73 ideas
| 21616 | Truth and falsity apply to suppositions as well as to assertions [Williamson] |
| Full Idea: The notion of truth and falsity apply to suppositions as well as to assertions. | |
| From: Timothy Williamson (Vagueness [1994], 7.2) | |
| A reaction: This may not be obvious to those who emphasise pragmatics and ordinary language, but it is self-evident to anyone who emphasises logic. |
| 21623 | True and false are not symmetrical; false is more complex, involving negation [Williamson] |
| Full Idea: The concepts of truth and falsity are not symmetrical. The asymmetry is visible in the fundamental principles governing them, for F is essentially more complex than T, by its use of negation. | |
| From: Timothy Williamson (Vagueness [1994], 7.5) | |
| A reaction: If T and F are primitives, controlled by axioms, then they might be symmetrical in nature, but asymmetrical in use. However, if forced to choose just one primitive, I presume it would be T. |
| 9724 | Until the 1960s the only semantics was truth-tables [Enderton] |
| Full Idea: Until the 1960s standard truth-table semantics were the only ones that there were. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.10.1) | |
| A reaction: The 1960s presumably marked the advent of possible worlds. |
| 21602 | Many-valued logics don't solve vagueness; its presence at the meta-level is ignored [Williamson] |
| Full Idea: It is an illusion that many-valued logic constitutes a well-motivated and rigorously worked out theory of vagueness. ...[top] There has been a reluctance to acknowledge higher-order vagueness, or to abandon classical logic in the meta-language. | |
| From: Timothy Williamson (Vagueness [1994], 4.12) |
| 9703 | 'dom R' indicates the 'domain' of objects having a relation [Enderton] |
| Full Idea: 'dom R' indicates the 'domain' of a relation, that is, the set of all objects that are members of ordered pairs and that have that relation. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9705 | 'fld R' indicates the 'field' of all objects in the relation [Enderton] |
| Full Idea: 'fld R' indicates the 'field' of a relation, that is, the set of all objects that are members of ordered pairs on either side of the relation. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9704 | 'ran R' indicates the 'range' of objects being related to [Enderton] |
| Full Idea: 'ran R' indicates the 'range' of a relation, that is, the set of all objects that are members of ordered pairs and that are related to by the first objects. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9710 | We write F:A→B to indicate that A maps into B (the output of F on A is in B) [Enderton] |
| Full Idea: We write F : A → B to indicate that A maps into B, that is, the domain of relating things is set A, and the things related to are all in B. If we add that F = B, then A maps 'onto' B. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9707 | 'F(x)' is the unique value which F assumes for a value of x [Enderton] |
|
Full Idea:
F(x) is a 'function', which indicates the unique value which y takes in |
|
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9712 | A relation is 'symmetric' on a set if every ordered pair has the relation in both directions [Enderton] |
| Full Idea: A relation is 'symmetric' on a set if every ordered pair in the set has the relation in both directions. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9713 | A relation is 'transitive' if it can be carried over from two ordered pairs to a third [Enderton] |
| Full Idea: A relation is 'transitive' on a set if the relation can be carried over from two ordered pairs to a third. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9699 | The 'powerset' of a set is all the subsets of a given set [Enderton] |
| Full Idea: The 'powerset' of a set is all the subsets of a given set. Thus: PA = {x : x ⊆ A}. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9700 | Two sets are 'disjoint' iff their intersection is empty [Enderton] |
| Full Idea: Two sets are 'disjoint' iff their intersection is empty (i.e. they have no members in common). | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9702 | A 'domain' of a relation is the set of members of ordered pairs in the relation [Enderton] |
| Full Idea: The 'domain' of a relation is the set of all objects that are members of ordered pairs that are members of the relation. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9701 | A 'relation' is a set of ordered pairs [Enderton] |
| Full Idea: A 'relation' is a set of ordered pairs. The ordering relation on the numbers 0-3 is captured by - in fact it is - the set of ordered pairs {<0,1>,<0,2>,<0,3>,<1,2>,<1,3>,<2,3>}. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) | |
| A reaction: This can't quite be a definition of order among numbers, since it relies on the notion of a 'ordered' pair. |
