33 ideas
| 22270 | Frege changed philosophy by extending logic's ability to check the grounds of thinking [Potter on Frege] |
| Full Idea: Frege's 1879 logic transformed philosophy because it greatly expanded logic's reach - what thought can achieve unaided - and hence compelled a re-examination of everything previously said about the grounds of thought when logic gives out. | |
| From: comment on Gottlob Frege (Begriffsschrift [1879]) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 Intro | |
| A reaction: I loved the gloss on logic as 'what thought can achieve unaided'. I largely see logic in terms of what is mechanically computable. |
| 8939 | We should not describe human laws of thought, but how to correctly track truth [Frege, by Fisher] |
| Full Idea: Frege disagree that logic should merely describe the laws of thought - how people actually did reason. Logic is essentially normative, not descriptive. We want the one logic which successfully tracks the truth. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Jennifer Fisher - On the Philosophy of Logic 1.III | |
| A reaction: This explains Frege's sustained attack on psychologism, and it also explains we he ended up as a platonist about logic - because he wanted its laws to be valid independently of human thinking. A step too far, perhaps. Brains are truth machines. |
| 9143 | Implicit definitions must be satisfiable, creative definitions introduce things, contextual definitions build on things [Fine,K, by Cook/Ebert] |
| Full Idea: Fine distinguishes 'implicit definitions', where we must know it is satisfiable before it is deployed, 'creative definitions', where objects are introduced in virtue of the definition, ..and 'contextual definitions', based on established vocabulary. | |
| From: report of Kit Fine (The Limits of Abstraction [2002], 060) by R Cook / P Ebert - Notice of Fine's 'Limits of Abstraction' 3 | |
| A reaction: Fine is a fan of creative definition. This sounds something like the distinction between cutting nature at the perceived joints, and speculating about where new joints might be inserted. Quite a helpful thought. |
| 10143 | 'Creative definitions' do not presuppose the existence of the objects defined [Fine,K] |
| Full Idea: What I call 'creative definitions' are made from a standpoint in which the existence of the objects that are to be assigned to the terms is not presupposed. | |
| From: Kit Fine (The Limits of Abstraction [2002], II.1) |
| 4971 | I don't use 'subject' and 'predicate' in my way of representing a judgement [Frege] |
| Full Idea: A distinction of subject and predicate finds no place in my way of representing a judgement. | |
| From: Gottlob Frege (Begriffsschrift [1879], §03) | |
| A reaction: Perhaps this sentence could be taken as the beginning of modern analytical philosophy. The old view doesn't seem to me entirely redundant - merely replaced by a much more detailed analysis of what makes a 'subject' and what makes a 'predicate'. |
| 17745 | For Frege, 'All A's are B's' means that the concept A implies the concept B [Frege, by Walicki] |
| Full Idea: 'All A's are B's' meant for Frege that the concept A implies the concept B, or that to be A implies also to be B. Moreover this applies to arbitrary x which happens to be A. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Michal Walicki - Introduction to Mathematical Logic History D.2 | |
| A reaction: This seems to hit the renate/cordate problem. If all creatures with hearts also have kidneys, does that mean that being enhearted logically implies being kidneyfied? If all chimps are hairy, is that a logical requirement? Is inclusion implication? |
| 7728 | Frege has a judgement stroke (vertical, asserting or judging) and a content stroke (horizontal, expressing) [Frege, by Weiner] |
| Full Idea: Frege distinguished between asserting a proposition and expressing it, and he introduced the judgement stroke (a small vertical line, assertion) and the content stroke (a long horizontal line, expression) to represent them. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Joan Weiner - Frege Ch.3 | |
| A reaction: There are also strokes for conditional and denial. |
| 16881 | The laws of logic are boundless, so we want the few whose power contains the others [Frege] |
| Full Idea: Since in view of the boundless multitude of laws that can be enunciated we cannot list them all, we cannot achieve completeness except by searching out those that, by their power, contain all of them. | |
| From: Gottlob Frege (Begriffsschrift [1879], §13) | |
