Combining Texts

All the ideas for 'Substance and Individuation in Leibniz', 'There is no question of physicalism' and 'A Tour through Mathematical Logic'

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35 ideas

4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL

13520	A 'tautology' must include connectives [Wolf,RS]
	Full Idea: 'For every number x, x = x' is not a tautology, because it includes no connectives.
	From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.2)

4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL

13524	Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS]
	Full Idea: Deduction Theorem: If T ∪ {P} |- Q, then T |- (P → Q). This is the formal justification of the method of conditional proof (CPP). Its converse holds, and is essentially modus ponens.
	From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3)

4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / d. Universal quantifier ∀

13522	Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS]
	Full Idea: Universal Generalization: If we can prove P(x), only assuming what sort of object x is, we may conclude ∀xP(x) for the same x.
	From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3)
	A reaction: This principle needs watching closely. If you pick one person in London, with no presuppositions, and it happens to be a woman, can you conclude that all the people in London are women? Fine in logic and mathematics, suspect in life.

13521	Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS]
	Full Idea: Universal Specification: from ∀xP(x) we may conclude P(t), where t is an appropriate term. If something is true for all members of a domain, then it is true for some particular one that we specify.
	From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3)

4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / e. Existential quantifier ∃

13523	Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS]
	Full Idea: Existential Generalization (or 'proof by example'): From P(t), where t is an appropriate term, we may conclude ∃xP(x).
	From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3)
	A reaction: It is amazing how often this vacuous-sounding principles finds itself being employed in discussions of ontology, but I don't quite understand why.

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / e. Axiom of the Empty Set IV

13529	Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS]
	Full Idea: Empty Set Axiom: ∃x ∀y ¬ (y ∈ x). There is a set x which has no members (no y's). The empty set exists. There is a set with no members, and by extensionality this set is unique.
	From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.3)
	A reaction: A bit bewildering for novices. It says there is a box with nothing in it, or a pair of curly brackets with nothing between them. It seems to be the key idea in set theory, because it asserts the idea of a set over and above any possible members.

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / n. Axiom of Comprehension

13526	Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS]
	Full Idea: The comprehension axiom says that any collection of objects that can be clearly specified can be considered to be a set.
	From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.2)
	A reaction: This is virtually tautological, since I presume that 'clearly specified' means pinning down exact which items are the members, which is what a set is (by extensionality). The naïve version is, of course, not so hot.

5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic

13534	In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS]
	Full Idea: One of the most appealing features of first-order logic is that the two 'turnstiles' (the syntactic single |-, and the semantic double |=), which are the two reasonable notions of logical consequence, actually coincide.
	From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.3)
	A reaction: In the excitement about the possibility of second-order logic, plural quantification etc., it seems easy to forget the virtues of the basic system that is the target of the rebellion. The issue is how much can be 'expressed' in first-order logic.

13535	First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS]
	Full Idea: The 'completeness' of first order-logic does not mean that every sentence or its negation is provable in first-order logic. We have instead the weaker result that every valid sentence is provable.
	From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.3)
	A reaction: Peter Smith calls the stronger version 'negation completeness'.

5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models

13531	Model theory reveals the structures of mathematics [Wolf,RS]
	Full Idea: Model theory helps one to understand what it takes to specify a mathematical structure uniquely.
	From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.1)
	A reaction: Thus it is the development of model theory which has led to the 'structuralist' view of mathematics.

13532	Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS]
	Full Idea: A 'structure' in model theory has a non-empty set, the 'universe', as domain of variables, a subset for each 'relation', some 'functions', and 'constants'.
	From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.2)

13519	Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS]
	Full Idea: Model theory uses set theory to show that the theorem-proving power of the usual methods of deduction in mathematics corresponds perfectly to what must be true in actual mathematical structures.
	From: Robert S. Wolf (A Tour through Mathematical Logic [2005], Pref)
	A reaction: That more or less says that model theory demonstrates the 'soundness' of mathematics (though normal arithmetic is famously not 'complete'). Of course, he says they 'correspond' to the truths, rather than entailing them.