| 9706 | A 'function' is a relation in which each object is related to just one other object [Enderton] |
| Full Idea: A 'function' is a relation which is single-valued. That is, for each object, there is only one object in the function set to which that object is related. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9708 | A function 'maps A into B' if the relating things are set A, and the things related to are all in B [Enderton] |
| Full Idea: A function 'maps A into B' if the domain of relating things is set A, and the things related to are all in B. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9709 | A function 'maps A onto B' if the relating things are set A, and the things related to are set B [Enderton] |
| Full Idea: A function 'maps A onto B' if the domain of relating things is set A, and the things related to are set B. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9711 | A relation is 'reflexive' on a set if every member bears the relation to itself [Enderton] |
| Full Idea: A relation is 'reflexive' on a set if every member of the set bears the relation to itself. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9714 | A relation satisfies 'trichotomy' if all pairs are either relations, or contain identical objects [Enderton] |
| Full Idea: A relation satisfies 'trichotomy' on a set if every ordered pair is related (in either direction), or the objects are identical. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9717 | A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second [Enderton] |
| Full Idea: A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9715 | An 'equivalence relation' is a reflexive, symmetric and transitive binary relation [Enderton] |
| Full Idea: An 'equivalence relation' is a binary relation which is reflexive, and symmetric, and transitive. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9716 | We 'partition' a set into distinct subsets, according to each relation on its objects [Enderton] |
| Full Idea: Equivalence classes will 'partition' a set. That is, it will divide it into distinct subsets, according to each relation on the set. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
| 9722 | Inference not from content, but from the fact that it was said, is 'conversational implicature' [Enderton] |
| Full Idea: The process is dubbed 'conversational implicature' when the inference is not from the content of what has been said, but from the fact that it has been said. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.7.3) |
| 9718 | Validity is either semantic (what preserves truth), or proof-theoretic (following procedures) [Enderton] |
| Full Idea: The point of logic is to give an account of the notion of validity,..in two standard ways: the semantic way says that a valid inference preserves truth (symbol |=), and the proof-theoretic way is defined in terms of purely formal procedures (symbol |-). | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.1.3..) | |
| A reaction: This division can be mirrored in mathematics, where it is either to do with counting or theorising about things in the physical world, or following sets of rules from axioms. Language can discuss reality, or play word-games. |
| 21611 | Formal semantics defines validity as truth preserved in every model [Williamson] |
| Full Idea: An aim of formal semantics is to define in mathematical terms a set of models such that an argument is valid if and only if it preserves truth in every model in the set, for that will provide us with a precise standard of validity. | |
| From: Timothy Williamson (Vagueness [1994], 5.3) |
| 21606 | 'Bivalence' is the meta-linguistic principle that 'A' in the object language is true or false [Williamson] |
| Full Idea: The meta-logical law of excluded middle is the meta-linguistic principle that any statement 'A' in the object language is either truth or false; it is now known as the principle of 'bivalence'. | |
| From: Timothy Williamson (Vagueness [1994], 5.2) | |
| A reaction: [He cites Henryk Mehlberg 1958] See also Idea 21605. Without this way of distinguishing bivalence from excluded middle, most discussions of them strikes me as shockingly lacking in clarity. Personally I would cut the normativity from this one. |
| 21605 | Excluded Middle is 'A or not A' in the object language [Williamson] |
| Full Idea: The logical law of excluded middle (now the standard one) is the schema 'A or not A' in the object-language. | |
| From: Timothy Williamson (Vagueness [1994], 5.2) | |
| A reaction: [He cites Henryk Mehlberg 1958] See Idea 21606. The only sensible way to keep Excluded Middle and Bivalence distinct. I would say: (meta-) only T and F are available, and (object) each proposition must have one of them. Are they both normative? |
| 21612 | Or-elimination is 'Argument by Cases'; it shows how to derive C from 'A or B' [Williamson] |