| A reaction: He refers to these laws in the previous sentence as the 'core'. His talk of 'power' is music to my ears, since it implies a direction of explanation. Burge says the power is that of defining other concepts. |
| 7622 | In 1879 Frege developed second order logic [Frege, by Putnam] |
| Full Idea: By 1879 Frege had discovered an algorithm, a mechanical proof procedure, that embraces what is today standard 'second order logic'. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Hilary Putnam - Reason, Truth and History Ch.5 | |
| A reaction: Note that Frege did more than introduce quantifiers, and the logic of predicates. |
| 7729 | Frege replaced Aristotle's subject/predicate form with function/argument form [Frege, by Weiner] |
| Full Idea: Frege's regimentation is based on the view of the simplest sort of statement as having, not subject/predicate form (as in Aristotle), but function/argument form. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Joan Weiner - Frege | |
| A reaction: This looks like being a crucial move into the modern world, where one piece of information is taken in and dealt with, as in computer procedures. Have educated people reorganised their minds along Fregean lines? |
| 9950 | A quantifier is a second-level predicate (which explains how it contributes to truth-conditions) [Frege, by George/Velleman] |
| Full Idea: The contribution of the quantifier to the truth conditions of sentences of which it is a part cannot be adequately explained if it is treated as other than a second-level predicate (for instance, if it is viewed as name). | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2 | |
| A reaction: They suggest that this makes it something like a 'property of properties'. With this account it becomes plausible to think of numbers as quantifiers (since they do, after all, specify quantities). |
| 9991 | For Frege the variable ranges over all objects [Frege, by Tait] |
| Full Idea: For Frege the variable ranges over all objects. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by William W. Tait - Frege versus Cantor and Dedekind XII | |
| A reaction: The point is that Frege had not yet seen the necessity to define the domain of quantification, and this leads him into various difficulties. |
| 10536 | Frege's domain for variables is all objects, but modern interpretations first fix the domain [Dummett on Frege] |
| Full Idea: For Frege there is no need to specify the domain of the individual variables, which is taken as the totality of all objects. This contrasts with the standard notion of an interpretation, which demands that we first fix the domain. | |
| From: comment on Gottlob Frege (Begriffsschrift [1879]) by Michael Dummett - Frege Philosophy of Language (2nd ed) Ch.14 | |
| A reaction: What intrigues me is how domains of quantification shift according to context in ordinary usage, even in mid-sentence. I ought to go through every idea in this database, specifying its domain of quantification. Any volunteers? |
| 7730 | Frege introduced quantifiers for generality [Frege, by Weiner] |
| Full Idea: In order to express generality, Frege introduced quantifier notation. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Joan Weiner - Frege | |
| A reaction: This is the birth of predicate logic, beloved of analytical philosophers (but of no apparent interest to phenomenalists, deconstructionists, existentialists?). Generality is what you get from induction (which is, of course, problematic). |
| 7742 | Frege reduced most quantifiers to 'everything' combined with 'not' [Frege, by McCullogh] |
| Full Idea: Frege treated 'everything' as basic, and suggested ways of recasting propositions containing other quantifiers so that this was the only one remaining. He recast 'something' as 'at least one thing', and defined this in terms of 'everything' and 'not'. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Gregory McCullogh - The Game of the Name 1.6 | |
| A reaction: Extreme parsimony seems highly desirable in logic as well as ontology, but it can lead to frustrations, especially over the crucial question of the existence of things quantified over. See Idea 6068. |
| 13824 | Proof theory began with Frege's definition of derivability [Frege, by Prawitz] |
| Full Idea: Frege's formal definition of derivability is perhaps the first investigation in general proof theory. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Dag Prawitz - Gentzen's Analysis of First-Order Proofs 2 n2 | |
| A reaction: In 'On General Proof Theory §1' Prawitz says "proof theory originated with Hilbert" in 1900. Presumably Frege offered a theory, and then Hilbert saw it as a general project. |
