display all the ideas for this combination of texts
29 ideas
| 8077 | Stoic propositional logic is like chemistry - how atoms make molecules, not the innards of atoms [Chrysippus, by Devlin] |
| Full Idea: In Stoic logic propositions are treated the way atoms are treated in present-day chemistry, where the focus is on the way atoms fit together to form molecules, rather than on the internal structure of the atoms. | |
| From: report of Chrysippus (fragments/reports [c.240 BCE]) by Keith Devlin - Goodbye Descartes Ch.2 | |
| A reaction: A nice analogy to explain the nature of Propositional Logic, which was invented by the Stoics (N.B. after Aristotle had invented predicate logic). |
| 20791 | Chrysippus has five obvious 'indemonstrables' of reasoning [Chrysippus, by Diog. Laertius] |
| Full Idea: Chrysippus has five indemonstrables that do not need demonstration:1) If 1st the 2nd, but 1st, so 2nd; 2) If 1st the 2nd, but not 2nd, so not 1st; 3) Not 1st and 2nd, the 1st, so not 2nd; 4) 1st or 2nd, the 1st, so not 2nd; 5) 1st or 2nd, not 2nd, so 1st. | |
| From: report of Chrysippus (fragments/reports [c.240 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.80-81 | |
| A reaction: [from his lost text 'Dialectics'; squashed to fit into one quote] 1) is Modus Ponens, 2) is Modus Tollens. 4) and 5) are Disjunctive Syllogisms. 3) seems a bit complex to be an indemonstrable. |
| 9154 | Frege agreed with Euclid that the axioms of logic and mathematics are known through self-evidence [Frege, by Burge] |
| Full Idea: Frege maintained a sophisticated version of the Euclidean position that knowledge of the axioms and theorems of logic, geometry, and arithmetic rests on the self-evidence of the axioms, definitions, and rules of inference. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Tyler Burge - Frege on Apriority Intro | |
| A reaction: I am inclined to agree that they are indeed self-evident, but not in a purely a priori way. They are self-evident general facts about how reality is and how (it seems) that it must be. It seems to me closer to a perception than an insight. |
| 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. |
| 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] |
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Full Idea:
F(x) is a 'function', which indicates the unique value which y takes in |
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| 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) |
| 9157 | The null set is only defensible if it is the extension of an empty concept [Frege, by Burge] |
| Full Idea: Frege regarded the null set as an indefensible entity from the point of view of iterative set theory. It collects nothing. He thought a null entity (a null extension) is derivable only as the extension of an empty concept. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Tyler Burge - Frege on Apriority II | |
| A reaction: Frege is right, if you like sets. Othewise all the other sets are going to be defined simply by their extension, and the empty set has to be defined in a different way, which looks like appalling theory. Empty concepts bother me though! |
| 9835 | It is because a concept can be empty that there is such a thing as the empty class [Frege, by Dummett] |
| Full Idea: Since he thought of classes as extensions of concepts, ...it is because a concept can be empty that there is such a thing as the empty class. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege philosophy of mathematics Ch.8 | |
| A reaction: Frege was already up against the awaiting Russell Paradox, but this view also seems to imply that there are many empty classes, since the absences of sandwiches would be different from the absence of heroism. |
| 9854 | We can introduce new objects, as equivalence classes of objects already known [Frege, by Dummett] |
| Full Idea: We can introduce a new type of object from the obtaining of some equivalence relation between objects of some already known kind, by identifying the new objects as equivalence classes of the old ones under that equivalence relation. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege philosophy of mathematics Ch.14 | |
| A reaction: Some accounts of abstraction merely describe the concept, but this is a rival to the traditional pyschological abstractionism that Frege attacked so vigorously. Should we take a platonist or constructivist view of the new objects? |
| 9883 | Frege introduced the standard device, of defining logical objects with equivalence classes [Frege, by Dummett] |
| Full Idea: Frege decided that all logical objects, or at least all those needed for mathematics, could be defined by logical abstraction, except the classes needed for such definitions. ..This definition by equivalence classes has been adopted as a standard device. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §64-68) by Michael Dummett - Frege philosophy of mathematics | |
| A reaction: This means if we are to understand modern abstraction (instead of the psychological method of ignoring selected properties of objects), we must understand the presuppositions needed for a definition by equivalence. |
| 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) |
| 18104 | Frege, unlike Russell, has infinite individuals because numbers are individuals [Frege, by Bostock] |
| Full Idea: Frege was able to prove that there are infinitely many individuals by taking the numbers themselves to be individuals, but this course was not open to Russell. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by David Bostock - Philosophy of Mathematics 5.2 |
| 9834 | A class is, for Frege, the extension of a concept [Frege, by Dummett] |
| Full Idea: A class is, for Frege, the extension of a concept. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege philosophy of mathematics Ch.8 | |
| A reaction: This simple idea was the source of all his troubles, because there are concepts which can't have an extension, because of contradiction. ...And yet all intuition says Frege is right.. |