| Full Idea: Argument by Cases (or or-elimination) is the standard way of using disjunctive premises. If one can argue from A and some premises to C, and from B and some premises to C, one can argue from 'A or B' and the combined premises to C. | |
| From: Timothy Williamson (Vagueness [1994], 5.3) |
| 9721 | A logical truth or tautology is a logical consequence of the empty set [Enderton] |
| Full Idea: A is a logical truth (tautology) (|= A) iff it is a semantic consequence of the empty set of premises (φ |= A), that is, every interpretation makes A true. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.3.4) | |
| A reaction: So the final column of every line of the truth table will be T. |
| 9994 | A truth assignment to the components of a wff 'satisfy' it if the wff is then True [Enderton] |
| Full Idea: A truth assignment 'satisfies' a formula, or set of formulae, if it evaluates as True when all of its components have been assigned truth values. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.2) | |
| A reaction: [very roughly what Enderton says!] The concept becomes most significant when a large set of wff's is pronounced 'satisfied' after a truth assignment leads to them all being true. |
| 9719 | A proof theory is 'sound' if its valid inferences entail semantic validity [Enderton] |
| Full Idea: If every proof-theoretically valid inference is semantically valid (so that |- entails |=), the proof theory is said to be 'sound'. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.1.7) |
| 9720 | A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity [Enderton] |
| Full Idea: If every semantically valid inference is proof-theoretically valid (so that |= entails |-), the proof-theory is said to be 'complete'. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.1.7) |
| 9995 | Proof in finite subsets is sufficient for proof in an infinite set [Enderton] |
| Full Idea: If a wff is tautologically implied by a set of wff's, it is implied by a finite subset of them; and if every finite subset is satisfiable, then so is the whole set of wff's. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 2.5) | |
| A reaction: [Enderton's account is more symbolic] He adds that this also applies to models. It is a 'theorem' because it can be proved. It is a major theorem in logic, because it brings the infinite under control, and who doesn't want that? |
| 9996 | Expressions are 'decidable' if inclusion in them (or not) can be proved [Enderton] |
| Full Idea: A set of expressions is 'decidable' iff there exists an effective procedure (qv) that, given some expression, will decide whether or not the expression is included in the set (i.e. doesn't contradict it). | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.7) | |
| A reaction: This is obviously a highly desirable feature for a really reliable system of expressions to possess. All finite sets are decidable, but some infinite sets are not. |
| 9997 | For a reasonable language, the set of valid wff's can always be enumerated [Enderton] |
| Full Idea: The Enumerability Theorem says that for a reasonable language, the set of valid wff's can be effectively enumerated. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 2.5) | |
| A reaction: There are criteria for what makes a 'reasonable' language (probably specified to ensure enumerability!). Predicates and functions must be decidable, and the language must be finite. |
| 21599 | A sorites stops when it collides with an opposite sorites [Williamson] |
| Full Idea: A sorites paradox is stopped when it collides with a sorites paradox going in the opposite direction. That account will not strike a logician as solving the sorites paradox. | |
| From: Timothy Williamson (Vagueness [1994], 3.3) |
| 21589 | When bivalence is rejected because of vagueness, we lose classical logic [Williamson] |
| Full Idea: The principle of bivalence (that every statement is either true or false) has been rejected for vague languages. To reject bivalence is to reject classical logic or semantics. | |
| From: Timothy Williamson (Vagueness [1994], Intro) | |
| A reaction: His example is specifying a moment when Rembrandt became 'old'. This is the number one reason why the problem of vagueness is seen as important. Is the rejection of classical logic a loss of our grip on the world? |
| 21596 | Vagueness undermines the stable references needed by logic [Williamson] |
| Full Idea: Logic requires expressions to have the same referents wherever they occur; vague natural languages violate this contraint. | |
| From: Timothy Williamson (Vagueness [1994], 2.2) | |
| A reaction: This doesn't mean that logic has to win. Maybe it is important for philosophers who see logic as central to be always aware of vagueness as the gulf between their precision and the mess of reality. Precision is worth trying for, though. |