| 13609 | Frege produced axioms for logic, though that does not now seem the natural basis for logic [Frege, by Kaplan] |
| Full Idea: Frege's work supplied a set of axioms for logic itself, at least partly because it was a well-known way of presenting the foundations in other disciplines, especially mathematics, but it does not nowadays strike us as natural for logic. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by David Kaplan - Dthat 5.1 | |
| A reaction: What Bostock has in mind is the so-called 'natural' deduction systems, which base logic on rules of entailment, rather than on a set of truths. The axiomatic approach uses a set of truths, plus the idea of possible contradictions. |
| 17855 | It may be possible to define induction in terms of the ancestral relation [Frege, by Wright,C] |
| Full Idea: Frege's account of the ancestral has made it possible, in effect, to define the natural numbers as entities for which induction holds. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Crispin Wright - Frege's Concept of Numbers as Objects 4.xix | |
| A reaction: This is the opposite of the approach in the Peano Axioms, where induction is used to define the natural numbers. |
| 10607 | Frege's logic has a hierarchy of object, property, property-of-property etc. [Frege, by Smith,P] |
| Full Idea: Frege's general logical system involves a type hierarchy, distinguishing objects from properties from properties-of-properties etc., with every item belonging to a determinate level. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Peter Smith - Intro to Gödel's Theorems 14.1 | |
| A reaction: The Theory of Types went on to apply this hierarchy to classes, where Frege's disastrous Basic Law V flattens the hierarchy of classes, putting them on the same level (Smith p.119) |
| 11008 | Existence is not a first-order property, but the instantiation of a property [Frege, by Read] |
| Full Idea: When Kant said that existence was not a property, what he meant was, according to Frege, that existence is not a first-order property - it is not a property of individuals but a property of properties, that the property has an instance. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Stephen Read - Thinking About Logic Ch.5 |
| 10145 | Abstracts cannot be identified with sets [Fine,K] |
| Full Idea: It is impossible for a proponent of both sets and abstracts to identify the abstracts, in any reasonable manner, with the sets. | |
| From: Kit Fine (The Limits of Abstraction [2002], IV.1) | |
| A reaction: [This observation emerges from a proof Fine has just completed] Cf Idea 10137. The implication is that there is no compromise view available, and one must choose between abstraction or sets as one's account of numbers and groups of concepts. |
| 10136 | Points in Euclidean space are abstract objects, but not introduced by abstraction [Fine,K] |
| Full Idea: Points in abstract Euclidean space are abstract objects, and yet are not objects of abstraction, since they are not introduced through a principle of abstraction of the sort envisaged by Frege. | |
| From: Kit Fine (The Limits of Abstraction [2002], I.1) | |
| A reaction: The point seems to be that they are not abstracted 'from' anything, but are simpy posited as basic constituents. I suggest that points are idealisations (of smallness) rather than abstractions. They are idealised 'from' substances. |
| 10144 | Postulationism says avoid abstract objects by giving procedures that produce truth [Fine,K] |
| Full Idea: A procedural form of postulationism says that instead of stipulating that certain statements are true, one specifies certain procedures for extending the domain to one in which the statement will in fact be true, without invoking an abstract ontology. | |
| From: Kit Fine (The Limits of Abstraction [2002], II.5) | |
| A reaction: The whole of philosophy might go better if it was founded on procedures and processes, rather than on objects. The Hopi Indians were right. |
| 9144 | Fine's 'procedural postulationism' uses creative definitions, but avoids abstract ontology [Fine,K, by Cook/Ebert] |
| Full Idea: Fine says creative definitions can found mathematics. His 'procedural postulationism' says one stipulates not truths, but certain procedures for extending a domain. The procedures can be stated without invoking an abstract ontology. | |
| From: report of Kit Fine (The Limits of Abstraction [2002], 100) by R Cook / P Ebert - Notice of Fine's 'Limits of Abstraction' 4 | |
