Combining Texts

All the ideas for 'On the Natural Faculties', 'Virtue Ethics: an Introduction' and 'Model Theory'

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17 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
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'.
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,
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.
15. Nature of Minds / C. Capacities of Minds / 1. Faculties
21799 We just use the word 'faculty' when we don't know the psychological cause [Galen]
     Full Idea: So long as we are ignorant of the true essence of the cause which is operating, we call it a 'faculty'.
     From: Galen (On the Natural Faculties [c.170], I.iv), quoted by Dominik Perler - Intro to The Faculties: a History 2
     A reaction: This is probably the view of most modern neuroscientists. I want to defend the idea that we need the concept of a faculty in philosophy, even if the psychologists and neuroscientists say it is too vague for their purposes.
22. Metaethics / A. Ethics Foundations / 1. Nature of Ethics / d. Ethical theory
5078 Kant and Mill both try to explain right and wrong, without a divine lawgiver [Taylor,R]
     Full Idea: Kant and Mill were in total agreement in trying to give content to the distinction between moral right and wrong, without recourse to any divine lawgiver.
     From: Richard Taylor (Virtue Ethics: an Introduction [2002], Ch.14)
     A reaction: A nice analysis, in tune with MacIntyre and others, who see such attempts as failures. It is hard, however, to deny the claims of rational principles, or of suffering, in our moral framework. I agree with Taylor's move back to virtue, but it ain't simple.
5067 Morality based on 'forbid', 'permit' and 'require' implies someone who does these things [Taylor,R]
     Full Idea: If morality is based on wrong (meaning 'forbidden'), right ('permitted'), and obligatory ('required'), we are led to ask 'Who is it that thus permits, forbids or requires that certain things be done or not done?'
     From: Richard Taylor (Virtue Ethics: an Introduction [2002], Ch.2)
     A reaction: Clear reinforcement for Nietzsche's attack on conventional morals, which Taylor sees as a relic of medieval religious attitudes. Taylor says Kant offered a non-religious version of the same authority. I agree. Back to the Greek pursuit of excellence!
22. Metaethics / C. The Good / 2. Happiness / a. Nature of happiness
5079 Pleasure can have a location, and be momentary, and come and go - but happiness can't [Taylor,R]
     Full Idea: Pleasures can be located in a particular part of the body, and can be momentary, and come and go, but this is not the case with happiness.
     From: Richard Taylor (Virtue Ethics: an Introduction [2002], Ch.16)
     A reaction: Probably no one ever thought that pleasure and happiness were actually identical - merely that pleasure is the only cause and source of happiness. These are good objections to that hypothesis. Pleasure simply isn't 'the good'.
22. Metaethics / C. The Good / 2. Happiness / b. Eudaimonia
5068 'Eudaimonia' means 'having a good demon', implying supreme good fortune [Taylor,R]
     Full Idea: The word 'eudaimonia' means literally 'having a good demon', which is apt, because it suggests some kind of supreme good fortune, of the sort which might be thought of as a bestowal.
     From: Richard Taylor (Virtue Ethics: an Introduction [2002], Ch.5)
     A reaction: Beware of etymology. This implies that eudaimonia is almost entirely beyond a person's control, but Aristotle doesn't think that. A combination of education and effort can build on some natural gifts to create a fully successful life.
23. Ethics / C. Virtue Theory / 1. Virtue Theory / b. Basis of virtue
5076 To Greeks it seemed obvious that the virtue of anything is the perfection of its function [Taylor,R]
     Full Idea: To the Greeks it seemed obvious that the virtue of anything is the perfection of its function.
     From: Richard Taylor (Virtue Ethics: an Introduction [2002], Ch.10)
     A reaction: A problem case might be a work of art, but one might reply that there is no obvious perfection there because there is no clear function. For artefacts and organisms the principle seems very good. But 'Is the Cosmos good?'
23. Ethics / D. Deontological Ethics / 1. Deontology
5077 The modern idea of obligation seems to have lost the idea of an obligation 'to' something [Taylor,R]
     Full Idea: In modern moral thinking, obligation is something every responsible person is supposed to have, but it is not an obligation to the state, or society, or humanity, or even to God. It is an obligation standing by itself.
     From: Richard Taylor (Virtue Ethics: an Introduction [2002], Ch.12)
     A reaction: This nicely pinpoints how some our moral attitudes are relics of religion. Taylor wants a return to virtue, but one could respond by opting for the social contract (with very clear obligations) or Kantian 'contractualism' (answering to rational beings).
23. Ethics / D. Deontological Ethics / 2. Duty
5066 If we are made in God's image, pursuit of excellence is replaced by duty to obey God [Taylor,R]
     Full Idea: Once people are declared to be images of God, just by virtue of minimal humanity, they have, therefore, no greater individual excellence to aspire to, and their purpose became one of obligation, that is, obedience to God's will.
     From: Richard Taylor (Virtue Ethics: an Introduction [2002], Ch.2)
     A reaction: An interesting and plausible historical analysis. There is a second motivation for the change, though, in Grotius's desire to develop a more legalistic morality, focusing on actions rather than character. Taylor's point is more interesting, though.
5065 The ethics of duty requires a religious framework [Taylor,R]
     Full Idea: The ethics of duty cannot be sustained independently of a religious framework.
     From: Richard Taylor (Virtue Ethics: an Introduction [2002], Ch.2)
     A reaction: This is a big challenge to Kant, echoing Nietzsche's jibe that Kant just wanted to be 'obedient'. The only options are either 'natural duties', or 'duties of reason'. Reason may have a pull (like pleasure), but a 'duty'? Difficult.