| 21601 | A vague term can refer to very precise elements [Williamson] |
| Full Idea: Both 30° and 60° are clearly acute angles. 'Acute' is precise in all relevant respects. Nevertheless, 30° is acuter than 60°. | |
| From: Timothy Williamson (Vagueness [1994], 4.11) | |
| A reaction: A very nice example of something which is vague, despite involving precise ingredients. But then 'bald' is vague, while 'this is a hair on his head' is fairly precise. |
| 21629 | Equally fuzzy objects can be identical, so fuzziness doesn't entail vagueness [Williamson] |
| Full Idea: Fuzzy boundaries do not in any way require vague identity. Objects are identical only if their boundaries have exactly the same fuzziness. | |
| From: Timothy Williamson (Vagueness [1994], 9.2) | |
| A reaction: This all rests on the Fregean idea that determinate existence requires the ability to participate in an identity statement. |
| 21591 | Vagueness is epistemic. Statements are true or false, but we often don't know which [Williamson] |
| Full Idea: My thesis is that vagueness is an epistemic phenomenon. In cases of unclarity, statements remain true or false, but speakers of the language have no way of knowing which. Higher-order vagueness consists in ignorance about ignorance. | |
| From: Timothy Williamson (Vagueness [1994], Intro) | |
| A reaction: He has plumped for the intuitively least plausible theory. It means that a hair dropping out of someone's head triggers a situation where they are 'bald', but none of us know when that was. And Rembrandt became 'old' in an instant. |
| 21619 | If a heap has a real boundary, omniscient speakers would agree where it is [Williamson] |
| Full Idea: If, in judging a heap as grains are removed, omniscient speakers all stop at the same point, it must does mark some sort of previously hidden boundary. ...If there is no hidden boundary, then different omniscient speakers would stop at different points. | |
| From: Timothy Williamson (Vagueness [1994], 7.3) | |
| A reaction: A very nice thought experiment, which obviously won't settle anything, but brings out nicely the view the vagueness is a sort of ignorance. God is never vague in the application of terms (though God might withhold the application if there is no boundary). |
| 21620 | The epistemic view says that the essence of vagueness is ignorance [Williamson] |
| Full Idea: The epistemic view is that ignorance is the real essence of the phenomenon ostensively identified as vagueness. ...[203] According to the epistemic view, I am either thin or not thin, ...and we have no idea how to find out out which. | |
| From: Timothy Williamson (Vagueness [1994], 7.4) | |
| A reaction: Presumably this implies that there is often a real border (of which we may be ignorant), but it doesn't seem to rule out cases where there just is no border. Where does the east Atlantic meet the west Atlantic? |
| 21622 | If there is a true borderline of which we are ignorant, this drives a wedge between meaning and use [Williamson] |
| Full Idea: A common complaint against the epistemic view is that to postulate a matter of fact in borderline cases is to suppose, incoherently, that the meanings of our words draw a line where our use of them does not. | |
| From: Timothy Williamson (Vagueness [1994], 7.5) | |
| A reaction: This doesn't necessarily seem to require the view that the meaning of words is their usage. Just that if there is one consensus on usage, it seems unlikely that there is a different underlying reality about the true meaning. Externalist meanings? |
| 9120 | Vagueness in a concept is its indiscriminability from other possible concepts [Williamson] |
| Full Idea: Vagueness in a concept is its indiscriminability from other possible concepts; this can be reconciled with our knowledge of vague terms. | |
| From: Timothy Williamson (Vagueness [1994], 8.1) | |
| A reaction: Sorensen objects that this makes vagueness too relative to members of a speech community. He prefers 'absolute borderline cases'. If you like the epistemic view, then Williamson seems more plausible. My 'vague' might differ from yours. |
| 21625 | The vagueness of 'heap' can remain even when the context is fixed [Williamson] |
| Full Idea: Vagueness remains even when the context is fixed. In principle, a vague word might exhibit no context dependence whatsoever. ...For example, a dispute over whether someone has left a 'heap' of sand on the floor. | |
| From: Timothy Williamson (Vagueness [1994], 7.7) | |
| A reaction: A fairly devastating rebuttal of what seems to be David Lewis's view. He talks of something being 'smooth' depending on context. |
| 21614 | The 'nihilist' view of vagueness says that 'heap' is not a legitimate concept [Williamson] |
| Full Idea: The 'nihilist' view is that no genuine distinction can be vaguely drawn; since vague expressions are not properly meaningful, there is nothing for sorites reasoning to betray; they are empty. | |
