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All the ideas for 'works', 'Killing in War' and 'A Tour through Mathematical Logic'

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44 ideas

4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL
A 'tautology' must include connectives [Wolf,RS]
     Full Idea: 'For every number x, x = x' is not a tautology, because it includes no connectives.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.2)
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL
Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS]
     Full Idea: Deduction Theorem: If T ∪ {P} |- Q, then T |- (P → Q). This is the formal justification of the method of conditional proof (CPP). Its converse holds, and is essentially modus ponens.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3)
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / d. Universal quantifier ∀
Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS]
     Full Idea: Universal Generalization: If we can prove P(x), only assuming what sort of object x is, we may conclude ∀xP(x) for the same x.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3)
     A reaction: This principle needs watching closely. If you pick one person in London, with no presuppositions, and it happens to be a woman, can you conclude that all the people in London are women? Fine in logic and mathematics, suspect in life.
Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS]
     Full Idea: Universal Specification: from ∀xP(x) we may conclude P(t), where t is an appropriate term. If something is true for all members of a domain, then it is true for some particular one that we specify.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3)
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / e. Existential quantifier ∃
Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS]
     Full Idea: Existential Generalization (or 'proof by example'): From P(t), where t is an appropriate term, we may conclude ∃xP(x).
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3)
     A reaction: It is amazing how often this vacuous-sounding principles finds itself being employed in discussions of ontology, but I don't quite understand why.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / e. Axiom of the Empty Set IV
Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS]
     Full Idea: Empty Set Axiom: ∃x ∀y ¬ (y ∈ x). There is a set x which has no members (no y's). The empty set exists. There is a set with no members, and by extensionality this set is unique.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.3)
     A reaction: A bit bewildering for novices. It says there is a box with nothing in it, or a pair of curly brackets with nothing between them. It seems to be the key idea in set theory, because it asserts the idea of a set over and above any possible members.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / n. Axiom of Comprehension
Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS]
     Full Idea: The comprehension axiom says that any collection of objects that can be clearly specified can be considered to be a set.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.2)
     A reaction: This is virtually tautological, since I presume that 'clearly specified' means pinning down exact which items are the members, which is what a set is (by extensionality). The naïve version is, of course, not so hot.
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS]
     Full Idea: One of the most appealing features of first-order logic is that the two 'turnstiles' (the syntactic single |-, and the semantic double |=), which are the two reasonable notions of logical consequence, actually coincide.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.3)
     A reaction: In the excitement about the possibility of second-order logic, plural quantification etc., it seems easy to forget the virtues of the basic system that is the target of the rebellion. The issue is how much can be 'expressed' in first-order logic.
First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS]
     Full Idea: The 'completeness' of first order-logic does not mean that every sentence or its negation is provable in first-order logic. We have instead the weaker result that every valid sentence is provable.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.3)
     A reaction: Peter Smith calls the stronger version 'negation completeness'.
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
Model theory reveals the structures of mathematics [Wolf,RS]
     Full Idea: Model theory helps one to understand what it takes to specify a mathematical structure uniquely.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.1)
     A reaction: Thus it is the development of model theory which has led to the 'structuralist' view of mathematics.
Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS]
     Full Idea: A 'structure' in model theory has a non-empty set, the 'universe', as domain of variables, a subset for each 'relation', some 'functions', and 'constants'.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.2)
Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS]
     Full Idea: Model theory uses set theory to show that the theorem-proving power of the usual methods of deduction in mathematics corresponds perfectly to what must be true in actual mathematical structures.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], Pref)
     A reaction: That more or less says that model theory demonstrates the 'soundness' of mathematics (though normal arithmetic is famously not 'complete'). Of course, he says they 'correspond' to the truths, rather than entailing them.
First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS]
     Full Idea: The three foundations of first-order model theory are the Completeness theorem, the Compactness theorem, and the Löwenheim-Skolem-Tarski theorem.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.3)
     A reaction: On p.180 he notes that Compactness and LST make no mention of |- and are purely semantic, where Completeness shows the equivalence of |- and |=. All three fail for second-order logic (p.223).
