30 ideas
| 9376 | A sentence may simultaneously define a term, and also assert a fact [Boghossian] |
| Full Idea: It doesn't follow from the fact that a given sentence is being used to implicitly define one of its ingredient terms, that it is not a factual statement. 'This stick is a meter long at t' may define an ingredient terms and express something factual. | |
| From: Paul Boghossian (Analyticity Reconsidered [1996], §III) | |
| A reaction: This looks like a rather good point, but it is tied in with a difficulty about definition, which is deciding which sentences are using a term, and which ones are defining it. If I say 'this stick in Paris is a meter long', I'm not defining it. |
| 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) |
| 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. |
| 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. |
| 9375 | Conventionalism agrees with realists that logic has truth values, but not over the source [Boghossian] |
| Full Idea: Conventualism is a factualist view: it presupposes that sentences of logic have truth values. It differs from a realist view in its conception of the source of those truth values, not on their existence. I call the denial of truths Non-Factualism. | |
| From: Paul Boghossian (Analyticity Reconsidered [1996], §III) | |
| A reaction: It barely seems to count as truth is we say 'p is true because we say so'. It is a truth about an agreement, not a truth about logic. Driving on the left isn't a truth about which side of the road is best. |
| 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. |
| 11976 | Aristotelian essentialism says essences are not relative to specification [Lewis] |
| Full Idea: So-called 'Aristotelian essentialism' is the doctrine of essences not relative to specifications. | |
| From: David Lewis (Counterpart theory and Quant. Modal Logic [1968], III) | |
| A reaction: In other words, they are so-called 'real essences', understood as de re. Quine says essences are all de dicto, and relative to some specification. I vote for Aristotle. |
| 11978 | Causal necessities hold in all worlds compatible with the laws of nature [Lewis] |
| Full Idea: Just as a sentence is necessary if it holds in all worlds, so it is causally necessary if it holds in all worlds compatible with the laws of nature. | |
| From: David Lewis (Counterpart theory and Quant. Modal Logic [1968], V) | |
| A reaction: I don't believe in the so-called 'laws of nature', so I'm not buying that. Is there no distinction in Lewis's view between those sentences which must hold, and those which happen to hold universally? |
| 11979 | It doesn't take the whole of a possible Humphrey to win the election [Lewis] |
| Full Idea: Even if Humphrey is a modal continuant, it doesn't take the whole of him to do such things as winning. | |
| From: David Lewis (Counterpart theory and Quant. Modal Logic [1968], Post B) | |
| A reaction: This responds to Kripke's famous example, that people only care about what happens to themselves, and not to some 'counterpart' of themselves. |
| 16994 | Counterpart theory is bizarre, as no one cares what happens to a mere counterpart [Kripke on Lewis] |
| Full Idea: Probably Humphrey could not care less whether someone else, no matter how much resembling him, would have been victorious in another possible world. Thus Lewis's view seems even more bizarre that the usual transworld identification it replaces. | |
| From: comment on David Lewis (Counterpart theory and Quant. Modal Logic [1968]) by Saul A. Kripke - Naming and Necessity notes and addenda note 13 | |
| A reaction: I begin to see this as a devastating reply to a theory I previously found quite congenial. |
| 11974 | Counterparts are not the original thing, but resemble it more than other things do [Lewis] |
| Full Idea: Your counterparts resemble you closely in content and context in important respects. They resemble you more closely than do the other things in their worlds. But they are not really you. | |
| From: David Lewis (Counterpart theory and Quant. Modal Logic [1968], I) | |
| A reaction: It is a dilemma. If my counterpart were exactly me, I couldn't contemplate possibly losing a leg, or my sanity. But if my counterpart isn't exactly me, then I don't have much interest in its fate. Only essences can save us here. Cf. me tomorrow. |
| 11975 | If the closest resembler to you is in fact quite unlike you, then you have no counterpart [Lewis] |
| Full Idea: If whatever thing in world w6 it is that resembles you more closely than anything else in w6 is nevertheless quite unlike you; nothing in w6 resembles you at all closely. If so, you have no counterpart in w6. | |
| From: David Lewis (Counterpart theory and Quant. Modal Logic [1968], I) | |
| A reaction: This is the nub, because the whole theory rests on deciding whether two things resemble sufficiently 'closely'. But then we need a criterion of closeness, so we must start talking about which properties matter. Essences loom. |
| 11977 | Essential attributes are those shared with all the counterparts [Lewis] |
| Full Idea: An essential attribute of something is an attribute it shares with all its counterparts. | |
| From: David Lewis (Counterpart theory and Quant. Modal Logic [1968], III) | |
| A reaction: I don't like this. It ties essence entirely to identity, but I think essence precedes identity. Essence is a nexus of causal and explanatory powers which bestows an identity on each thing. But essence might be unstable, and identity with it. |
| 9369 | 'Snow is white or it isn't' is just true, not made true by stipulation [Boghossian] |
| Full Idea: Isn't it overwhelmingly obvious that 'Either snow is white or it isn't' was true before anyone stipulated a meaning for it, and that it would have been true even if no one had thought about it, or chosen it to be expressed by one of our sentences? | |
| From: Paul Boghossian (Analyticity Reconsidered [1996], §I) | |
| A reaction: Boghossian would have to believe in propositions (unexpressed truths) to hold this - which he does. I take the notion of truth to only have relevance when there are minds around. Otherwise the so-called 'truths' are just the facts. |
| 9367 | The a priori is explained as analytic to avoid a dubious faculty of intuition [Boghossian] |