| From: Timothy Williamson (Vagueness [1994], 6.1) | |
| A reaction: He cites Frege as holding this view. The thought is that 'heap' is not a legitimate concept, so fussing over what qualifies as one is pointless. This seems to be a semantic view of vagueness, of which the main rival is the contextual view. |
| 21617 | We can say propositions are bivalent, but vague utterances don't express a proposition [Williamson] |
| Full Idea: A philosopher might endorse bivalence for propositions, while treating vagueness as the failure of an utterance to express a unique proposition. | |
| From: Timothy Williamson (Vagueness [1994], 7.2) | |
| A reaction: This idea jumps at out me as an extremely promising approach to vagueness, because I am a fan of propositions (and have written a paper on them). The whole point of propositions is that they are not ambiguous (and probably not vague). |
| 21618 | If the vague 'TW is thin' says nothing, what does 'TW is thin if his perfect twin is thin' say? [Williamson] |
| Full Idea: If vague utterances in borderline cases fail to say anything, then if 'TW is thin' is vague, and TW has a twin of identical dimensions, it still seems that 'If TW is thin then his twin is thin' must be true, and so it must have said something. | |
| From: Timothy Williamson (Vagueness [1994], 7.2 (d)) | |
| A reaction: This an objection to the Fregean 'nihilistic' view of Idea 21614. I am inclined to a solution based on the proposition expressed, rather than the sentence. The first question is whether you are willing to assert 'TW is thin'. |
| 21590 | Asking when someone is 'clearly' old is higher-order vagueness [Williamson] |
| Full Idea: Difficulties of vagueness are presented by the question 'When did Rembrandt become clearly old?', and the iterating question 'When did he become clearly clearly old?'. This is the phenomenon of higher-order vagueness. The language of vagueness is vague. | |
| From: Timothy Williamson (Vagueness [1994], Intro) | |
| A reaction: [compressed] I presume the bottom level is a question about Rembrandt, the second level is about this use of the word 'old', and the third level is about this particular application of the word 'clearly'. Meta-languages. |
| 21592 | Supervaluation keeps classical logic, but changes the truth in classical semantics [Williamson] |
| Full Idea: Supervaluationism preserves almost all of classical logic, at the expense of classical semantics, but giving a non-standard account of truth. I argue that its treatment of higher-order vagueness undermines the non-standard account of truth. | |
| From: Timothy Williamson (Vagueness [1994], Intro) |
| 21603 | You can't give a precise description of a language which is intrinsically vague [Williamson] |
| Full Idea: If a vague language is made precise, its expressions change in meaning, so an accurate semantic description of the precise language is inaccurate as a description of the vague one. | |
| From: Timothy Williamson (Vagueness [1994], 5.1) | |
| A reaction: Kind of obvious, really, but it clarifies the nature of any project (starting with Leibniz) to produce a wholly precise language. That is usually seen as a specialist language for science. |
| 21604 | Supervaluation assigns truth when all the facts are respected [Williamson] |
| Full Idea: 'Admissible' interpretations respect all the theoretical and ostensive connections. ...'Supervaluation' is the assignment of truth to the statements true on all admissible valuations, falsity to the false one, and neither to the rest. | |
| From: Timothy Williamson (Vagueness [1994], 5.2) | |
| A reaction: So 'he is bald' is true if when faced with all observations and definitions it is acceptable. Prima facie, that doesn't sound like a solution to the problem. Supervaluation started in philosophy of science. [p.156 'Admissible seems vague'] |
| 21607 | Supervaluation has excluded middle but not bivalence; 'A or not-A' is true, even when A is undecided [Williamson] |
| Full Idea: The supervaluationist denies bivalence but accepts excluded middle. The statement 'A or not-A' is true on each admissible interpretation, and therefore true, even if 'A' (and hence 'not-A') are true and some and false on others, so neither T nor F. | |
| From: Timothy Williamson (Vagueness [1994], 5.2) | |
| A reaction: See Ideas 21605 and 21606 for the distinction being used here. Denying bivalence allows 'A' to be neither true nor false. It seems common sense that 'he is either bald or not-bald' is true, without being sure about the disjuncts. |
| 21608 | Truth-functionality for compound statements fails in supervaluation [Williamson] |
| Full Idea: A striking fearure of supervaluations is the failure of truth-functionality for compound statements. | |
| From: Timothy Williamson (Vagueness [1994], 5.3) | |
