7786 | Propositional logic handles negation, disjunction, conjunction; predicate logic adds quantifiers, predicates, relations |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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) |
7789 | Necessary implication is called 'strict implication'; if successful, it is called 'entailment' |
Full Idea: Necessary implication is often called 'strict implication'. The sort of strict implication found in valid arguments, where the conjunction of the premises necessarily implies the conclusion, is often called 'entailment'. | |
From: Rod Girle (Modal Logics and Philosophy [2000], 1.2) | |
A reaction: These are basic concept for all logic. |
7790 | If an argument is invalid, a truth tree will indicate a counter-example |
Full Idea: The truth trees method for establishing the validity of arguments and formulas is easy to use, and has the advantage that if an argument or formula is not valid, then a counter-example can be retrieved from the tree. | |
From: Rod Girle (Modal Logics and Philosophy [2000], 1.4) |
7800 | Analytic truths are divided into logically and conceptually necessary |
Full Idea: It has been customary to see analytic truths as dividing into the logically necessary and the conceptually necessary. | |
From: Rod Girle (Modal Logics and Philosophy [2000], 7.3) | |
A reaction: I suspect that this neglected distinction is important in discussions of Quine's elimination of the analytic/synthetic distinction. Was Quine too influenced by what is logically necessary, which might shift with a change of axioms? |
7801 | Possibilities can be logical, theoretical, physical, economic or human |
Full Idea: Qualified modalities seem to form a hierarchy, if we say that 'the possibility that there might be no hunger' is possible logically, theoretically, physically, economically, and humanly. | |
From: Rod Girle (Modal Logics and Philosophy [2000], 7.3) | |
A reaction: Girle also mentions conceptual possibility. I take 'physically' to be the same as 'naturally'. I would take 'metaphysically' possible to equate to 'theoretically' rather than 'logically'. Almost anything might be logically possible, with bizarre logic. |
7792 | A world has 'access' to a world it generates, which is important in possible worlds semantics |
Full Idea: When one world generates another then it has 'access' to the world it generated. The accessibility relation between worlds is very important in possible worlds semantics. | |
From: Rod Girle (Modal Logics and Philosophy [2000], 3.2) | |
A reaction: This invites the obvious question what is meant by 'generates'. |