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All the ideas for 'Substance and Individuation in Leibniz', 'Model Theory' and 'Centring'

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28 ideas

2. Reason / D. Definition / 7. Contextual Definition
10476 The idea that groups of concepts could be 'implicitly defined' was abandoned [Hodges,W]
     Full Idea: Late nineteenth century mathematicians said that, although plus, minus and 0 could not be precisely defined, they could be partially 'implicitly defined' as a group. This nonsense was rejected by Frege and others, as expressed in Russell 1903.
     From: Wilfrid Hodges (Model Theory [2005], 2)
     A reaction: [compressed] This is helpful in understanding what is going on in Frege's 'Grundlagen'. I won't challenge Hodges's claim that such definitions are nonsense, but there is a case for understanding groups of concepts together.
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
10478 Since first-order languages are complete, |= and |- have the same meaning [Hodges,W]
     Full Idea: In first-order languages the completeness theorem tells us that T |= φ holds if and only if there is a proof of φ from T (T |- φ). Since the two symbols express the same relationship, theorist often just use |- (but only for first-order!).
     From: Wilfrid Hodges (Model Theory [2005], 3)
     A reaction: [actually no spaces in the symbols] If you are going to study this kind of theory of logic, the first thing you need to do is sort out these symbols, which isn't easy!
5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
10477 |= in model-theory means 'logical consequence' - it holds in all models [Hodges,W]
     Full Idea: If every structure which is a model of a set of sentences T is also a model of one of its sentences φ, then this is known as the model-theoretic consequence relation, and is written T |= φ. Not to be confused with |= meaning 'satisfies'.
     From: Wilfrid Hodges (Model Theory [2005], 3)
     A reaction: See also Idea 10474, which gives the other meaning of |=, as 'satisfies'. The symbol is ALSO used in propositional logical, to mean 'tautologically implies'! Sort your act out, logicians.
5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction
10474 |= should be read as 'is a model for' or 'satisfies' [Hodges,W]
     Full Idea: The symbol in 'I |= S' reads that if the interpretation I (about word meaning) happens to make the sentence S state something true, then I 'is a model for' S, or I 'satisfies' S.
     From: Wilfrid Hodges (Model Theory [2005], 1)
     A reaction: Unfortunately this is not the only reading of the symbol |= [no space between | and =!], so care and familiarity are needed, but this is how to read it when dealing with models. See also Idea 10477.
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
10481 Models in model theory are structures, not sets of descriptions [Hodges,W]
     Full Idea: The models in model-theory are structures, but there is also a common use of 'model' to mean a formal theory which describes and explains a phenomenon, or plans to build it.
     From: Wilfrid Hodges (Model Theory [2005], 5)
     A reaction: Hodges is not at all clear here, but the idea seems to be that model-theory offers a set of objects and rules, where the common usage offers a set of descriptions. Model-theory needs homomorphisms to connect models to things,
10473 Model theory studies formal or natural language-interpretation using set-theory [Hodges,W]
     Full Idea: Model theory is the study of the interpretation of any language, formal or natural, by means of set-theoretic structures, with Tarski's truth definition as a paradigm.
     From: Wilfrid Hodges (Model Theory [2005], Intro)
     A reaction: My attention is caught by the fact that natural languages are included. Might we say that science is model theory for English? That sounds like Quine's persistent message.
10475 A 'structure' is an interpretation specifying objects and classes of quantification [Hodges,W]
     Full Idea: A 'structure' in model theory is an interpretation which explains what objects some expressions refer to, and what classes some quantifiers range over.
     From: Wilfrid Hodges (Model Theory [2005], 1)
     A reaction: He cites as examples 'first-order structures' used in mathematical model theory, and 'Kripke structures' used in model theory for modal logic. A structure is also called a 'universe'.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
10480 First-order logic can't discriminate between one infinite cardinal and another [Hodges,W]
     Full Idea: First-order logic is hopeless for discriminating between one infinite cardinal and another.
     From: Wilfrid Hodges (Model Theory [2005], 4)
     A reaction: This seems rather significant, since mathematics largely relies on first-order logic for its metatheory. Personally I'm tempted to Ockham's Razor out all these super-infinities, but mathematicians seem to make use of them.