| A reaction: Supervaluations has the initial appearance of enhancing classical logic, but turns out to somewhat undermine it. Hence Williamson's lack of sympathy. But see Idea 21610. |
| 21609 | Supervaluationism defines 'supertruth', but neglects it when defining 'valid' [Williamson] |
| Full Idea: Supervaluationists identify truth with 'supertruth'; since validity is necessary preservation of truth, they should identify it with necessary preservation of supertruth. But it plays no role in their definition of 'local' validity. | |
| From: Timothy Williamson (Vagueness [1994], 5.3) | |
| A reaction: [See text for 'local'] Generally Williamson's main concern with attempts to sort out vagueness is that higher-order and meta-language issues are neglected. |
| 21610 | Supervaluation adds a 'definitely' operator to classical logic [Williamson] |
| Full Idea: Supervaluation seems to inherit the power of classical logic, ...but also enables it to be extended. It makes room for a new operator 'definitely' to express supertruth in the object-language. | |
| From: Timothy Williamson (Vagueness [1994], 5.3) | |
| A reaction: Once you mention higher-order vagueness you can see a regress looming over the horizon. 'He is definitely definitely definitely bald'. [p.164 he says 'definitely' has no analysis, and is an uninteresting primitive] |
| 21613 | Supervaluationism cannot eliminate higher-order vagueness [Williamson] |
| Full Idea: Supervaluationism cannot eliminate higher-order vagueness. It must conduct its business in a vague meta-language. ...[162] All truth is at least disquotational, and supertruth is not. | |
| From: Timothy Williamson (Vagueness [1994], 5.6) | |
| A reaction: This is Williamson's final verdict on the supervaluation strategy for vagueness. Intuitively, it looks as if merely narrowing down the vagueness (by some sort of consensus) is no solution to the problem of vagueness. |
| 21633 | Nominalists suspect that properties etc are our projections, and could have been different [Williamson] |
| Full Idea: The nominalist suspects that properties, relations and states of affairs are mere projections onto the world of our forms of speech. One source of the suspicion is a sense that we could just as well have classified things differently. | |
| From: Timothy Williamson (Vagueness [1994], 9.3) | |
| A reaction: I know it is very wicked to say so, but I'm afraid I have some sympathy with this view. But I like the primary/secondary distinction, so there is more 'projection' in the latter case. Classification is not random; it is a response to reality. |
| 21630 | If fuzzy edges are fine, then why not fuzzy temporal, modal or mereological boundaries? [Williamson] |
| Full Idea: If objects can have fuzzy spatial boundaries, surely they can have fuzzy temporal, modal or mereological boundaries too. | |
| From: Timothy Williamson (Vagueness [1994], 9.2) | |
| A reaction: Fair point. I think there is a distinction between parts of the thing, such as its edges, being fuzzy, and the whole thing being fuzzy, in the temporal case. |
| 21632 | A river is not just event; it needs actual and counterfactual boundaries [Williamson] |
| Full Idea: A river is not just an event. One would need to specify counterfactual as well as actual boundaries. | |
| From: Timothy Williamson (Vagueness [1994], 9.3) | |
| A reaction: In other words the same river can change its course a bit, but it can't head off in the opposite direction. |
| 9723 | Sentences with 'if' are only conditionals if they can read as A-implies-B [Enderton] |
| Full Idea: Not all sentences using 'if' are conditionals. Consider 'if you want a banana, there is one in the kitchen'. The rough test is that a conditional can be rewritten as 'that A implies that B'. | |
| From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.6.4) |
| 21621 | We can't infer metaphysical necessities to be a priori knowable - or indeed knowable in any way [Williamson] |
| Full Idea: The inference from metaphysical necessity to a priori knowlability is, as Kripke has emphasized, fallacious. Indeed, metaphysical necessities cannot be assumed knowable in any way at all. | |
| From: Timothy Williamson (Vagueness [1994], 7.4) | |
| A reaction: The second sentence sounds like common sense. He cites Goldbach's Conjecture. A nice case of the procedural rule of keeping your ontology firmly separated from your epistemology. How is it? is not How do we know it? |
| 21627 | We have inexact knowledge when we include margins of error [Williamson] |
| Full Idea: Inexact knowledge is a widespread and easily recognised cognitive phenomenon, whose underlying nature turns out to be characterised by the holding of margin of error principles. | |
| From: Timothy Williamson (Vagueness [1994], 8.3) | |
