18 ideas
| 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. |
| 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! |
| 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. |
| 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. |
| 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, |
| 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. |
| 22167 | Our images of bodies are not produced by the bodies, but by our own minds [Augustine, by Aquinas] |
| Full Idea: Augustine says bodies don't form images in our spirit; our spirit does that itself with amazing quickness. ...So the appearances under which mind knows things aren't drawn from the things themselves. | |
| From: report of Augustine (works [c.415]) by Thomas Aquinas - Quodlibeta 8.2.1 | |
| A reaction: This is Augustine's theory of 'illumination' - that God creates experience within us. His theory was soon discarded by the early scholastics. |
| 22117 | Our minds grasp reality by direct illumination (rather than abstraction from experience) [Augustine, by Matthews] |
| Full Idea: Instead of supposing that what we know can be abstracted from sensible particulars that instantiate such knowledge, Augustine insists that our mind is so constituted as to see 'intelligible realities' directly by inner illumination. | |
| From: report of Augustine (works [c.415]) by Gareth B. Matthews - Augustine p.74 | |
| A reaction: His 'theory of illumination'. This seems to be a sort of super-rationalism. This doesn't make clear the role of sensations. Surely he doesn't thing that we just bypass them? |
| 22118 | Augustine created the modern concept of the will [Augustine, by Matthews] |
| Full Idea: The modern concept of the will is often said to originate with Augustine. | |
| From: report of Augustine (works [c.415]) by Gareth B. Matthews - Augustine p.74 | |
| A reaction: I'm beginning to think that this is the source of the trouble. How can a thing be intrinsically free? Surely freedom is always a contextual concept? |
| 4348 | Love, and do what you will [Augustine] |
| Full Idea: Love, and do what you will. | |
| From: Augustine (works [c.415]) | |
| A reaction: This sounds libertarian, but Augustine had a stern concept of what love required. It nicely captures one of the essential ideas of virtue ethics. |
| 7821 | Pagans produced three hundred definitions of the highest good [Augustine, by Grayling] |
| Full Idea: Augustine claimed that the pagan schools between them had produced nearly three hundred different definitions of the highest good. | |
| From: report of Augustine (works [c.415]) by A.C. Grayling - What is Good? Ch.5 | |
| A reaction: I would expect the right definition to be in there somewhere, but no doubt Augustine's definition made it 301. Perhaps the biggest problem of human life is that (as with the Kennedy assassination) proliferating stories obscure the true story. |
| 22119 | Augustine said (unusually) that 'ought' does not imply 'can' [Augustine, by Matthews] |
| Full Idea: Augustine insisted that 'ought' does not, in any straightforward way, imply 'can' - which distinguishes him from most modern ethicists. | |
| From: report of Augustine (works [c.415]) by Gareth B. Matthews - Augustine p.74 | |
| A reaction: Not unreasonable. I ought to help my ailing friend who lives abroad, but I haven't the time or money to do it. We can experience impossibilities as duties. Impossibilities are just excuses. Augustine is opposing the Pelagian heresy. |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |
| 22116 | Augustine identified Donatism, Pelagianism and Manicheism as the main heresies [Augustine, by Matthews] |
| Full Idea: Augustine did the most to define Christian heresy. The three most prominent were Donatism, Pelagianism (that humans are perfectible), and Manicheism (that good and evil are equally basic metaphysical realities). | |
| From: report of Augustine (works [c.415]) by Gareth B. Matthews - Augustine p.73 | |
| A reaction: Manicheans had presumably been studying Empedocles. (I suppose it's too late to identify Christianity as a heresy?). |
| 19338 | Augustine said evil does not really exist, and evil is a limitation in goodness [Augustine, by Perkins] |
| Full Idea: Augustine solution to the problem of evil was to say that, strictly speaking, evil does not exist. Human beings are not part evil and part good, but rather just a limited amount of goodness. | |
| From: report of Augustine (works [c.415]) by Franklin Perkins - Leibniz: Guide for the Perplexed 2.III | |
| A reaction: Augustine was rebelling against Manicheanism, which he espoused when young, which proposed a good and an evil force. An apathetic slob seems devoid of goodness, but is not evil. It takes extra effort to perform active evil. |