13533	First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS]
	Full Idea: The three foundations of first-order model theory are the Completeness theorem, the Compactness theorem, and the Löwenheim-Skolem-Tarski theorem.
	From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.3)
	A reaction: On p.180 he notes that Compactness and LST make no mention of |- and are purely semantic, where Completeness shows the equivalence of |- and |=. All three fail for second-order logic (p.223).

5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms

13537	An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS]
	Full Idea: An 'isomorphism' is a bijection between two sets that preserves all structural components. The interpretations of each constant symbol are mapped across, and functions map the relation and function symbols.
	From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.4)

5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems

13539	The LST Theorem is a serious limitation of first-order logic [Wolf,RS]
	Full Idea: The Löwenheim-Skolem-Tarski theorem demonstrates a serious limitation of first-order logic, and is one of primary reasons for considering stronger logics.
	From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.7)

5. Theory of Logic / K. Features of Logics / 4. Completeness

13538	If a theory is complete, only a more powerful language can strengthen it [Wolf,RS]
	Full Idea: It is valuable to know that a theory is complete, because then we know it cannot be strengthened without passing to a more powerful language.
	From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.5)

5. Theory of Logic / K. Features of Logics / 10. Monotonicity

13525	Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS]
	Full Idea: Deductive logic, including first-order logic and other types of logic used in mathematics, is 'monotonic'. This means that we never retract a theorem on the basis of new givens. If T|-φ and T⊆SW, then S|-φ. Ordinary reasoning is nonmonotonic.
	From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.7)
	A reaction: The classic example of nonmonotonic reasoning is the induction that 'all birds can fly', which is retracted when the bird turns out to be a penguin. He says nonmonotonic logic is a rich field in computer science.

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers

13530	An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS]
	Full Idea: Less theoretically, an ordinal is an equivalence class of well-orderings. Formally, we say a set is 'transitive' if every member of it is a subset of it, and an ordinal is a transitive set, all of whose members are transitive.
	From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.4)
	A reaction: He glosses 'transitive' as 'every member of a member of it is a member of it'. So it's membership all the way down. This is the von Neumann rather than the Zermelo approach (which is based on singletons).

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

13518	Modern mathematics has unified all of its objects within set theory [Wolf,RS]
	Full Idea: One of the great achievements of modern mathematics has been the unification of its many types of objects. It began with showing geometric objects numerically or algebraically, and culminated with set theory representing all the normal objects.
	From: Robert S. Wolf (A Tour through Mathematical Logic [2005], Pref)
	A reaction: His use of the word 'object' begs all sorts of questions, if you are arriving from the street, where an object is something which can cause a bruise - but get used to it, because the word 'object' has been borrowed for new uses.

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

7855	Some suggest that materialism is empty, because 'physical' cannot be properly characterized [Mellor/Crane, by Papineau]
	Full Idea: It is sometimes contended that the whole idea of materialism is somehow empty, on the grounds that there is no proper way to characterize the 'physical' realm.
	From: report of DH Mellor / T Crane (There is no question of physicalism [1990]) by David Papineau - Thinking about Consciousness 1.1
	A reaction: [Papineau also cites Gabriel Segal] I agree with Papineau in rejecting this. Uncertainties about quantum states do not pose a problem, and the position tends to have an implicit dualism buried in it somewhere.

8. Modes of Existence / A. Relations / 1. Nature of Relations

13076	Scholastics treat relations as two separate predicates of the relata [Cover/O'Leary-Hawthorne]
	Full Idea: The scholastics treated it as a step in the right explanatory direction to analyze a relational statement of the form 'aRb' into two subject-predicate statements, attributing different relational predicates to a and to b.
	From: Cover,J/O'Leary-Hawthorne,J (Substance and Individuation in Leibniz [1999], 2.2.1)
	A reaction: The only alternative seems to be Russell's view of relations as pure universals, having a life of their own, quite apart from their relata. Or you could take them as properties of space, time (and powers?), external to the relata?

9. Objects / A. Existence of Objects / 5. Individuation / a. Individuation

13102	If you individuate things by their origin, you still have to individuate the origins themselves [Cover/O'Leary-Hawthorne]
	Full Idea: If we go for the necessity-of-origins view, A and B are different if the origin of A is different from the origin of B. But one is left with the further question 'When is the origin of A distinct from the origin of B?'
	From: Cover,J/O'Leary-Hawthorne,J (Substance and Individuation in Leibniz [1999], 7.4.1)
	A reaction: There may be an answer to this, in a regress of origins that support one another, but in the end the objection is obviously good. You can't begin to refer to an 'origin' if you can't identify anything in the first place.