7. Existence / A. Nature of Existence / 3. Being / h. Dasein (being human)
8228 Being is revealed at the point between waking and sleep [Anon (Cent)]
     Full Idea: At the point of sleep when sleep has not yet come and external wakefulness vanishes, at this point being is revealed.
     From: Anon (Cent) (Centring [c.500 BCE], 50)
     A reaction: One for Heidegger. The problem with eastern philosophy is that no one expects that this revelation could be put into words, so we cannot compare our revelations of being, or pass them on to others. Which means they can't be criticised.
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 / 4. Other Minds / a. Other minds
8232 Neglect your self, and feel the consciousness of each other being [Anon (Cent)]
     Full Idea: Feel the consciousness of each person as your own consciousness. So, leaving aside concern for self, become each being.
     From: Anon (Cent) (Centring [c.500 BCE], 83)
     A reaction: Good. The western scepticism about other minds puts up a barrier to this sort of thought. The best of Zen is when it encourages the use of imagination to explore what is real. But imagination struggles to distinguish what is true.
20. Action / B. Preliminaries of Action / 2. Willed Action / a. Will to Act
8230 Just as you have the impulse to do something, stop [Anon (Cent)]
     Full Idea: Just as you have the impulse to do something, stop.
     From: Anon (Cent) (Centring [c.500 BCE], 64)
     A reaction: Nice. You might train your own will like a dog in this way. It is a counterexample to the simplistic idea that all you need is a belief and a desire, and you have got an action. But (pace Searle, Ideas 3817 + 3818) this does not prove free will.
25. Social Practice / A. Freedoms / 5. Freedom of lifestyle
8234 Bondage and liberation are relative terms, which only frighten those already terrified of the universe [Anon (Cent)]
     Full Idea: Since in truth, bondage and freedom are relative, these words are only for those terrified with the universe. The universe is a reflection of minds. As you see many suns in water from one sun, so see bondage and liberation.
     From: Anon (Cent) (Centring [c.500 BCE], 110)
     A reaction: Since most easterners have experienced permanent social repression, their philosophies all encourage them not to worry about it. No wonder marxism, proved popular, when it suggested you could actually change things.
29. Religion / C. Spiritual Disciplines / 4. Zen Buddhism
8231 Abandon attachment to body, and feel the joy of being everywhere [Anon (Cent)]
     Full Idea: Toss attachment for body aside, realising I am everywhere. One who is everywhere is joyous.
     From: Anon (Cent) (Centring [c.500 BCE], 79)
     A reaction: I really don't think it is wisdom to spend your time imagining things which are not true. I might imagine that some gorgeous film star is in love with me, but I don't recommend it. Live according to nature. We are physical beings.
8227 Feel your whole body saturated with cosmic essence [Anon (Cent)]
     Full Idea: Feel your substance, bones, flesh, blood, saturated with cosmic essence.
     From: Anon (Cent) (Centring [c.500 BCE], 23)
     A reaction: I find this unsatisfactory. Being a sad victim of materialistic western scientific culture, I don't think 'cosmic essence' exists. I could imagine myself full of the stuff, and that might feel good, but I could also imagine I had won the lottery.
8229 The serenity in blue sky beyond clouds [Anon (Cent)]
     Full Idea: Simply by looking into blue sky beyond clouds - the serenity.
     From: Anon (Cent) (Centring [c.500 BCE], 59)
     A reaction: Philip Larkin finds the same experience looking through high windows. If we articulate the experience, it seems to combine humility and detachment. The sky makes us insignificant, but the mind can project into the sky.
29. Religion / D. Religious Issues / 2. Immortality / a. Immortality
8233 Imagine fire burning up your whole body, but not you [Anon (Cent)]
     Full Idea: Focus on fire rising through your form from the toes up until the body burns to ashes, but not you.
     From: Anon (Cent) (Centring [c.500 BCE], 28)
     A reaction: Try very hard to persuade yourself that you are immortal, even if you are not. Hm. Zen Buddhism actually contains very little commitment to what is true or false. It explores interesting beliefs like children playing with toys. Very post-modern.