25 ideas
| 9766 | Study vagueness first by its logic, then by its truth-conditions, and then its metaphysics [Fine,K] |
| Full Idea: My investigation of vagueness began with the question 'What is the correct logic of vagueness?', which led to the further question 'What are the correct truth-conditions for a vague language?', which led to questions of meaning and existence. | |
| From: Kit Fine (Vagueness, Truth and Logic [1975], Intro) | |
| A reaction: This is the most perfect embodiment of the strategy of analytical philosophy which I have ever read. It is the strategy invented by Frege in the 'Grundlagen'. Is this still the way to go, or has this pathway slowly sunk into the swamp? |
| 13886 | Later Frege held that definitions must fix a function's value for every possible argument [Frege, by Wright,C] |
| Full Idea: Frege later became fastidious about definitions, and demanded that they must provide for every possible case, and that no function is properly determined unless its value is fixed for every conceivable object as argument. | |
| From: report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Crispin Wright - Frege's Concept of Numbers as Objects 3.xiv | |
| A reaction: Presumably definitions come in degrees of completeness, but it seems harsh to describe a desire for the perfect definition as 'fastidious', especially if we are talking about mathematics, rather than defining 'happiness'. |
| 9845 | We can't define a word by defining an expression containing it, as the remaining parts are a problem [Frege] |
| Full Idea: Given the reference (bedeutung) of an expression and a part of it, obviously the reference of the remaining part is not always determined. So we may not define a symbol or word by defining an expression in which it occurs, whose remaining parts are known | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §66) | |
| A reaction: Dummett cites this as Frege's rejection of contextual definitions, which he had employed in the Grundlagen. I take it not so much that they are wrong, as that Frege decided to set the bar a bit higher. |
| 10019 | Only what is logically complex can be defined; what is simple must be pointed to [Frege] |
| Full Idea: Only what is logically complex can be defined; what is simple can only be pointed to. | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §180), quoted by Harold Hodes - Logicism and Ontological Commits. of Arithmetic p.137 | |
| A reaction: Frege presumably has in mind his treasured abstract objects, such as cardinal numbers. It is hard to see how you could 'point to' anything in the phenomenal world that had atomic simplicity. Hodes calls this a 'desperate Kantian move'. |
| 9775 | Excluded Middle, and classical logic, may fail for vague predicates [Fine,K] |
| Full Idea: Maybe classical logic fails for vagueness in Excluded Middle. If 'H bald ∨ ¬(H bald)' is true, then one disjunct is true. But if the second is true the first is false, and the sentence is either true or false, contrary to the borderline assumption. | |
| From: Kit Fine (Vagueness, Truth and Logic [1975], 4) | |
| A reaction: Fine goes on to argue against the implication that we need a special logic for vague predicates. |
| 9771 | Logic holding between indefinite sentences is the core of all language [Fine,K] |
| Full Idea: If language is like a tree, then penumbral connection (logic holding among indefinite sentences) is the seed from which the tree grows, for it provides an initial repository of truths that are to be retained throughout all growth. | |
| From: Kit Fine (Vagueness, Truth and Logic [1975], 2) | |
| A reaction: A nice incidental insight arising from his investigation of vagueness. People accept one another's reasons even when they are confused, or hopeless at expressing themselves. Nice. |
| 9886 | Cardinals say how many, and reals give measurements compared to a unit quantity [Frege] |
| Full Idea: The cardinals and the reals are completely disjoint domains. The cardinal numbers answer the question 'How many objects of a given kind are there?', but the real numbers are for measurement, saying how large a quantity is compared to a unit quantity. | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §157), quoted by Michael Dummett - Frege philosophy of mathematics Ch.19 | |
| A reaction: We might say that cardinals are digital and reals are analogue. Frege is unusual in totally separating them. They map onto one another, after all. Cardinals look like special cases of reals. Reals are dreams about the gaps between cardinals. |