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS]
     Full Idea: An 'isomorphism' is a bijection between two sets that preserves all structural components. The interpretations of each constant symbol are mapped across, and functions map the relation and function symbols.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.4)
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
The LST Theorem is a serious limitation of first-order logic [Wolf,RS]
     Full Idea: The Löwenheim-Skolem-Tarski theorem demonstrates a serious limitation of first-order logic, and is one of primary reasons for considering stronger logics.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.7)
5. Theory of Logic / K. Features of Logics / 4. Completeness
If a theory is complete, only a more powerful language can strengthen it [Wolf,RS]
     Full Idea: It is valuable to know that a theory is complete, because then we know it cannot be strengthened without passing to a more powerful language.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.5)
5. Theory of Logic / K. Features of Logics / 10. Monotonicity
Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS]
     Full Idea: Deductive logic, including first-order logic and other types of logic used in mathematics, is 'monotonic'. This means that we never retract a theorem on the basis of new givens. If T|-φ and T⊆SW, then S|-φ. Ordinary reasoning is nonmonotonic.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.7)
     A reaction: The classic example of nonmonotonic reasoning is the induction that 'all birds can fly', which is retracted when the bird turns out to be a penguin. He says nonmonotonic logic is a rich field in computer science.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS]
     Full Idea: Less theoretically, an ordinal is an equivalence class of well-orderings. Formally, we say a set is 'transitive' if every member of it is a subset of it, and an ordinal is a transitive set, all of whose members are transitive.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.4)
     A reaction: He glosses 'transitive' as 'every member of a member of it is a member of it'. So it's membership all the way down. This is the von Neumann rather than the Zermelo approach (which is based on singletons).
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Modern mathematics has unified all of its objects within set theory [Wolf,RS]
     Full Idea: One of the great achievements of modern mathematics has been the unification of its many types of objects. It began with showing geometric objects numerically or algebraically, and culminated with set theory representing all the normal objects.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], Pref)
     A reaction: His use of the word 'object' begs all sorts of questions, if you are arriving from the street, where an object is something which can cause a bruise - but get used to it, because the word 'object' has been borrowed for new uses.
20. Action / C. Motives for Action / 4. Responsibility for Actions
Legal excuses are duress, ignorance, and diminished responsibility [McMahan]
     Full Idea: The common legal practice is to distinguish three broad categories of excuse: duress, epistemic limitation, and diminished responsibility.
     From: Jeff McMahan (Killing in War [2009], 3.2.1)
     A reaction: McMahan cites these with reference to soldiers in wartime, but they have general application. The third one seems particularly open to very wide interpretation. Presumably I can't be excused by just being irresponsible.
25. Social Practice / C. Rights / 1. Basis of Rights
Liberty Rights are permissions, and Claim Rights are freedom from intervention [McMahan]
     Full Idea: There are two types of right. A Liberty right is merely a permission, meaning it is not wrong to do it. But a Claim right is a right against intervention, meaning no one has a liberty right to prevent it.
     From: Jeff McMahan (Killing in War [2009], 2.3)
     A reaction: There must also be a third type of right, which requires other people to perform actions on your behalf. If you pay for a book in a shop, you must then be given the book.
25. Social Practice / E. Policies / 1. War / a. Just wars
A person or state may be attacked if they are responsible for an unjustified threat [McMahan]
     Full Idea: It is a necessary condition of liability to defensive attack that one be morally responsible for posing an objectively unjustified threat.
     From: Jeff McMahan (Killing in War [2009], 4.1.1)
     A reaction: This implies that one may not actually be doing the threatening (but merely ordering it, or enabling it). McMahan aims to have the same criteria for wartime as for peacetime. He denies Anscombe's claim that merely posing the threat is enough.
