quantifiers.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], Pref)

6. Mathematics / A. Nature of Mathematics / 1. Mathematics

15907	Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine]
	Full Idea: Mathematics is today thought of as the study of abstract structure, not the study of quantity. That point of view arose directly out of the development of the set-theoretic notion of abstract structure.
	From: Shaughan Lavine (Understanding the Infinite [1994], III.2)
	A reaction: It sounds as if Structuralism, which is a controversial view in philosophy, is a fait accompli among mathematicians.

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number

15942	Every rational number, unlike every natural number, is divisible by some other number [Lavine]
	Full Idea: One reason to introduce the rational numbers is that it simplifes the theory of division, since every rational number is divisible by every nonzero rational number, while the analogous statement is false for the natural numbers.
	From: Shaughan Lavine (Understanding the Infinite [1994], VI.3)
	A reaction: That is, with rations every division operation has an answer.

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers

15922	For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine]
	Full Idea: The chief importance of the Continuum Hypothesis for Cantor (I believe) was that it would show that the real numbers form a set, and hence that they were encompassed by his theory.
	From: Shaughan Lavine (Understanding the Infinite [1994], IV.2)

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy

18250	Cauchy gave a necessary condition for the convergence of a sequence [Lavine]
	Full Idea: The Cauchy convergence criterion for a sequence: the sequence S0,S1,... has a limit if |S(n+r) - S(n)| is less than any given quantity for every value of r and sufficiently large values of n. He proved this necessary, but not sufficient.
	From: Shaughan Lavine (Understanding the Infinite [1994], 2.5)

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts

15904	The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine]
	Full Idea: Roughly speaking, the upper and lower parts of the Dedekind cut correspond to the commensurable ratios greater than and less than a given incommensurable ratio.
	From: Shaughan Lavine (Understanding the Infinite [1994], II.6)
	A reaction: Thus there is the problem of whether the contents of the gap are one unique thing, or many.

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure

15912	Counting results in well-ordering, and well-ordering makes counting possible [Lavine]
	Full Idea: Counting a set produces a well-ordering of it. Conversely, if one has a well-ordering of a set, one can count it by following the well-ordering.
	From: Shaughan Lavine (Understanding the Infinite [1994], III.4)
	A reaction: Cantor didn't mean that you could literally count the set, only in principle.

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite

15949	The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine]
	Full Idea: The indiscernibility of indefinitely large sizes will be a critical part of the theory of indefinitely large sizes.
	From: Shaughan Lavine (Understanding the Infinite [1994], VIII.2)

15947	The infinite is extrapolation from the experience of indefinitely large size [Lavine]
	Full Idea: My proposal is that the concept of the infinite began with an extrapolation from the experience of indefinitely large size.
	From: Shaughan Lavine (Understanding the Infinite [1994], VIII.2)
	A reaction: I think it might be better to talk of an 'abstraction' than an 'extrapolition', since the latter is just more of the same, which doesn't get you to concept. Lavine spends 100 pages working out his proposal.

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / c. Potential infinite

15940	The intuitionist endorses only the potential infinite [Lavine]
	Full Idea: The intuitionist endorse the actual finite, but only the potential infinite.
	From: Shaughan Lavine (Understanding the Infinite [1994], VI.2)

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / f. Uncountable infinities

15909	'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine]
	Full Idea: The symbol 'aleph-nought' denotes the cardinal number of the set of natural numbers. The symbol 'aleph-one' denotes the next larger cardinal number. 'Aleph-omega' denotes the omega-th cardinal number.
	From: Shaughan Lavine (Understanding the Infinite [1994], III.3)

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity

15915	Ordinals are basic to Cantor's transfinite, to count the sets [Lavine]
	Full Idea: The ordinals are basic because the transfinite sets are those that can be counted, or (equivalently for Cantor), those that can be numbered by an ordinal or are well-ordered.
	From: Shaughan Lavine (Understanding the Infinite [1994], III.4)
	A reaction: Lavine observes (p.55) that for Cantor 'countable' meant 'countable by God'!

15917	Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine]
	Full Idea: The paradox of the largest ordinal (the 'Burali-Forti') is that the class of all ordinal numbers is apparently well-ordered, and so it has an ordinal number as order type, which must be the largest ordinal - but all ordinals can be increased by one.
	From: Shaughan Lavine (Understanding the Infinite [1994], III.5)

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity

15918	Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine]
	Full Idea: The paradox of the largest cardinal ('Cantor's Paradox') says the diagonal argument shows there is no largest cardinal, but the class of all individuals (including the classes) must be the largest cardinal number.
	From: Shaughan Lavine (Understanding the Infinite [1994], III.5)

6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory

15929	Set theory will found all of mathematics - except for the notion of proof [Lavine]
	Full Idea: Every theorem of mathematics has a counterpart with set theory - ...but that theory cannot serve as a basis for the notion of proof.
	From: Shaughan Lavine (Understanding the Infinite [1994], V.3)

6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism

15935	Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine]
	Full Idea: In modern mathematics virtually all work is only up to isomorphism and no one cares what the numbers or points and lines 'really are'.
	From: Shaughan Lavine (Understanding the Infinite [1994], VI.1)
	A reaction: At least that leaves the field open for philosophers, because we do care what things really are. So should everybody else, but there is no persuading some people.

