19 ideas
| 6396 | A sentence is held true because of a combination of meaning and belief [Davidson] |
| Full Idea: A sentence is held true because of two factors: what the holder takes the sentence to mean, and what he believes. | |
| From: Donald Davidson (Thought and Talk [1975], p.20) | |
| A reaction: A key question is whether a belief (e.g. an imagistic one, or one held by an animal) could be true, even though no sentence is involved. Linguistic philosophers tend to avoid this question, or assume the answer is 'no'. |
| 14239 | The empty set is usually derived from Separation, but it also seems to need Infinity [Oliver/Smiley] |
| Full Idea: The empty set is usually derived via Zermelo's axiom of separation. But the axiom of separation is conditional: it requires the existence of a set in order to generate others as subsets of it. The original set has to come from the axiom of infinity. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) | |
| A reaction: They charge that this leads to circularity, as Infinity depends on the empty set. |
| 14240 | The empty set is something, not nothing! [Oliver/Smiley] |
| Full Idea: Some authors need to be told loud and clear: if there is an empty set, it is something, not nothing. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) | |
| A reaction: I'm inclined to think of a null set as a pair of brackets, so maybe that puts it into a metalanguage. |
| 14241 | We don't need the empty set to express non-existence, as there are other ways to do that [Oliver/Smiley] |
| Full Idea: The empty set is said to be useful to express non-existence, but saying 'there are no Us', or ¬∃xUx are no less concise, and certainly less roundabout. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) |
| 14242 | Maybe we can treat the empty set symbol as just meaning an empty term [Oliver/Smiley] |
| Full Idea: Suppose we introduce Ω not as a term standing for a supposed empty set, but as a paradigm of an empty term, not standing for anything. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) | |
| A reaction: This proposal, which they go on to explore, seems to mean that Ω (i.e. the traditional empty set symbol) is no longer part of set theory but is part of semantics. |
| 14243 | The unit set may be needed to express intersections that leave a single member [Oliver/Smiley] |
| Full Idea: Thomason says with no unit sets we couldn't call {1,2}∩{2,3} a set - but so what? Why shouldn't the intersection be the number 2? However, we then have to distinguish three different cases of intersection (common subset or member, or disjoint). | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 2.2) |
| 14234 | If you only refer to objects one at a time, you need sets in order to refer to a plurality [Oliver/Smiley] |
| Full Idea: A 'singularist', who refers to objects one at a time, must resort to the language of sets in order to replace plural reference to members ('Henry VIII's wives') by singular reference to a set ('the set of Henry VIII's wives'). | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], Intro) | |
| A reaction: A simple and illuminating point about the motivation for plural reference. Null sets and singletons give me the creeps, so I would personally prefer to avoid set theory when dealing with ontology. |
| 14237 | We can use plural language to refer to the set theory domain, to avoid calling it a 'set' [Oliver/Smiley] |
| Full Idea: Plurals earn their keep in set theory, to answer Skolem's remark that 'in order to treat of 'sets', we must begin with 'domains' that are constituted in a certain way'. We can speak in the plural of 'the objects', not a 'domain' of objects. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], Intro) | |
| A reaction: [Skolem 1922:291 in van Heijenoort] Zermelo has said that the domain cannot be a set, because every set belongs to it. |
| 14245 | Logical truths are true no matter what exists - but predicate calculus insists that something exists [Oliver/Smiley] |
| Full Idea: Logical truths should be true no matter what exists, so true even if nothing exists. The classical predicate calculus, however, makes it logically true that something exists. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.1) |
| 14246 | If mathematics purely concerned mathematical objects, there would be no applied mathematics [Oliver/Smiley] |
| Full Idea: If mathematics was purely concerned with mathematical objects, there would be no room for applied mathematics. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.1) | |
| A reaction: Love it! Of course, they are using 'objects' in the rather Fregean sense of genuine abstract entities. I don't see why fictionalism shouldn't allow maths to be wholly 'pure', although we have invented fictions which actually have application. |