You (e.g. a police officer) are not liable to attack just because you pose a threat [McMahan]
     Full Idea: It is false that by posing a threat to another, one necessarily makes oneself liable to defensive action. A police officer who shoots an active murderer does not thereby by make herself liable to defensive action.
     From: Jeff McMahan (Killing in War [2009], 1.2)
     A reaction: This is one of his arguments against the moral equality of combatants. It is not morally OK to shoot all the local soldiers when you unjustly invade a territory. Sounds right to me.
Wars can be unjust, despite a just cause, if they are unnecessary or excessive or of mixed cause [McMahan]
     Full Idea: Wars can be unjust despite having a just cause, because they are not actually needed, or they will cause excessive harm, or they also pursue some unjust causes.
     From: Jeff McMahan (Killing in War [2009], 1.1)
     A reaction: [compressed] The point is that older writers often think that a 'just cause' is sufficient. He is obviously right.
Just war theory says all and only persons posing a threat are liable to attack [McMahan]
     Full Idea: In mainstream just war theory (Anscombe, Nagel, Walzer) the criterion of liability to attack is simply posing a threat. Since all combatants pose a threat to each other, they are morally liable to attack; because noncombatants do not, they are not liable.
     From: Jeff McMahan (Killing in War [2009], 1.2)
     A reaction: McMahan says that the distinction between legitimate and illegitimate targets rests mostly on this basis. The problem is that a huge range of unarmed people can also pose various degrees of threat.
The worst unjustified wars have no aim at all [McMahan]
     Full Idea: The most serious reason why a war might be unjustified is that it lacks any justifying aim at all.
     From: Jeff McMahan (Killing in War [2009], 1.1)
     A reaction: It seems that Louis XIV invaded the Netherlands in around 1674 purely to enhance his own glory. That strikes me as worse. I supposed Ghenghis Khan invaded places simply because he enjoyed fighting.
A defensive war is unjust, if it is responding to a just war [McMahan]
     Full Idea: It is possible for a defensive war to be unjust, when the defensive war to which it is a response is a just war.
     From: Jeff McMahan (Killing in War [2009], 3.3.3)
     A reaction: An example might be a state resisting an intervention from outside, when the state is in the process of exterminating some unwanted minority. Or perhaps the invaders are crossing the state's territory to achieve some admirable end.
25. Social Practice / E. Policies / 1. War / b. Justice in war
Proportionality in fighting can't be judged independently of the justice of each side [McMahan]
     Full Idea: There is simply no satisfactory understanding of proportionality in war that can be applied independently of whether the acts that are evaluated support a just or an unjust cause.
     From: Jeff McMahan (Killing in War [2009], 1.3)
     A reaction: He rejects traditional just war theory, which sees both sides as morally equal in combat, and hence equally subject to the principles of proportional response. But the just can then be harsher, when their just principles should make them milder.
Can an army start an unjust war, and then fight justly to defend their own civilians? [McMahan]
     Full Idea: There is a paradox if the unjust are justified in fighting the just in order to protect their own civilians who have been endangered by the starting of an unjust war.
     From: Jeff McMahan (Killing in War [2009], 2.1)
     A reaction: [my summary of MacMahan pp.48-49] It suggests that in a war there may be local concepts of justice which are at odds with the general situation - which is the ad bellum/in bello distinction. But this is the justice of fighting, not how it is conducted.
Soldiers cannot freely fight in unjust wars, just because they behave well when fighting [McMahan]
     Full Idea: We must stop reassuring soldiers that they act permissibly when they fight in an unjust war, provided that they conduct themselve honorably on the battlefield by fighting in accordance with the rules of engagement.
     From: Jeff McMahan (Killing in War [2009], 2.8)
     A reaction: This culminates McMahan's arguments against the moral equality of combatants, and against the sharp division of justice of war from justice in war. How rare it is for philosophy to culminate in a policy recommendation!
The law of war differs from criminal law; attacking just combatants is immoral, but legal [McMahan]
     Full Idea: Unlike domestic criminal law, the law of war is designed not to protect moral rights but to prevent harm. …This means when unjust combatants attack just combatants they violate their moral rights, yet they act within their legal rights.
