65 ideas
18835 | Logic doesn't have a metaphysical basis, but nor can logic give rise to the metaphysics [Rumfitt] |
18819 | The idea that there are unrecognised truths is basic to our concept of truth [Rumfitt] |
18826 | 'True at a possibility' means necessarily true if what is said had obtained [Rumfitt] |
18803 | Semantics for propositions: 1) validity preserves truth 2) non-contradition 3) bivalence 4) truth tables [Rumfitt] |
18814 | 'Absolute necessity' would have to rest on S5 [Rumfitt] |
18799 | Intuitionists can accept Double Negation Elimination for decidable propositions [Rumfitt] |
18798 | It is the second-order part of intuitionistic logic which actually negates some classical theorems [Rumfitt] |
9672 | Free logic is one of the few first-order non-classical logics [Priest,G] |
18830 | Most set theorists doubt bivalence for the Continuum Hypothesis, but still use classical logic [Rumfitt] |
9697 | X1 x X2 x X3... x Xn indicates the 'cartesian product' of those sets [Priest,G] |
9685 | <a,b&62; is a set whose members occur in the order shown [Priest,G] |
9675 | a ∈ X says a is an object in set X; a ∉ X says a is not in X [Priest,G] |
9674 | {x; A(x)} is a set of objects satisfying the condition A(x) [Priest,G] |
9673 | {a1, a2, ...an} indicates that a set comprising just those objects [Priest,G] |
9677 | Φ indicates the empty set, which has no members [Priest,G] |
9676 | {a} is the 'singleton' set of a (not the object a itself) [Priest,G] |
9679 | X⊂Y means set X is a 'proper subset' of set Y [Priest,G] |
9678 | X⊆Y means set X is a 'subset' of set Y [Priest,G] |
9681 | X = Y means the set X equals the set Y [Priest,G] |
9683 | X ∩ Y indicates the 'intersection' of sets X and Y, the objects which are in both sets [Priest,G] |
9682 | X∪Y indicates the 'union' of all the things in sets X and Y [Priest,G] |
9684 | Y - X is the 'relative complement' of X with respect to Y; the things in Y that are not in X [Priest,G] |
9694 | The 'relative complement' is things in the second set not in the first [Priest,G] |
9693 | The 'intersection' of two sets is a set of the things that are in both sets [Priest,G] |
9692 | The 'union' of two sets is a set containing all the things in either of the sets [Priest,G] |
9698 | The 'induction clause' says complex formulas retain the properties of their basic formulas [Priest,G] |
9687 | A 'member' of a set is one of the objects in the set [Priest,G] |
9695 | An 'ordered pair' (or ordered n-tuple) is a set with its members in a particular order [Priest,G] |
9696 | A 'cartesian product' of sets is the set of all the n-tuples with one member in each of the sets [Priest,G] |
9686 | A 'set' is a collection of objects [Priest,G] |
9689 | The 'empty set' or 'null set' has no members [Priest,G] |
9690 | A set is a 'subset' of another set if all of its members are in that set [Priest,G] |
9691 | A 'proper subset' is smaller than the containing set [Priest,G] |
9688 | A 'singleton' is a set with only one member [Priest,G] |
9680 | The empty set Φ is a subset of every set (including itself) [Priest,G] |
18843 | The iterated conception of set requires continual increase in axiom strength [Rumfitt] |
18836 | A set may well not consist of its members; the empty set, for example, is a problem [Rumfitt] |
18837 | A set can be determinate, because of its concept, and still have vague membership [Rumfitt] |
18845 | If the totality of sets is not well-defined, there must be doubt about the Power Set Axiom [Rumfitt] |
18815 | Logic is higher-order laws which can expand the range of any sort of deduction [Rumfitt] |
18804 | The case for classical logic rests on its rules, much more than on the Principle of Bivalence [Rumfitt] |
18805 | Classical logic rules cannot be proved, but various lines of attack can be repelled [Rumfitt] |
18827 | If truth-tables specify the connectives, classical logic must rely on Bivalence [Rumfitt] |
18813 | Logical consequence is a relation that can extended into further statements [Rumfitt] |
18808 | Normal deduction presupposes the Cut Law [Rumfitt] |
18840 | When faced with vague statements, Bivalence is not a compelling principle [Rumfitt] |
18802 | In specifying a logical constant, use of that constant is quite unavoidable [Rumfitt] |
18800 | Introduction rules give deduction conditions, and Elimination says what can be deduced [Rumfitt] |
18809 | Logical truths are just the assumption-free by-products of logical rules [Rumfitt] |
18807 | Monotonicity means there is a guarantee, rather than mere inductive support [Rumfitt] |
18842 | Maybe an ordinal is a property of isomorphic well-ordered sets, and not itself a set [Rumfitt] |
18834 | Infinitesimals do not stand in a determinate order relation to zero [Rumfitt] |
18846 | Cantor and Dedekind aimed to give analysis a foundation in set theory (rather than geometry) [Rumfitt] |
18839 | An object that is not clearly red or orange can still be red-or-orange, which sweeps up problem cases [Rumfitt] |
18838 | The extension of a colour is decided by a concept's place in a network of contraries [Rumfitt] |
18816 | Metaphysical modalities respect the actual identities of things [Rumfitt] |
18825 | S5 is the logic of logical necessity [Rumfitt] |
18824 | Since possibilities are properties of the world, calling 'red' the determination of a determinable seems right [Rumfitt] |
18828 | If two possibilities can't share a determiner, they are incompatible [Rumfitt] |
18821 | Possibilities are like possible worlds, but not fully determinate or complete [Rumfitt] |
18831 | Medieval logicians said understanding A also involved understanding not-A [Rumfitt] |
18820 | In English 'evidence' is a mass term, qualified by 'little' and 'more' [Rumfitt] |
18817 | We understand conditionals, but disagree over their truth-conditions [Rumfitt] |
18829 | The truth grounds for 'not A' are the possibilities incompatible with truth grounds for A [Rumfitt] |
22489 | 'Good' is an attributive adjective like 'large', not predicative like 'red' [Geach, by Foot] |