54 ideas
| 17749 | Post proved the consistency of propositional logic in 1921 [Walicki] |
| Full Idea: A proof of the consistency of propositional logic was given by Emil Post in 1921. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.2.1) |
| 17765 | Propositional language can only relate statements as the same or as different [Walicki] |
| Full Idea: Propositional language is very rudimentary and has limited powers of expression. The only relation between various statements it can handle is that of identity and difference. As are all the same, but Bs can be different from As. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 7 Intro) | |
| A reaction: [second sentence a paraphrase] In predicate logic you could represent two statements as being the same except for one element (an object or predicate or relation or quantifier). |
| 17764 | Boolean connectives are interpreted as functions on the set {1,0} [Walicki] |
| Full Idea: Boolean connectives are interpreted as functions on the set {1,0}. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 5.1) | |
| A reaction: 1 and 0 are normally taken to be true (T) and false (F). Thus the functions output various combinations of true and false, which are truth tables. |
| 15375 | If terms change their designations in different states, they are functions from states to objects [Fitting] |
| Full Idea: The common feature of every designating term is that designation may change from state to state - thus it can be formalized by a function from states to objects. | |
| From: Melvin Fitting (Intensional Logic [2007], 3) | |
| A reaction: Specifying the objects sounds OK, but specifying states sounds rather tough. |
| 15376 | Intensional logic adds a second type of quantification, over intensional objects, or individual concepts [Fitting] |
| Full Idea: To first order modal logic (with quantification over objects) we can add a second kind of quantification, over intensions. An intensional object, or individual concept, will be modelled by a function from states to objects. | |
| From: Melvin Fitting (Intensional Logic [2007], 3.3) |
| 15378 | Awareness logic adds the restriction of an awareness function to epistemic logic [Fitting] |
| Full Idea: Awareness logic enriched Hintikka's epistemic models with an awareness function, mapping each state to the set of formulas we are aware of at that state. This reflects some bound on the resources we can bring to bear. | |
| From: Melvin Fitting (Intensional Logic [2007], 3.6.1) | |
| A reaction: [He cites Fagin and Halpern 1988 for this] |
| 15379 | Justication logics make explicit the reasons for mathematical truth in proofs [Fitting] |
| Full Idea: In justification logics, the logics of knowledge are extended by making reasons explicit. A logic of proof terms was created, with a semantics. In this, mathematical truths are known for explicit reasons, and these provide a measure of complexity. | |
| From: Melvin Fitting (Intensional Logic [2007], 3.6.1) |
| 17752 | The empty set is useful for defining sets by properties, when the members are not yet known [Walicki] |
| Full Idea: The empty set is mainly a mathematical convenience - defining a set by describing the properties of its members in an involved way, we may not know from the very beginning what its members are. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 1.1) |
| 17753 | The empty set avoids having to take special precautions in case members vanish [Walicki] |
| Full Idea: Without the assumption of the empty set, one would often have to take special precautions for the case where a set happened to contain no elements. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 1.1) | |
| A reaction: Compare the introduction of the concept 'zero', where special precautions are therefore required. ...But other special precautions are needed without zero. Either he pays us, or we pay him, or ...er. Intersecting sets need the empty set. |
| 17759 | Ordinals play the central role in set theory, providing the model of well-ordering [Walicki] |
| Full Idea: Ordinals play the central role in set theory, providing the paradigmatic well-orderings. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) | |
| A reaction: When you draw the big V of the iterative hierarchy of sets (built from successive power sets), the ordinals are marked as a single line up the middle, one ordinal for each level. |
| 17741 | To determine the patterns in logic, one must identify its 'building blocks' [Walicki] |
