display all the ideas for this combination of texts
14 ideas
| 7786 | Propositional logic handles negation, disjunction, conjunction; predicate logic adds quantifiers, predicates, relations [Girle] |
| Full Idea: Propositional logic can deal with negation, disjunction and conjunction of propositions, but predicate logic goes beyond it to deal with quantifiers, predicates and relations. | |
| From: Rod Girle (Modal Logics and Philosophy [2000], 1.1) | |
| A reaction: This is on the first page of an introduction to the next stage, which is to include modal notions like 'must' and 'possibly'. |
| 7798 | There are three axiom schemas for propositional logic [Girle] |
| Full Idea: The axioms of propositional logic are: A→(B→A); A→(B→C)→(A→B)→(A→C) ; and (¬A→¬B)→(B→A). | |
| From: Rod Girle (Modal Logics and Philosophy [2000], 6.5) |
| 7799 | Proposition logic has definitions for its three operators: or, and, and identical [Girle] |
| Full Idea: The operators of propositional logic are defined as follows: 'or' (v) is not-A implies B; 'and' (ampersand) is not A-implies-not-B; and 'identity' (three line equals) is A-implies-B and B-implies-A. | |
| From: Rod Girle (Modal Logics and Philosophy [2000], 6.5) |
| 7797 | Axiom systems of logic contain axioms, inference rules, and definitions of proof and theorems [Girle] |
| Full Idea: An axiom system for a logic contains three elements: a set of axioms; a set of inference rules; and definitions for proofs and theorems. There are also definitions for the derivation of conclusions from sets of premises. | |
| From: Rod Girle (Modal Logics and Philosophy [2000], 6.5) |
| 7794 | There are seven modalities in S4, each with its negation [Girle] |
| Full Idea: In S4 there are fourteen modalities: no-operator; necessarily; possibly; necessarily-possibly; possibly-necessarily; necessarily-possibly-necessarily; and possibly-necessarily-possibly (each with its negation). | |
| From: Rod Girle (Modal Logics and Philosophy [2000], 3.5) | |
| A reaction: This is said to be 'more complex' than S5, but also 'weaker'. |
| 7793 | ◊p → □◊p is the hallmark of S5 [Girle] |
| Full Idea: The critical formula that distinguishes S5 from all others is: ◊p → □◊p. | |
| From: Rod Girle (Modal Logics and Philosophy [2000], 3.3) | |
| A reaction: If it is possible that it is raining, then it is necessary that it is possible that it is raining. But if it is possible in this world, how can that possibility be necessary in all possible worlds? |
| 7795 | S5 has just six modalities, and all strings can be reduced to those [Girle] |
| Full Idea: In S5 there are six modalities: no-operator; necessarily; and possibly (and their negations). In any sequence of operators we may delete all but the last to gain an equivalent formula. | |
| From: Rod Girle (Modal Logics and Philosophy [2000], 3.5) | |
| A reaction: Such drastic simplification seems attractive. Is there really no difference, though, between 'necessarily-possibly', 'possibly-possibly' and just 'possibly'? Could p be contingently possible in this world, and necessarily possible in another? |
| 7787 | Possible worlds logics use true-in-a-world rather than true [Girle] |
| Full Idea: In possible worlds logics a statement is true-in-a-world rather than just true. | |
| From: Rod Girle (Modal Logics and Philosophy [2000], 1.1) | |
| A reaction: This sounds relativist, but I don't think it is. It is the facts which change, not the concept of truth. So 'donkeys can talk' may be true in a world, but not in the actual one. |
| 7788 | Modal logic has four basic modal negation equivalences [Girle] |
| Full Idea: The four important logical equivalences in modal logic (the Modal Negation equivalences) are: ¬◊p↔□¬p, ◊¬p↔¬□p, □p↔¬◊¬p, and ◊p↔¬□¬p. | |
| From: Rod Girle (Modal Logics and Philosophy [2000], 1.2) | |
| A reaction: [Possibly is written as a diamond, necessarily a square] These are parallel to a set of equivalences between quantifiers in predicate logic. They are called the four 'modal negation (MN) equivalences'. |
| 7796 | Modal logics were studied in terms of axioms, but now possible worlds semantics is added [Girle] |
| Full Idea: Modal logics were, for a long time, studied in terms of axiom systems. The advent of possible worlds semantics made it possible to study them in a semantic way as well. | |
| From: Rod Girle (Modal Logics and Philosophy [2000], 6.5) |
| 10482 | The logic of ZF is classical first-order predicate logic with identity [Boolos] |
| Full Idea: The logic of ZF Set Theory is classical first-order predicate logic with identity. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.121) | |
| A reaction: This logic seems to be unable to deal with very large cardinals, precisely those that are implied by set theory, so there is some sort of major problem hovering here. Boolos is fairly neutral. |
| 10492 | A few axioms of set theory 'force themselves on us', but most of them don't [Boolos] |
| Full Idea: Maybe the axioms of extensionality and the pair set axiom 'force themselves on us' (Gödel's phrase), but I am not convinced about the axioms of infinity, union, power or replacement. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.130) | |
| A reaction: Boolos is perfectly happy with basic set theory, but rather dubious when very large cardinals come into the picture. |
| 10485 | Naïve sets are inconsistent: there is no set for things that do not belong to themselves [Boolos] |
| Full Idea: The naïve view of set theory (that any zero or more things form a set) is natural, but inconsistent: the things that do not belong to themselves are some things that do not form a set. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.127) | |
| A reaction: As clear a summary of Russell's Paradox as you could ever hope for. |
| 10484 | The iterative conception says sets are formed at stages; some are 'earlier', and must be formed first [Boolos] |
| Full Idea: According to the iterative conception, every set is formed at some stage. There is a relation among stages, 'earlier than', which is transitive. A set is formed at a stage if and only if its members are all formed before that stage. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.126) | |
| A reaction: He gives examples of the early stages, and says the conception is supposed to 'justify' Zermelo set theory. It is also supposed to make the axioms 'natural', rather than just being selected for convenience. And it is consistent. |