| A reaction: Williamson is invoking this as a tool in developing his epistemic view of vagueness. It obviously invites the question of how it can be knowledge if error is a possibility. A very large margin of error would obviously invalidate it. |
| 21626 | Knowing you know (KK) is usually denied if the knowledge concept is missing, or not considered [Williamson] |
| Full Idea: The failure of the KK principle is not news. The standard counterexamples involve knowing subjects who lack the concept of knowledge, or have not reflected on their knowledge, and therefore do not know that they know. | |
| From: Timothy Williamson (Vagueness [1994], 8.2) | |
| A reaction: There is also the timid but knowledgeable pupil, who can't believe they know so much. The simplest case would be if we accept that animals know lots of things, but are largely devoid of any metathinking. |
| 21631 | To know, believe, hope or fear, one must grasp the thought, but not when you fail to do them [Williamson] |
| Full Idea: To know, believe, hope, or fear that A, one must grasp the thought that A. In contrast, to fail to know, believe, hope or fear that A, one need not grasp the thought that A. | |
| From: Timothy Williamson (Vagueness [1994], 9.3 c) | |
| A reaction: A simple point, which at least shows that propositional attitudes are a two-stage operation. |
| 21600 | 'Blue' is not a family resemblance, because all the blues resemble in some respect [Williamson] |
| Full Idea: 'Blue' is vague by some standards, for it has borderline cases, but that does not make it a family resemblance term, for all the shades of blue resemble each other in some respect. | |
| From: Timothy Williamson (Vagueness [1994], 3.3) | |
| A reaction: Presumably the point of family resemblance is that fringe members as still linked to the family, despite having lost the main features. A bit of essentialism seems needed here. |
| 21615 | References to the 'greatest prime number' have no reference, but are meaningful [Williamson] |
| Full Idea: The predicate 'is a prime number greater than all other prime numbers' is necessarily not true of anything, but it is not semantically defective, for it occurs in sentences that constitute a sound proof that there is no such number. | |
| From: Timothy Williamson (Vagueness [1994], 6.2) | |
| A reaction: One might reply that the description can be legitimately mentioned, but not legitimately used. |
| 18038 | The 't' and 'f' of formal semantics has no philosophical interest, and may not refer to true and false [Williamson] |
| Full Idea: In a formal semantics we can label two properties 't' and 'f' and suppose that some sentences have neither (or both). Such a manoeuvre shows nothing of philosophical interest. No connection has been made between 't' and 'f' and truth and falsity. | |
| From: Timothy Williamson (Vagueness [1994], 7.2) | |
| A reaction: This is right, and means there is a huge gulf between 'formal' semantics (which could be implemented on a computer), and seriously interesting semantics about how language refers to and describes the world. |
| 21624 | It is known that there is a cognitive loss in identifying propositions with possible worlds [Williamson] |
| Full Idea: It is well known that when a proposition is identified with the set of possible worlds at which it is true, a region in the space of possible worlds, cognitively significant distinctions are lost. | |
| From: Timothy Williamson (Vagueness [1994], 7.6) | |
| A reaction: Alas, he doesn't specify which distinctions get lost, so this is just a pointer. It would seem likely that two propositions could have identical sets of possible worlds, while not actually saying the same thing. Equilateral/equiangular. |
| 22405 | Negative utilitarianism implies that the world should be destroyed, to avoid future misery [Smart] |
| Full Idea: The doctrine of negative utilitarianism (that we should concern ourselves with the minimisation of suffering, rather than the maximisation of happiness) ...means we should support a tyrant who explodes the world, to prevent infinite future misery. | |
| From: J.J.C. Smart (Outline of a System of Utilitarianism [1973], 5) | |
| A reaction: That only seems to imply that the negative utilitarian rule needs supplementary rules. We are too fond of looking for one single moral rule that guides everything. |
| 22404 | Any group interested in ethics must surely have a sentiment of generalised benevolence [Smart] |
| Full Idea: A utilitarian can appeal to the sentiment of generalised benevolence, which is surely present in any group with whom it is profitable to discuss ethical questions. | |
| From: J.J.C. Smart (Outline of a System of Utilitarianism [1973], I) | |
| A reaction: But ethics is not intended only for those who are interested in ethics. If this is the basics of ethics, then we must leave the mafia to pursue its sordid activities without criticism. Their lack of sympathy seems to be their good fortune. |