| Full Idea: In order to construct precise and valid patterns of arguments one has to determine their 'building blocks'. One has to identify the basic terms, their kinds and means of combination. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History Intro) | |
| A reaction: A deceptively simple and important idea. All explanation requires patterns and levels, and it is the idea of building blocks which makes such things possible. It is right at the centre of our grasp of everything. |
| 11026 | Classical logic is deliberately extensional, in order to model mathematics [Fitting] |
| Full Idea: Mathematics is typically extensional throughout (we write 3+2=2+3 despite the two terms having different meanings). ..Classical first-order logic is extensional by design since it primarily evolved to model the reasoning of mathematics. | |
| From: Melvin Fitting (Intensional Logic [2007], §1) |
| 11028 | λ-abstraction disambiguates the scope of modal operators [Fitting] |
| Full Idea: λ-abstraction can be used to abstract and disambiguate a predicate. De re is [λx◊P(x)](f) - f has the possible-P property - and de dicto is ◊[λxP(x)](f) - possibly f has the P-property. Also applies to □. | |
| From: Melvin Fitting (Intensional Logic [2007], §3.3) | |
| A reaction: Compare the Barcan formula. Originated with Church in the 1930s, and Carnap 1947, but revived by Stalnaker and Thomason 1968. Because it refers to the predicate, it has a role in intensional versions of logic, especially modal logic. |
| 17747 | A 'model' of a theory specifies interpreting a language in a domain to make all theorems true [Walicki] |
| Full Idea: A specification of a domain of objects, and of the rules for interpreting the symbols of a logical language in this domain such that all the theorems of the logical theory are true is said to be a 'model' of the theory. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.1.3) | |
| A reaction: The basic ideas of this emerged 1915-30, but it needed Tarski's account of truth to really get it going. |
| 17748 | The L-S Theorem says no theory (even of reals) says more than a natural number theory [Walicki] |
| Full Idea: The L-S Theorem is ...a shocking result, since it implies that any consistent formal theory of everything - even about biology, physics, sets or the real numbers - can just as well be understood as being about natural numbers. It says nothing more. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.2) | |
| A reaction: Illuminating. Particularly the point that no theory about the real numbers can say anything more than a theory about the natural numbers. So the natural numbers contain all the truths we can ever express? Eh????? |
| 17761 | A compact axiomatisation makes it possible to understand a field as a whole [Walicki] |
| Full Idea: Having such a compact [axiomatic] presentation of a complicated field [such as Euclid's], makes it possible to relate not only to particular theorems but also to the whole field as such. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1) |
| 17763 | Axiomatic systems are purely syntactic, and do not presuppose any interpretation [Walicki] |
| Full Idea: Axiomatic systems, their primitive terms and proofs, are purely syntactic, that is, do not presuppose any interpretation. ...[142] They never address the world directly, but address a possible semantic model which formally represents the world. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1) |
| 17758 | Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [Walicki] |
| Full Idea: An ordinal can be defined as a transitive set of transitive sets, or else, as a transitive set totally ordered by set inclusion. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) |
| 17755 | Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals [Walicki] |
| Full Idea: The collection of ordinals is defined inductively: Basis: the empty set is an ordinal; Ind: for an ordinal x, the union with its singleton is also an ordinal; and any arbitrary (possibly infinite) union of ordinals is an ordinal. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) | |
| A reaction: [symbolism translated into English] Walicki says they are called 'ordinal numbers', but are in fact a set. |
| 17756 | The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki] |
| Full Idea: We can form infinite ordinals by taking unions of ordinals. We can thus form 'limit ordinals', which have no immediate predecessor. ω is the first (the union of all finite ordinals), ω + ω = sω is second, 3ω the third.... | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) |