| 9889 | Real numbers are ratios of quantities [Frege, by Dummett] |
| Full Idea: Frege fixed on construing real numbers as ratios of quantities (in agreement with Newton). | |
| From: report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Michael Dummett - Frege philosophy of mathematics Ch.20 | |
| A reaction: If 3/4 is the same real number as 6/8, which is the correct ratio? Why doesn't the square root of 9/16 also express it? Why should irrationals be so utterly different from rationals? In what sense are they both 'numbers'? |
| 10553 | A number is a class of classes of the same cardinality [Frege, by Dummett] |
| Full Idea: For Frege, in 'Grundgesetze', a number is a class of classes of the same cardinality. | |
| From: report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Michael Dummett - Frege Philosophy of Language (2nd ed) Ch.14 |
| 10020 | Frege's biggest error is in not accounting for the senses of number terms [Hodes on Frege] |
| Full Idea: The inconsistency of Grundgesetze was only a minor flaw. Its fundamental flaw was its inability to account for the way in which the senses of number terms are determined. It leaves the reference-magnetic nature of the standard numberer a mystery. | |
| From: comment on Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Harold Hodes - Logicism and Ontological Commits. of Arithmetic p.139 | |
| A reaction: A point also made by Hofweber. As a logician, Frege was only concerned with the inferential role of number terms, and he felt he had captured their logical form, but it is when you come to look at numbers in natural language that he seem in trouble. |
| 9887 | Formalism misunderstands applications, metatheory, and infinity [Frege, by Dummett] |
| Full Idea: Frege's three main objections to radical formalism are that it cannot account for the application of mathematics, that it confuses a formal theory with its metatheory, and it cannot explain an infinite sequence. | |
| From: report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §86-137) by Michael Dummett - Frege philosophy of mathematics | |
| A reaction: The application is because we don't design maths randomly, but to be useful. The third objection might be dealt with by potential infinities (from formal rules). The second objection sounds promising. |
| 8751 | Only applicability raises arithmetic from a game to a science [Frege] |
| Full Idea: It is applicability alone which elevates arithmetic from a game to the rank of a science. | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §91), quoted by Stewart Shapiro - Thinking About Mathematics 6.1.2 | |
| A reaction: This is the basic objection to Formalism. It invites the question of why it is applicable, which platonists like Frege don't seem to answer (though Plato himself has reality modelled on the Forms). This is why I like structuralism. |
| 9768 | Vagueness is semantic, a deficiency of meaning [Fine,K] |
| Full Idea: I take vagueness to be a semantic feature, a deficiency of meaning. It is to be distinguished from generality, undecidability, and ambiguity. | |
| From: Kit Fine (Vagueness, Truth and Logic [1975], Intro) | |
| A reaction: Sounds good. If we cut nature at the joints with our language, then nature is going to be too subtle and vast for our finite and gerrymandered language, and so it will break down in tricky situations. But maybe epistemology precedes semantics? |
| 9776 | A thing might be vaguely vague, giving us higher-order vagueness [Fine,K] |
| Full Idea: There is a possibility of 'higher-order vagueness'. The vague may be vague, or vaguely vague, and so on. If J has few hairs on his head than H, then he may be a borderline case of a borderline case. | |
| From: Kit Fine (Vagueness, Truth and Logic [1975], 5) | |
| A reaction: Such slim grey areas can also be characterised as those where you think he is definitely bald, but I am not so sure. |
| 9767 | A vague sentence is only true for all ways of making it completely precise [Fine,K] |
| Full Idea: A vague sentence is (roughly stated) true if and only if it is true for all ways of making it completely precise (the 'super-truth theory'). | |
| From: Kit Fine (Vagueness, Truth and Logic [1975], Intro) | |
| A reaction: Intuitively this sounds quite promising. Personally I think we should focus on the 'proposition' rather than the 'sentence' (where fifteen sentences might be needed before we can agree on the one proposition). |
| 9770 | Logical connectives cease to be truth-functional if vagueness is treated with three values [Fine,K] |