13103	Numerical difference is a symmetrical notion, unlike proper individuation [Cover/O'Leary-Hawthorne]
	Full Idea: Scholastics distinguished criteria of numerical difference from questions of individuation proper, since numerical difference is a symmetrical notion.
	From: Cover,J/O'Leary-Hawthorne,J (Substance and Individuation in Leibniz [1999], 7.4.1)
	A reaction: This apparently old-fashioned point appears to be conclusively correct. Modern thinkers, though, aren't comfortable with proper individuation, because they don't believe in concepts like 'essence' and 'substance' that are needed for the job.

9. Objects / A. Existence of Objects / 5. Individuation / d. Individuation by haecceity

13104	Haecceity as property, or as colourless thisness, or as singleton set [Cover/O'Leary-Hawthorne]
	Full Idea: There is a contemporary property construal of haecceities, ...and a Scotistic construal as primitive, 'colourless' thisnesses which, unlike singleton-set haecceities, are aimed to do some explanatory work.
	From: Cover,J/O'Leary-Hawthorne,J (Substance and Individuation in Leibniz [1999], 7.4.4)
	A reaction: [He associates the contemporary account with David Kaplan] I suppose I would say that individuation is done by properties, but not by some single property, so I take it that I don't believe in haecceities at all. What individuates a haecceity?

9. Objects / B. Unity of Objects / 2. Substance / a. Substance

13100	Maybe 'substance' is more of a mass-noun than a count-noun [Cover/O'Leary-Hawthorne]
	Full Idea: We could think of 'substance' on the model of a mass noun, rather than a count noun.
	From: Cover,J/O'Leary-Hawthorne,J (Substance and Individuation in Leibniz [1999], 7.3)
	A reaction: They offer this to help Leibniz out of a mess, but I think he would be appalled. The proposal seems close to 'prime matter' in Aristotle, which never quite does the job required of it. The idea is nice, though, and should be taken seriously.

9. Objects / B. Unity of Objects / 2. Substance / c. Types of substance

13068	We can ask for the nature of substance, about type of substance, and about individual substances [Cover/O'Leary-Hawthorne]
	Full Idea: In the 'blueprint' approach to substance, we confront at least three questions: What is it for a thing to be an individual substance? What is it for a thing to be the kind of substance that it is? What is it to be that very individual substance?
	From: Cover,J/O'Leary-Hawthorne,J (Substance and Individuation in Leibniz [1999], 1.1.1)
	A reaction: My working view is that the answer to the first question is that substance is essence, that the second question is overrated and parasitic on the third, and that the third is the key question, and also reduces to essence.

9. Objects / B. Unity of Objects / 2. Substance / d. Substance defined

13069	The general assumption is that substances cannot possibly be non-substances [Cover/O'Leary-Hawthorne]
	Full Idea: There is a widespread assumption, now and in the past, that substances are essentially substances: nothing is actually a substance but possibly a non-substance.
	From: Cover,J/O'Leary-Hawthorne,J (Substance and Individuation in Leibniz [1999], 1.1.2)
	A reaction: It seems to me that they clearly mean, in this context, that substances are 'necessarily' substances, not that they are 'essentially' substances. I would just say that substances are essences, and leave the necessity question open.

9. Objects / D. Essence of Objects / 7. Essence and Necessity / a. Essence as necessary properties

13072	Modern essences are sets of essential predicate-functions [Cover/O'Leary-Hawthorne]
	Full Idea: The modern view of essence is that the essence of a particular thing is given by the set of predicate-functions essential to it, and the essence of any kind is given by the set of predicate-functions essential to every possible member of that kind.
	From: Cover,J/O'Leary-Hawthorne,J (Substance and Individuation in Leibniz [1999], 1.2.2)
	A reaction: Thus the modern view has elided the meanings of 'essential' and 'necessary' when talking of properties. They are said to be 'functions' from possible worlds to individuals. The old view (and mine) demands real essences, not necessary properties.

