101 ideas
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] |
13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
19906 | All countries are in a mutual state of nature [Locke] |
19882 | We are not created for solitude, but are driven into society by our needs [Locke] |
19864 | In nature men can dispose of possessions and their persons in any way that is possible [Locke] |
19865 | There is no subjection in nature, and all creatures of the same species are equal [Locke] |
19866 | The rational law of nature says we are all equal and independent, and should show mutual respect [Locke] |
19872 | The animals and fruits of the earth belong to mankind [Locke] |
19907 | There is a natural right to inheritance within a family [Locke] |
19863 | Politics is the right to make enforceable laws to protect property and the state, for the common good [Locke] |
5654 | The Second Treatise explores the consequences of the contractual view of the state [Locke, by Scruton] |
19888 | A society only begins if there is consent of all the individuals to join it [Locke] |
6702 | If anyone enjoys the benefits of government (even using a road) they give tacit assent to its laws [Locke] |
19909 | A politic society is created from a state of nature by a unanimous agreement [Locke] |
19910 | A single will creates the legislature, which is duty-bound to preserve that will [Locke] |
19893 | Anyone who enjoys the benefits of a state has given tacit consent to be part of it [Locke] |
19894 | You can only become an actual member of a commonwealth by an express promise [Locke] |
19892 | Children are not born into citizenship of a state [Locke] |
19885 | Absolute monarchy is inconsistent with civil society [Locke] |
19886 | The idea that absolute power improves mankind is confuted by history [Locke] |
19903 | Despotism is arbitrary power to kill, based neither on natural equality, nor any social contract [Locke] |
19905 | People stripped of their property are legitimately subject to despotism [Locke] |
19904 | Legitimate prisoners of war are subject to despotism, because that continues the state of war [Locke] |
19895 | Even the legislature must be preceded by a law which gives it power to make laws [Locke] |
19900 | The executive must not be the legislature, or they may exempt themselves from laws [Locke] |
19902 | Any obstruction to the operation of the legislature can be removed forcibly by the people [Locke] |
19908 | Rebelling against an illegitimate power is no sin [Locke] |
19911 | If legislators confiscate property, or enslave people, they are no longer owed obedience [Locke] |
19901 | The people have supreme power, to depose a legislature which has breached their trust [Locke] |
19887 | Unanimous consent makes a united community, which is then ruled by the majority [Locke] |
19913 | A master forfeits ownership of slaves he abandons [Locke] |
19883 | Slaves captured in a just war have no right to property, so are not part of civil society [Locke] |
19870 | If you try to enslave me, you have declared war on me [Locke] |
19871 | Freedom is not absence of laws, but living under laws arrived at by consent [Locke] |
19880 | All value depends on the labour involved [Locke] |
19884 | There is only a civil society if the members give up all of their natural executive rights [Locke] |
19873 | We all own our bodies, and the work we do is our own [Locke] |
6580 | Locke (and Marx) held that ownership of objects is a natural relation, based on the labour put into it [Locke, by Fogelin] |
20520 | Locke says 'mixing of labour' entitles you to land, as well as nuts and berries [Wolff,J on Locke] |
19875 | A man's labour gives ownership rights - as long as there are fair shares for all [Locke] |
19874 | If a man mixes his labour with something in Nature, he thereby comes to own it [Locke] |
19877 | Fountain water is everyone's, but a drawn pitcher of water has an owner [Locke] |
19876 | Gathering natural fruits gives ownership; the consent of other people is irrelevant [Locke] |
19878 | Mixing labour with a thing bestows ownership - as long as the thing is not wasted [Locke] |
19879 | A man owns land if he cultivates it, to the limits of what he needs [Locke] |
19898 | Soldiers can be commanded to die, but not to hand over their money [Locke] |
19881 | The aim of law is not restraint, but to make freedom possible [Locke] |
19868 | It is only by a law of Nature that we can justify punishing foreigners [Locke] |
19867 | Reparation and restraint are the only justifications for punishment [Locke] |
19912 | Self-defence is natural, but not the punishment of superiors by inferiors [Locke] |
19869 | Punishment should make crime a bad bargain, leading to repentance and deterrence [Locke] |
19899 | The consent of the people is essential for any tax [Locke] |