| 17760 | Two infinite ordinals can represent a single infinite cardinal [Walicki] |
| Full Idea: There may be several ordinals for the same cardinality. ...Two ordinals can represent different ways of well-ordering the same number (aleph-0) of elements. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) | |
| A reaction: This only applies to infinite ordinals and cardinals. For the finite, the two coincide. In infinite arithmetic the rules are different. |
| 17757 | Members of ordinals are ordinals, and also subsets of ordinals [Walicki] |
| Full Idea: Every member of an ordinal is itself an ordinal, and every ordinal is a transitive set (its members are also its subsets; a member of a member of an ordinal is also a member of the ordinal). | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) |
| 17762 | In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate [Walicki] |
| Full Idea: Since non-Euclidean geometry preserves all Euclid's postulates except the fifth one, all the theorems derived without the use of the fifth postulate remain valid. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1) |
| 17754 | Inductive proof depends on the choice of the ordering [Walicki] |
| Full Idea: Inductive proof is not guaranteed to work in all cases and, particularly, it depends heavily on the choice of the ordering. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.1.1) | |
| A reaction: There has to be an well-founded ordering for inductive proofs to be possible. |
| 17742 | Scotus based modality on semantic consistency, instead of on what the future could allow [Walicki] |
| Full Idea: The link between time and modality was severed by Duns Scotus, who proposed a notion of possibility based purely on the notion of semantic consistency. 'Possible' means for him logically possible, that is, not involving contradiction. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History B.4) |
| 15377 | Definite descriptions pick out different objects in different possible worlds [Fitting] |
| Full Idea: Definite descriptions pick out different objects in different possible worlds quite naturally. | |
| From: Melvin Fitting (Intensional Logic [2007], 3.4) | |
| A reaction: A definite description can pick out the same object in another possible world, or a very similar one, or an object which has almost nothing in common with the others. |
| 23590 | Criminal responsibility can be fully assigned to each member of a group [Walzer] |
| Full Idea: It is a feature of criminal responsibility that it can be distributed without being divided. We can, that is, blame more than one person for a particular act without splitting up the blame we assign. | |
| From: Michael Walzer (Just and Unjust Wars [1977], 19) | |
| A reaction: How far can this extend? To a large violent mob? To an entire nation? In court the responsibility is usually adjusted in the sentencing, rather than in the initial verdict. |
| 23578 | Double Effect needs a double intention - to achieve the good, and minimise the evil [Walzer] |
| Full Idea: Double effect is defensible, I want to argue, only when the two outcomes are the product of a double intention - that 'good' be achieved, and that the foreseeable evil be reduced as far as possible. | |
| From: Michael Walzer (Just and Unjust Wars [1977], 09) | |
| A reaction: A good proposal, I think. We have to accept evil side effects sometimes, but it is immoral to pursue some good 'whatever the cost'. |
| 23564 | Deep ethical theory is very controversial, but we have to live with higher ethical practice [Walzer] |
| Full Idea: The substructure of the ethical world is a matter of deep and unending controversy, Meanwhile, however, we are living in the superstructure. | |
| From: Michael Walzer (Just and Unjust Wars [1977], Pref) | |
| A reaction: This may be the best approach to ethics. Nearly all applied ethics takes the common sense consensus on values for granted. Personally I think that is because the substructure is the obvious success and failure of human functioning. |
| 20595 | You can't distribute goods from behind a veil, because their social meaning is unclear [Walzer, by Tuckness/Wolf] |
| Full Idea: Walzer says behind the veil of ignorance there would be no way to know how a particular good should be distributed, because we would not know the social meaning of the good in question. | |
| From: report of Michael Walzer (Spheres of Justice [1983]) by Tuckness,A/Wolf,C - This is Political Philosophy 4 'Communitarian' | |