| A reaction: (For creative definitions, see Idea 9143) This sounds close in spirit to fictionalism, but with the emphasis on the procedure (which can presumably be formalized) rather than a pure act of imaginative creation. |
| 10141 | Many different kinds of mathematical objects can be regarded as forms of abstraction [Fine,K] |
| Full Idea: Many different kinds of mathematical objects (natural numbers, the reals, points, lines, figures, groups) can be regarded as forms of abstraction, with special theories having their basis in a general theory of abstraction. | |
| From: Kit Fine (The Limits of Abstraction [2002], I.4) | |
| A reaction: This result, if persuasive, would be just the sort of unified account which the whole problem of abstact ideas requires. |
| 10135 | We can abstract from concepts (e.g. to number) and from objects (e.g. to direction) [Fine,K] |
| Full Idea: A principle of abstraction is 'conceptual' when the items upon which it abstracts are concepts (e.g. a one-one correspondence associated with a number), and 'objectual' if they are objects (parallel lines associated with a direction). | |
| From: Kit Fine (The Limits of Abstraction [2002], I) |
| 9142 | Fine considers abstraction as reconceptualization, to produce new senses by analysing given senses [Fine,K, by Cook/Ebert] |
| Full Idea: Fine considers abstraction principles as instances of reconceptualization (rather than implicit definition, or using the Context Principle). This centres not on reference, but on new senses emerging from analysis of a given sense. | |
| From: report of Kit Fine (The Limits of Abstraction [2002], 035) by R Cook / P Ebert - Notice of Fine's 'Limits of Abstraction' 2 | |
| A reaction: Fine develops an argument against this view, because (roughly) the procedure does not end in a unique result. Intuitively, the idea that abstraction is 'reconceptualization' sounds quite promising to me. |
| 10137 | Abstractionism can be regarded as an alternative to set theory [Fine,K] |
| Full Idea: The uncompromising abstractionist rejects set theory, seeing the theory of abstractions as an alternative, rather than as a supplement, to the standard theory of sets. | |
| From: Kit Fine (The Limits of Abstraction [2002], I.1) | |
| A reaction: There is also a 'compromising' version. Presumably you still have equivalence classes to categorise the objects, which are defined by their origin rather than by what they are members of... Cf. Idea 10145. |
| 10138 | An object is the abstract of a concept with respect to a relation on concepts [Fine,K] |
| Full Idea: We can see an object as being the abstract of a concept with respect to a relation on concepts. For example, we may say that 0 is the abstract of the empty concept with respect to the relation of one-one correspondence. | |
| From: Kit Fine (The Limits of Abstraction [2002], I.2) | |
| A reaction: This is Fine's attempt to give a modified account of the Fregean approach to abstraction. He says that the reference to a relation will solve the problem of identity between abstractions. |
| 22280 | Frege's account was top-down and decompositional, not bottom-up and compositional [Frege, by Potter] |
| Full Idea: Frege's account was top-down, not bottom-up: he aimed to decompose and discern function-argument structure in already existing sentences, not to explain how those sentences acquired their meanings in the first place. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 03 'Func' | |
| A reaction: This goes with the holistic account of meaning, which leads to Quine's gavagai and Kuhn's obfuscation of science. I recommend compositionality for everthing. |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
| From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
| A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
| Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
| From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
| A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |
| 7741 | The predicate 'exists' is actually a natural language expression for a quantifier [Frege, by Weiner] |
| Full Idea: On Frege's logical analysis, the predicate 'exists' is actually a natural language expression for a quantifier. | |
| From: report of Gottlob Frege (Begriffsschrift [1879]) by Joan Weiner - Frege Ch.8 | |
| A reaction: However see Idea 6067, for McGinn's alternative view of quantifiers. In the normal conventions of predicate logic it may be that existence is treated as a quantifier, but that is not the same as saying that existence just IS a quantifier. |