| Full Idea: The central impetus behind the analytic explanation of the a priori is a desire to explain the possibility of a priori knowledge without having to postulate a special evidence-gathering faculty of intuition. | |
| From: Paul Boghossian (Analyticity Reconsidered [1996], §I) | |
| A reaction: I don't see at all why one has to postulate a 'faculty' in order to talk about intuition. I take an intuition to be an apprehension of a probable truth, combined with an inability to articulate how the conclusion was arrived at. |
| 9373 | That logic is a priori because it is analytic resulted from explaining the meaning of logical constants [Boghossian] |
| Full Idea: The analytic theory of the apriority of logic arose indirectly, as a by-product of the attempt to explain in what a grasp of the meaning of the logical constants consists. | |
| From: Paul Boghossian (Analyticity Reconsidered [1996], §III) | |
| A reaction: Preumably he is referring to Wittgenstein's anguish over the meaning of the word 'not' in his World War I notebooks. He first defined the constants by truth tables, then asserted that they were purely conventional - so logic is conventional. |
| 9380 | We can't hold a sentence true without evidence if we can't agree which sentence is definitive of it [Boghossian] |
| Full Idea: If there is no sentence I must hold true if it is to mean what it does, then there is no basis on which to argue that I am entitled to hold it true without evidence. | |
| From: Paul Boghossian (Analyticity Reconsidered [1996], §III) | |
| A reaction: He is exploring Quine's view. Truth by convention depends on agreeing which part of the usage of a term constitutes its defining sentence(s), and that may be rather tricky. Boghossian says this slides into the 'dreaded indeterminacy of meaning'. |
| 9384 | We may have strong a priori beliefs which we pragmatically drop from our best theory [Boghossian] |
| Full Idea: It is consistent with a belief's being a priori in the strong sense that we should have pragmatic reasons for dropping it from our best overall theory. | |
| From: Paul Boghossian (Analyticity Reconsidered [1996], n 6) | |
| A reaction: Does 'dropping it' from the theory mean just ignoring it, or actually denying it? C.I. Lewis is the ancestor of this view. Could it be our 'best' theory, while conflicting with beliefs that were strongly a priori? Pragmatism can embrace falsehoods. |
| 9374 | If we learn geometry by intuition, how could this faculty have misled us for so long? [Boghossian] |
| Full Idea: If we learn geometrical truths by intuition, how could this faculty have misled us for so long? | |
| From: Paul Boghossian (Analyticity Reconsidered [1996], §III) | |
| A reaction: This refers to the development of non-Euclidean geometries, though the main misleading concerns parallels, which involves infinity. Boghossian cites 'distance' as a concept the Euclideans had misunderstood. Why shouldn't intuitions be wrong? |
| 9378 | If meaning depends on conceptual role, what properties are needed to do the job? [Boghossian] |
| Full Idea: Conceptual Role Semantics must explain what properties an inference or sentence involving a logical constant must have, if that inference or sentence is to be constitutive of its meaning. | |
| From: Paul Boghossian (Analyticity Reconsidered [1996], §III) | |
| A reaction: This is my perennial request that if something is to be defined by its function (or role), we must try to explain what properties it has that make its function possible, and those properties will be the more basic explanation. |
| 9377 | 'Conceptual role semantics' says terms have meaning from sentences and/or inferences [Boghossian] |
| Full Idea: 'Conceptual role semantics' says the logical constants mean what they do by virtue of figuring in certain inferences and/or sentences involving them and not others, ..so some inferences and sentences are constitutive of an expression's meaning. | |
| From: Paul Boghossian (Analyticity Reconsidered [1996], §III) | |
| A reaction: If the meaning of the terms derives from the sentences in which they figure, that seems to be meaning-as-use. The view that it depends on the inferences seems very different, and is a more interesting but more risky claim. |
| 9372 | Could expressions have meaning, without two expressions possibly meaning the same? [Boghossian] |
| Full Idea: Could there be a fact of the matter about what each expression means, but no fact of the matter about whether they mean the same? | |
| From: Paul Boghossian (Analyticity Reconsidered [1996], §II) | |
| A reaction: He is discussing Quine's attack on synonymy, and his scepticism about meaning. Boghossian and I believe in propositions, so we have no trouble with two statements having the same meaning. Denial of propositions breeds trouble. |
| 17721 | There are no truths in virtue of meaning, but there is knowability in virtue of understanding [Boghossian, by Jenkins] |
| Full Idea: Boghossian distinguishes metaphysical analyticity (truth purely in virtue of meaning, debunked by Quine, he says) from epistemic analyticity (knowability purely in virtue of understanding - a notion in good standing). | |
| From: report of Paul Boghossian (Analyticity Reconsidered [1996]) by Carrie Jenkins - Grounding Concepts 2.4 | |
| A reaction: [compressed] This fits with Jenkins's claim that we have a priori knowledge just through understanding and relating our concepts. She, however, rejects that idea that a priori is analytic. |
| 9368 | Epistemological analyticity: grasp of meaning is justification; metaphysical: truth depends on meaning [Boghossian] |
| Full Idea: The epistemological notion of analyticity: a statement is 'true by virtue of meaning' provided that grasp of its meaning alone suffices for justified belief in its truth; the metaphysical reading is that it owes its truth to its meaning, not to facts. | |
| From: Paul Boghossian (Analyticity Reconsidered [1996], §I) | |
| A reaction: Kripke thinks it is neither, but is a purely semantic notion. How could grasp of meaning alone be a good justification if it wasn't meaning which was the sole cause of the statement's truth? I'm not convinced by his distinction. |