84 ideas
8558 | One system has properties, powers, events, similarity and substance [Shoemaker] |
8559 | Analysis aims at internal relationships, not reduction [Shoemaker] |
13634 | Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro] |
13643 | Aristotelian logic is complete [Shapiro] |
13651 | A set is 'transitive' if contains every member of each of its members [Shapiro] |
13647 | Choice is essential for proving downward Löwenheim-Skolem [Shapiro] |
13631 | Are sets part of logic, or part of mathematics? [Shapiro] |
13654 | It is central to the iterative conception that membership is well-founded, with no infinite descending chains [Shapiro] |
13640 | Russell's paradox shows that there are classes which are not iterative sets [Shapiro] |
13666 | Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets [Shapiro] |
13653 | 'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element [Shapiro] |
13627 | There is no 'correct' logic for natural languages [Shapiro] |
13642 | Logic is the ideal for learning new propositions on the basis of others [Shapiro] |
13668 | Bernays (1918) formulated and proved the completeness of propositional logic [Shapiro] |
13669 | Can one develop set theory first, then derive numbers, or are numbers more basic? [Shapiro] |
13667 | Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order [Shapiro] |
13662 | First-order logic was an afterthought in the development of modern logic [Shapiro] |
13624 | The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed [Shapiro] |
13660 | Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable [Shapiro] |
13673 | The notion of finitude is actually built into first-order languages [Shapiro] |
15944 | Second-order logic is better than set theory, since it only adds relations and operations, and nothing else [Shapiro, by Lavine] |
13629 | Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? [Shapiro] |
13650 | Henkin semantics has separate variables ranging over the relations and over the functions [Shapiro] |
13645 | In standard semantics for second-order logic, a single domain fixes the ranges for the variables [Shapiro] |
13649 | Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics [Shapiro] |
13626 | Semantic consequence is ineffective in second-order logic [Shapiro] |
13637 | If a logic is incomplete, its semantic consequence relation is not effective [Shapiro] |
13632 | Finding the logical form of a sentence is difficult, and there are no criteria of correctness [Shapiro] |
13674 | We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro] |
13633 | 'Satisfaction' is a function from models, assignments, and formulas to {true,false} [Shapiro] |
13644 | Semantics for models uses set-theory [Shapiro] |
13636 | An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro] |
13670 | Categoricity can't be reached in a first-order language [Shapiro] |
13658 | Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [Shapiro] |
13659 | Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro] |
13648 | The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro] |
13675 | Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails [Shapiro] |
13635 | 'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence [Shapiro] |
13628 | We can live well without completeness in logic [Shapiro] |
13630 | Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro] |
13646 | Compactness is derived from soundness and completeness [Shapiro] |
13661 | A language is 'semantically effective' if its logical truths are recursively enumerable [Shapiro] |
13641 | Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro] |
13676 | Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro] |
13677 | Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro] |
13652 | The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro] |
13657 | First-order arithmetic can't even represent basic number theory [Shapiro] |
13656 | Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro] |
13664 | Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro] |
13625 | Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro] |
13663 | Some reject formal properties if they are not defined, or defined impredicatively [Shapiro] |
15092 | Formerly I said properties are individuated by essential causal powers and causing instantiation [Shoemaker, by Shoemaker] |
8543 | Genuine properties are closely related to genuine changes [Shoemaker] |
8551 | Properties must be essentially causal if we can know and speak about them [Shoemaker] |
8557 | To ascertain genuine properties, examine the object directly [Shoemaker] |
15761 | We should abandon the idea that properties are the meanings of predicate expressions [Shoemaker] |
15756 | Some truths are not because of a thing's properties, but because of the properties of related things [Shoemaker] |
13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
15758 | Things have powers in virtue of (which are entailed by) their properties [Shoemaker] |
8547 | One power can come from different properties; a thing's powers come from its properties [Shoemaker] |
8549 | Properties are functions producing powers, and powers are functions producing effects [Shoemaker] |
12678 | Shoemaker says all genuine properties are dispositional [Shoemaker, by Ellis] |
8545 | A causal theory of properties focuses on change, not (say) on abstract properties of numbers [Shoemaker] |
15757 | 'Square', 'round' and 'made of copper' show that not all properties are dispositional [Shoemaker] |
15759 | The identity of a property concerns its causal powers [Shoemaker] |
15760 | Properties are clusters of conditional powers [Shoemaker] |
15762 | Could properties change without the powers changing, or powers change without the properties changing? [Shoemaker] |
8552 | If properties are separated from causal powers, this invites total elimination [Shoemaker] |
4040 | The notions of property and of causal power are parts of a single system of related concepts [Shoemaker] |
15765 | Actually, properties are individuated by causes as well as effects [Shoemaker] |
8548 | Dispositional predicates ascribe powers, and the rest ascribe properties [Shoemaker] |
9485 | Universals concern how things are, and how they could be [Shoemaker, by Bird] |
8550 | Triangular and trilateral are coextensive, but different concepts; but powers and properties are the same [Shoemaker] |
8555 | There is no subset of properties which guarantee a thing's identity [Shoemaker] |
8554 | Possible difference across worlds depends on difference across time in the actual world [Shoemaker] |
15764 | 'Conceivable' is either not-provably-false, or compatible with what we know? [Shoemaker] |
8562 | It is possible to conceive what is not possible [Shoemaker] |
8556 | Grueness is not, unlike green and blue, associated with causal potential [Shoemaker] |
5655 | Happiness is not satisfaction of desires, but fulfilment of values [Bradley, by Scruton] |
8542 | If causality is between events, there must be reference to the properties involved [Shoemaker] |
8560 | If causal laws describe causal potentialities, the same laws govern properties in all possible worlds [Shoemaker] |
15763 | If properties are causal, then causal necessity is a species of logical necessity [Shoemaker] |
8561 | If a world has different causal laws, it must have different properties [Shoemaker] |
8553 | It looks as if the immutability of the powers of a property imply essentiality [Shoemaker] |