17080	Modern essentialists express essence as functions from worlds to extensions for predicates [Cover/O'Leary-Hawthorne]
	Full Idea: The modern essentialist gives the same metaphysical treatment to every grammatical predicate - by associating a function from worlds to extensions for each.
	From: Cover,J/O'Leary-Hawthorne,J (Substance and Individuation in Leibniz [1999], 2.2)
	A reaction: I take this to mean that essentialism is the view that if some predicate attaches to an object then that predicate is essential if there is an extension of that predicate in all possible worlds. In English, essential predicates are necessary predicates.

9. Objects / E. Objects over Time / 12. Origin as Essential

13101	Necessity-of-origin won't distinguish ex nihilo creations, or things sharing an origin [Cover/O'Leary-Hawthorne]
	Full Idea: A necessity-of-origins approach cannot work to distinguish things that come into being genuinely ex nihilo, and cannot work to distinguish things sharing a single origin.
	From: Cover,J/O'Leary-Hawthorne,J (Substance and Individuation in Leibniz [1999], 7.4.1)
	A reaction: Since I am deeply suspicious of essentiality or necessity of origin (and they are not, I presume, the same thing) I like these two. Twins have always bothered me with the second case (where order of birth seems irrelevant).

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

13081	Even extreme modal realists might allow transworld identity for abstract objects [Cover/O'Leary-Hawthorne]
	Full Idea: It might be suggested that even the extreme modal realist can countenance transworld identity for abstract objects.
	From: Cover,J/O'Leary-Hawthorne,J (Substance and Individuation in Leibniz [1999], 3.2.2 n46)
	A reaction: This may sound right for uncontroversial or well-defined abstracta such as numbers and circles, but even 'or' is ambiguous, and heaven knows what the transworld identity of 'democracy' is!

14. Science / D. Explanation / 2. Types of Explanation / c. Explanations by coherence

13071	We can go beyond mere causal explanations if we believe in an 'order of being' [Cover/O'Leary-Hawthorne]
	Full Idea: The philosopher comfortable with an 'order of being' has richer resources to make sense of the 'in virtue of' relation than that provided only by causal relations between states of affairs, positing in addition other sorts of explanatory relationships.
	From: Cover,J/O'Leary-Hawthorne,J (Substance and Individuation in Leibniz [1999], 1.1.2)
	A reaction: This might best be characterised as 'ontological dependence', and could be seen as a non-causal but fundamental explanatory relationship, and not one that has to depend on a theistic world view.

15. Nature of Minds / A. Nature of Mind / 3. Mental Causation

6120	Causation depends on intrinsic properties [Mellor/Crane]
	Full Idea: The problem thoughts pose for causation is that causation depends directly only on intrinsic properties, whereas the causal powers of token thoughts depend on their contents, which are not intrinsic.
	From: DH Mellor / T Crane (There is no question of physicalism [1990], p.194)
	A reaction: This, as we find later in the paper, depends on an externalist account of thoughts. Could a relational property not be causal? Edinburgh's being wetter than London is caused by its being further north?

17. Mind and Body / D. Property Dualism / 2. Anomalous Monism

6121	There are many psychophysicals laws - about the effects of sweets, colours and soft cushions [Mellor/Crane]
	Full Idea: There are many psychophysical laws, linking sensations to non-mental features of those who have them; the industries of anaesthetics, scents, narcotics, sweeteners, coloured paints, loudspeakers and soft cushions depend on them.
	From: DH Mellor / T Crane (There is no question of physicalism [1990], p.198)
	A reaction: It may be expressing it a little strongly to call these 'laws', but they are certainly reliable regularities, and they are probably enough to undermine Davidson's claim that such laws don't exist.

17. Mind and Body / E. Mind as Physical / 7. Anti-Physicalism / a. Physicalism critique

6122	No defences of physicalism can deprive psychology of the ontological authority of other sciences [Mellor/Crane]
	Full Idea: Neither laws nor causation nor claims of supervenience (the last refuge of the physicalist) deprive psychology of the ontological authority of non-mental science.
	From: DH Mellor / T Crane (There is no question of physicalism [1990], p.203)
	A reaction: There is no need to defend the practice of psychologists (or garden designers) from the depradations of bandit physicalists. But what is the ontology of psychology if it isn't dualist or physicalist?