| 14247 | Sets might either represent the numbers, or be the numbers, or replace the numbers [Oliver/Smiley] |
| Full Idea: Identifying numbers with sets may mean one of three quite different things: 1) the sets represent the numbers, or ii) they are the numbers, or iii) they replace the numbers. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.2) | |
| A reaction: Option one sounds the most plausible to me. I will take numbers to be patterns embedded in nature, and sets are one way of presenting them in shorthand form, in order to bring out what is repeated. |
| 11145 | Having a belief involves the possibility of being mistaken [Davidson] |
| Full Idea: Someone cannot have a belief unless he understands the possibility of being mistaken. | |
| From: Donald Davidson (Thought and Talk [1975], p.170) | |
| A reaction: If you pretend to throw a ball for a dog, but don't release it, the dog experiences being mistaken very dramatically. |
| 6397 | The concept of belief can only derive from relationship to a speech community [Davidson] |
| Full Idea: We have the idea of belief from its role in the interpretation of language; as a private attitude it is not intelligible except in relation to public language. So a creature must be a member of a speech community to have the concept of belief. | |
| From: Donald Davidson (Thought and Talk [1975], p.22) | |
| A reaction: This shows how Wittgenstein's Private Language Argument (e.g. Idea 4152) hovers behind Davidson's philosophy. The idea is quite persuasive. A solitary creature just follows its mental states. The question of whether it believes them is a meta-thought. |
| 6392 | Thought depends on speech [Davidson] |
| Full Idea: The thesis I want to refine and then argue for is that thought depends on speech. | |
| From: Donald Davidson (Thought and Talk [1975], p.8) | |
| A reaction: This has the instant and rather implausible implication that animals don't think. He is not, of course, saying that all thought is speech, which would leave out thinking in images. You can't do much proper thought without concepts and propositions. |
| 6393 | A creature doesn't think unless it interprets another's speech [Davidson] |
| Full Idea: A creature cannot have a thought unless it is an interpreter of the speech of another. | |
| From: Donald Davidson (Thought and Talk [1975], p.9) | |
| A reaction: His use of the word 'creature' shows that he is perfectly aware of the issue of whether animals think, and he is, presumably, denying it. At first glance this sounds silly, but maybe animals don't really 'think', in our sense of the word. |
| 11144 | Concepts are only possible in a language community [Davidson] |
| Full Idea: A private attitude is not intelligible except as an adjustment to the public norms provided by language. It follows that a creature must be a member of speech community if it is to have the concept of belief. | |
| From: Donald Davidson (Thought and Talk [1975], p.170) | |
| A reaction: This obviously draws on Wittgenstein's private language argument, and strikes me as blatantly wrong, because I take higher animals to have concepts without language. Pure vision gives rise to concepts. I don't even think they are necessarily conscious. |
| 6395 | An understood sentence can be used for almost anything; it isn't language if it has only one use [Davidson] |
| Full Idea: Once a sentence is understood, an utterance of it may be used to serve almost any extra-linguistic purpose; an instrument that could be put to only one use would lack autonomy of meaning, which means it should not be counted as language. | |
| From: Donald Davidson (Thought and Talk [1975], p.17) | |
| A reaction: I find this point very appealing, in opposition to the Wittgenstein view of meaning as use. Passwords seem to me a striking case of the separation of meaning and use. I like the phrase 'autonomy of meaning'. Random sticks can form a word. |
| 6394 | The pattern of sentences held true gives sentences their meaning [Davidson] |
| Full Idea: Although most utterances are not concerned with truth, it is the pattern of sentences held true that gives sentences their meaning. | |
| From: Donald Davidson (Thought and Talk [1975], p.14) | |
| A reaction: Davidson's distinctive version of meaning holism, as opposed to Quine's rather behaviouristic version. I agree that we relate to people through the pattern of sentences they hold true, but I am unconvinced that this 'gives sentences their meaning'. |
| 7489 | Joshua said, Sun, stand thou still [Anon (Josh)] |
| Full Idea: Then Joshua spake to the Lord, and he said in the sight of Israel, Sun, stand thou still upon Gibeon; and the Sun stood still. | |
| From: Anon (Josh) (06: Book of Joshua [c.540 BCE], 10.12) | |
| A reaction: This verse became highly significant during the controversies from Copernicus to Galileo about the heliocentric universe. |