| A reaction: Is Rawls actually proposing to decide details of distribution from behind the veil? There is just the maximin principle. What that means in practice would surely come once the society was under way. |
| 20592 | Complex equality restricts equalities from spilling over, like money influencing politics and law [Walzer, by Tuckness/Wolf] |
| Full Idea: Complex equality tries to keep advantages in one area (such as money) from translating into advantages in politics or before the law. | |
| From: report of Michael Walzer (Spheres of Justice [1983]) by Tuckness,A/Wolf,C - This is Political Philosophy 3 'Complex' | |
| A reaction: Put like that, Walzer's complex equality becomes very interesting, and pinpoints a major problem of our age, where discrepancies of wealth have become staggeringly large at the top end. |
| 20549 | Equality is complex, with different spheres of equality where different principles apply [Walzer, by Swift] |
| Full Idea: Michael Walzer argues for 'complex equality', saying different goods belong to different distributive 'spheres', each with its own distributive principles. | |
| From: report of Michael Walzer (Spheres of Justice [1983]) by Adam Swift - Political Philosophy (3rd ed) 3 'Egalitarian' | |
| A reaction: Sounds interesting. Equality seems to make different demands when it concerns basic food for survival, or fine wines. You can spend your money freely, but hording in a crisis is frowned on. |
| 23568 | If whole states possess rights, there can be social relations between states [Walzer] |
| Full Idea: If states possess rights more or less as individuals do, then it is possible to imagine a society among them more or less like the individuals. | |
| From: Michael Walzer (Just and Unjust Wars [1977], 04) | |
| A reaction: The state's rights must derive from the people. Plots of land don't have rights. In some states the people are in conflict. It can't just be the government which represents the rights of the state. |
| 23571 | States can rightly pre-empt real and serious threats [Walzer] |
| Full Idea: States can use force in the face of threats of war, if there is a serious risk to territory or independence. They are then forced to fight, and are the victims of aggression. | |
| From: Michael Walzer (Just and Unjust Wars [1977], 05) | |
| A reaction: [compressed] He uses this to justify Israeli pre-emptive strikes against Palestinians. I don't think his confident assertion of this principle is justified. It is open to massive abuse. There are, though, clearly situations where he is right. |
| 23572 | Just wars are self-defence, or a rightful intercession in another's troubles [Walzer] |
| Full Idea: Just wars may not be self-defence, if they are to help an independence struggle, or it is to save another country being invaded, or to prevent enslavement or massacre. | |
| From: Michael Walzer (Just and Unjust Wars [1977], 06) | |
| A reaction: [summary] Modern wars support some examples of these, but also suggest that without a long-term plan, or an understanding of the country they are entering, such intercessions may worsen the situation. |
| 23581 | The aim of reprisals is to enforce the rules of war [Walzer] |
| Full Idea: The purpose of reprisals is not to win the war or prevent defeat, but simply to enforce the rules [of war]. | |
| From: Michael Walzer (Just and Unjust Wars [1977], 13) | |
| A reaction: That may be wishful thinking, since reprisals are often vastly more ruthless than the original offence, and there is often injustice in the nature of the reprisals, since they cannot be precise. |
| 23582 | Reprisal is defensible, as an alternative to war [Walzer] |
| Full Idea: Reprisal is the first resort of force. It is an alternative to war, and that description is an important argument in its favour. | |
| From: Michael Walzer (Just and Unjust Wars [1977], 13) | |
| A reaction: Enduring wrongs with dignity might be another alternative. Successful reprisals may be acceptable, but how do you assess their prospects? |
| 23588 | With nuclear weapons we have a permanent supreme emergency (which is unstable) [Walzer] |
| Full Idea: With nuclear weapons, supreme emergency has become a permanent condition. …[283] But supreme emergency is never a stable position. | |
| From: Michael Walzer (Just and Unjust Wars [1977], 17) | |
| A reaction: The obvious instability of balanced mutual threat is a nuclear state which finds itself losing a war. |
| 23580 | States need not endure attacks passively, and successful reprisals are legitimate [Walzer] |