| Full Idea: With a three-value approach, if P is 'blob is pink' and R is 'blob is red', then P&P is indefinite, but P&R is false, and P∨P is indefinite, but P∨R is true. This means the connectives & and ∨ are not truth-functional. | |
| From: Kit Fine (Vagueness, Truth and Logic [1975], 1) | |
| A reaction: The point is that there could then be no logic in any way classical for vague sentences and three truth values. A powerful point. |
| 9772 | Meaning is both actual (determining instances) and potential (possibility of greater precision) [Fine,K] |
| Full Idea: The meaning of an expression is the product of both its actual meaning (what helps determine its instances and counter-instances), and its potential meaning (the possibilities for making it more precise). | |
| From: Kit Fine (Vagueness, Truth and Logic [1975], 2) | |
| A reaction: A modal approach to meaning is gloriously original. Being quite a fan of real modalities (the possibilities latent in actuality), I find this intuitively appealing. |
| 9773 | With the super-truth approach, the classical connectives continue to work [Fine,K] |
| Full Idea: With the super-truth approach, if P is 'blob is pink' and R is 'blob is red', then P&R is false, and P∨R is true, since one of P and R is true and one is false in any complete and admissible specification. It encompasses all 'penumbral truths'. | |
| From: Kit Fine (Vagueness, Truth and Logic [1975], 3) | |
| A reaction: [See Idea 9767 for the super-truth approach, and Idea 9770 for a contrasting view] The approach, which seems quite appealing, is that we will in no circumstances give up basic classical logic, but we will make maximum concessions to vagueness. |
| 9774 | Borderline cases must be under our control, as capable of greater precision [Fine,K] |
| Full Idea: Any borderline case must be under our control, in the sense that it can be settled by making the predicates more precise. | |
| From: Kit Fine (Vagueness, Truth and Logic [1975], 3) | |
| A reaction: Sounds good. Consider an abstract concept like the equator. It is precise on a map of the world, but vague when you are in the middle of the tropics. But we can always form a committee to draw a (widish) line on the ground delineating it. |
| 9769 | Vagueness can be in predicates, names or quantifiers [Fine,K] |
| Full Idea: There are three possible sources of vagueness: the predicates, the names, and the quantifiers. | |
| From: Kit Fine (Vagueness, Truth and Logic [1975], 1) | |
| A reaction: Presumably a vagueness about the domain of discussion would be a vagueness in the quantifier. This is a helpful preliminary division, in the semantic approach to vagueness. |
| 9891 | The first demand of logic is of a sharp boundary [Frege] |
| Full Idea: The first demand of logic is of a sharp boundary. | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §160), quoted by Michael Dummett - Frege philosophy of mathematics Ch.22 | |
| A reaction: Nothing I have read about vagueness has made me doubt Frege's view of this, although precisification might allow you to do logic with vague concepts without having to finally settle where the actual boundaries are. |
| 9890 | The modern account of real numbers detaches a ratio from its geometrical origins [Frege] |
| Full Idea: From geometry we retain the interpretation of a real number as a ratio of quantities or measurement-number; but in more recent times we detach it from geometrical quantities, and from all particular types of quantity. | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §159), quoted by Michael Dummett - Frege philosophy of mathematics | |
| A reaction: Dummett glosses the 'recent' version as by Cantor and Dedekind in 1872. This use of 'detach' seems to me startlingly like the sort of psychological abstractionism which Frege was so desperate to avoid. |
| 11846 | If we abstract the difference between two houses, they don't become the same house [Frege] |
| Full Idea: If abstracting from the difference between my house and my neighbour's, I were to regard both houses as mine, the defect of the abstraction would soon be made clear. It may, though, be possible to obtain a concept by means of abstraction... | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §99) | |
| A reaction: Note the important concession at the end, which shows Frege could never deny the abstraction process, despite all the modern protests by Geach and Dummett that he totally rejected it. |
| 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. |