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism

15928	Intuitionism rejects set-theory to found mathematics [Lavine]
	Full Idea: Intuitionism in philosophy of mathematics rejects set-theoretic foundations.
	From: Shaughan Lavine (Understanding the Infinite [1994], V.3 n33)

7. Existence / C. Structure of Existence / 3. Levels of Reality

16062	A necessary relation between fact-levels seems to be a further irreducible fact [Lynch/Glasgow]
	Full Idea: It seems unavoidable that the facts about logically necessary relations between levels of facts are themselves logically distinct further facts, irreducible to the microphysical facts.
	From: Lynch,MP/Glasgow,JM (The Impossibility of Superdupervenience [2003], C)
	A reaction: I'm beginning to think that rejecting every theory of reality that is proposed by carefully exposing some infinite regress hidden in it is a rather lazy way to do philosophy. Almost as bad as rejecting anything if it can't be defined.

7. Existence / C. Structure of Existence / 5. Supervenience / c. Significance of supervenience

16061	If some facts 'logically supervene' on some others, they just redescribe them, adding nothing [Lynch/Glasgow]
	Full Idea: Logical supervenience, restricted to individuals, seems to imply strong reduction. It is said that where the B-facts logically supervene on the A-facts, the B-facts simply re-describe what the A-facts describe, and the B-facts come along 'for free'.
	From: Lynch,MP/Glasgow,JM (The Impossibility of Superdupervenience [2003], C)
	A reaction: This seems to be taking 'logically' to mean 'analytically'. Presumably an entailment is logically supervenient on its premisses, and may therefore be very revealing, even if some people think such things are analytic.

7. Existence / D. Theories of Reality / 6. Physicalism

16060	Nonreductive materialism says upper 'levels' depend on lower, but don't 'reduce' [Lynch/Glasgow]
	Full Idea: The root intuition behind nonreductive materialism is that reality is composed of ontologically distinct layers or levels. …The upper levels depend on the physical without reducing to it.
	From: Lynch,MP/Glasgow,JM (The Impossibility of Superdupervenience [2003], B)
	A reaction: A nice clear statement of a view which I take to be false. This relationship is the sort of thing that drives people fishing for an account of it to use the word 'supervenience', which just says two things seem to hang out together. Fluffy materialism.

16064	The hallmark of physicalism is that each causal power has a base causal power under it [Lynch/Glasgow]
	Full Idea: Jessica Wilson (1999) says what makes physicalist accounts different from emergentism etc. is that each individual causal power associated with a supervenient property is numerically identical with a causal power associated with its base property.
	From: Lynch,MP/Glasgow,JM (The Impossibility of Superdupervenience [2003], n 11)
	A reaction: Hence the key thought in so-called (serious, rather than self-evident) 'emergentism' is so-called 'downward causation', which I take to be an idle daydream.

9. Objects / F. Identity among Objects / 7. Indiscernible Objects

13730	The Indiscernibility of Identicals has been a big problem for modal logic [Fitting/Mendelsohn]
	Full Idea: Equality has caused much grief for modal logic. Many of the problems, which have struck at the heart of the coherence of modal logic, stem from the apparent violations of the Indiscernibility of Identicals.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 7.1)
	A reaction: Thus when I say 'I might have been three inches taller', presumably I am referring to someone who is 'identical' to me, but who lacks one of my properties. A simple solution is to say that the person is 'essentially' identical.

10. Modality / E. Possible worlds / 3. Transworld Objects / a. Transworld identity

13725	□ must be sensitive as to whether it picks out an object by essential or by contingent properties [Fitting/Mendelsohn]
	Full Idea: If □ is to be sensitive to the quality of the truth of a proposition in its scope, then it must be sensitive as to whether an object is picked out by an essential property or by a contingent one.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 4.3)
	A reaction: This incredibly simple idea strikes me as being powerful and important. ...However, creating illustrative examples leaves me in a state of confusion. You try it. They cite '9' and 'number of planets'. But is it just nominal essence? '9' must be 9.

13731	Objects retain their possible properties across worlds, so a bundle theory of them seems best [Fitting/Mendelsohn]
	Full Idea: The property of 'possibly being a Republican' is as much a property of Bill Clinton as is 'being a democrat'. So we don't peel off his properties from world to world. Hence the bundle theory fits our treatment of objects better than bare particulars.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 7.3)
	A reaction: This bundle theory is better described in recent parlance as the 'modal profile'. I am reluctant to talk of a modal truth about something as one of its 'properties'. An objects, then, is a bundle of truths?

10. Modality / E. Possible worlds / 3. Transworld Objects / c. Counterparts

13726	Counterpart relations are neither symmetric nor transitive, so there is no logic of equality for them [Fitting/Mendelsohn]
	Full Idea: The main technical problem with counterpart theory is that the being-a-counterpart relation is, in general, neither symmetric nor transitive, so no natural logic of equality is forthcoming.
	From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 4.5)
	A reaction: That is, nothing is equal to a counterpart, either directly or indirectly.