| Full Idea: Whenever there is some substantial chance of success, reprisals are the legitimate resort of a victim state; for no state can be required passively to endure attacks upon its citizens. | |
| From: Michael Walzer (Just and Unjust Wars [1977], 13) | |
| A reaction: My concern is whether the reprisals have any direct connection to the attacks. They killed some of ours, so we will kill some of theirs is immoral. E.g. bombing Tripoli as reprisal for crashing the Lockerbie plane. |
| 23567 | Even non-violent intrusive acts between states count as aggression, if they justify resistance [Walzer] |
| Full Idea: Every violation of an independent state is called aggression, which fails to differentiate between a seizure or imposition, and an actual conquest. …But what they have in common is that all aggressive acts justify forceful resistance. | |
| From: Michael Walzer (Just and Unjust Wars [1977], 04) | |
| A reaction: [compressed] Walzer concedes that this makes 'aggression' rather imprecise, and small acts can be used as an excuse for desired violent resistance. Each entrant in August 1914 seems to have had a slightly different motive. |
| 23570 | The only good reason for fighting is in defence of rights [Walzer] |
| Full Idea: The defence of rights is a reason for fighting. I want now to stress again, and finally, that it is the only reason. | |
| From: Michael Walzer (Just and Unjust Wars [1977], 04) | |
| A reaction: Walzer states at the beginning, without discussion, that his moral assumptions are based on the notion of rights. This is tricky because rights are assigned by some people to other people, and claims of rights can be challenged. |
| 23587 | Nuclear bombs are not for normal war; they undermine the 'just war', with a new morality [Walzer] |
| Full Idea: Nuclear weapons are not designed for war at all. …They explode the idea of a just war. They are the first technological innovations that are simply not encompassable within the familiar moral world. | |
| From: Michael Walzer (Just and Unjust Wars [1977], 17) | |
| A reaction: A nuclear war can hardly lead to normal victory, if it destroys the thing you are trying to conquer. It is like bringing a machine gun to a boxing match. |
| 23573 | For moral reasons, a just war must be a limited war [Walzer] |
| Full Idea: Just wars are limited wars; there are moral reasons for the statesmen and soldiers who fight them to be prudent and realistic. | |
| From: Michael Walzer (Just and Unjust Wars [1977], 07) | |
| A reaction: This is rather profound, I think. Watch closely the behaviour of the good guys when they are winning the war. In general, to know someone's moral principles, the best indicator is how they behave when they have power. |
| 23577 | Napoleon said 'I don't care about the deaths of a million men' [Walzer] |
| Full Idea: Napoleon said 'Soldiers are made to be killed. …I do not care a fig for the lives of a million men'. | |
| From: Michael Walzer (Just and Unjust Wars [1977], 08) | |
| A reaction: [Two separate remarks attributed to Napoleon] He apparently often said things like this this later in his career. It strikes me as despicable, and anyone who still tries to present Napoleon as admirable should be ashamed. |
| 23593 | Jus ad bellum and Jus in bello are independent; unjust wars can be fought in a just way [Walzer] |
| Full Idea: Justice of war [ad bellum] and justice in war [in bello] are logically independent. It is perfectly possible for a just war to be fought unjustly, and for an unjust war to be fought in strict accordance with the rules. | |
| From: Michael Walzer (Just and Unjust Wars [1977], 02) | |
| A reaction: The perfect decorum of an unjust firing squad might even make the crime worse. There is something chilling about an evil army conducting itself perfectly within the ethics of warfare. Better than the other thing, though. McMahan disagrees. |
| 23574 | The duties and moral status of loyal and obedient soldiers is the same in defence and aggression [Walzer] |
| Full Idea: The duties of individual soldiers …are precisely the same in wars of aggression and wars of defence. …The moral status of soldiers on both sides is very much the same; they are led to fight by their loyalty and their lawful obedience. | |
| From: Michael Walzer (Just and Unjust Wars [1977], 08) | |
| A reaction: He excludes war crimes. This is the thesis which Jeff McMahan objects to. It would be very odd to think that mafiosi and the legitimate police were morally equal, because the former are loyal. We should all try hard to avoid supporting unjust causes. |
| 23575 | We can't blame soldiers for anything they do which clearly promotes victory [Walzer] |
| Full Idea: It would be difficult to condemn soldiers for anything they did in the course of a battle or a war that they honestly believed, and had good reason to believe, was necessary, or important, or simply useful in determining the outcome. | |
| From: Michael Walzer (Just and Unjust Wars [1977], 08) | |
| A reaction: We can't blame unjust aggressors if their own lives are at stake, but what about in a surprise attack on the first day of the war (such as Pearl Harbour)? Or if they massacre the enemy with safe and overwhelming superiority? |
| 23584 | Rejecting Combatant Equality allows just soldiers to be harsher, even to the extreme [Walzer] |
| Full Idea: Objections to combatant equality appeal to a sliding scale of 'the more justice, the more right'. …It allows the justice of one's cause to make a difference in the way one fights. …The extreme says soldiers fightly justly can do anything that is useful. | |
| From: Michael Walzer (Just and Unjust Wars [1977], 14) | |
| A reaction: This slippery slope fear seems to be Walzer's main argument in favour of the moral equality of combatants. See Jeff McMahan for the opposing view. |
| 23614 | Even aggressor soldiers are not criminals, so they have equal rights with their opponents [Walzer] |
| Full Idea: Soldiers fighting for an aggressor state are not themselves criminals: hence their war rights are the same as those of their opponents. | |
| From: Michael Walzer (Just and Unjust Wars [1977], 08) | |
| A reaction: Walzer's main support for this is that opposing armies never regard one another as intrinsically criminal. It seems inevitable, though, that even the invaders themselves see that they are a bit more criminal than the defenders. |
| 23589 | Kidnapped sailors and volunteers have different obligations to the passengers [Walzer] |
| Full Idea: Soldiers may stand to civilians like the crew of a liner to its passengers, for whom they must risk their lives. …But would they be so bound if the sailors had been kidnapped? | |
| From: Michael Walzer (Just and Unjust Wars [1977], 19) | |
| A reaction: The point, I assume, is that a conscripted army does not have the same obligations as volunteers. I can't imagine that principle being accepted in an army which is a mixture of the two. |
| 23579 | Soldiers will only protect civilians if they feel safe from them [Walzer] |
| Full Idea: Soldiers must feel safe among civilians if civilians are ever to be safe from soldiers. | |
| From: Michael Walzer (Just and Unjust Wars [1977], 11) | |
| A reaction: This is the great dilemma of any resistance movement. It is very easy for the soldiers to abuse their power, even if they do feel safe. Then what? |
| 23586 | What matters in war is unacceptable targets, not unacceptable weapons [Walzer] |
| Full Idea: The crucial distinction in the theory and practice of war is not between prohibited and acceptable weapons but between prohibited and acceptable targets. | |
| From: Michael Walzer (Just and Unjust Wars [1977], 17) | |
| A reaction: Walzer presents this idea as arising out of discussions about nuclear deterrence. Gas attacks were accepted in WW1 trenches, but modern gas attacks on civilians are a crime. Are nuclear attacks on strictly military targets OK? E.g a fleet. |
| 23591 | If the oppressor is cruel, nonviolence is either surrender, or a mere gesture [Walzer] |
| Full Idea: When one cannot count on the moral code of the oppressor, nonviolence is either a disguised form of surrender or a minimalist way of upholding communal values after a military defeat. | |
| From: Michael Walzer (Just and Unjust Wars [1977], Afterword) | |
| A reaction: The point is that ruthless conquerors may just kill the nonviolent, so it would achieve nothing. Nonviolence is only a plausible strategy in a fairly civilised world. Hard to disagree. |
| 23592 | We can only lead war towards peace if we firmly enforce the rules of war [Walzer] |
| Full Idea: We must begin by insisting upon the rules of war and by holding soldiers rigidly to the norms they set. The restraint of war is the beginning of peace. | |
| From: Michael Walzer (Just and Unjust Wars [1977], Afterword) | |
| A reaction: Last sentence of his book. Some cultures have a much greater tradition of ruthless cruelty than others, it seems. Most war ethics seems to concern how the good guys should respond to the bad guys (since the latter hardly care). |