     From: Jeff McMahan (Killing in War [2009], 3.1.1)
     A reaction: He says we must bring the law of war much closer to the morality of war. If there is any hope of slowly eliminating war, it may lie in reforms such as these.
If the unjust combatants are morally excused they are innocent, so how can they be killed? [McMahan]
     Full Idea: If most unjust combatants are morally innocent because they are excused, and if it is wrong to intentionally kill morally innocent people, then a contingent form of pacificism may be inescapable.
     From: Jeff McMahan (Killing in War [2009], 3.3.1)
     A reaction: A very nice argument against the moral equality of combatants. If I think we are the good guys, and the opposing troops are no morally different from us, how can I possibly kill them?
25. Social Practice / E. Policies / 1. War / c. Combatants
You don't become a legitimate target, just because you violently resist an unjust attack [McMahan]
     Full Idea: It is hard to see how just combatants could become legitimate targets simply by offering violent resistance to unjust attacks by unjust coombatants.
     From: Jeff McMahan (Killing in War [2009], 1.3)
     A reaction: It is, however, hard to criticise a soldier who is dragged into fighting for an unjust cause, and then kills just defenders in the course of the fight. Once the bullets fly, normal morality seems to be suspended. Just survive.
If all combatants are seen as morally equal, that facilitates starting unjust wars [McMahan]
     Full Idea: It would be naïve to doubt that the widespread acceptance of the moral equality of combatants has facilitated the ability of governments to fight unjust wars.
     From: Jeff McMahan (Killing in War [2009], 1.1)
     A reaction: The point is that their armies are both compliant and seeing their actions as guiltless, which makes them perfect tools for evil. McMahan's ideal is an army which asks sharp questions about the justification of the war, before they fight it.
Volunteer soldiers accept the risk of attack, but they don't agree to it, or to their deaths [McMahan]
     Full Idea: When soldiers go to war, they undoubtedly assume a certain risk. They voluntarily expose themselves to a significant risk of being attacked. But this is entirely different from consenting to being attacked.
     From: Jeff McMahan (Killing in War [2009], 2.2.1)
     A reaction: This is his response to Walzer's thought that soldiers resemble people who volunteer for a boxing match. The sailors at Pearl Harbour obviously didn't consent to the attack, or accept the Japanese right to kill them.
If being part of a big collective relieves soldiers of moral responsibility, why not the leaders too? [McMahan]
     Full Idea: If acting as an agent of a political collective justifies the combatants fighting an unjust war, that should also release the leaders from responsibility for their role in the fighting of that war. No one ever explains why this is not so.
     From: Jeff McMahan (Killing in War [2009], 2.5)
     A reaction: At the very least there seems to be a problem of the cut off point between innocent soldiers and culpable leaders. Which rank in the army or executive triggers the blame?
If soldiers can't refuse to fight in unjust wars, can they choose to fight in just wars? [McMahan]
     Full Idea: There is a certain symmetry here. The permissibility of disobeying a command to fight in an unjust war suggests the permissibility of disobeying a command not to fight in a just war.
     From: Jeff McMahan (Killing in War [2009], 2.7)
     A reaction: The argument considered here is that since we could never allow soldiers to choose to fight in their own wars, we similarly cannot let them opt out of the official wars. Implying obedience is absolute. Soldiers don't get to 'choose' anything!
Equality is both sides have permission, or both sides are justified, or one justified the other permitted [McMahan]
     Full Idea: Moral equality means either 1) because just combatants are permitted to fight in a just way, so are the unjust , or 2) because the just are justified, so are the unjust, or 3) because the just are justified, the unjust are therefore permitted.
     From: Jeff McMahan (Killing in War [2009], 3.1.2)
     A reaction: [summary] McMahan calls 1) the weak version, and 2) the strong. He suggests that although 3) is unusual, it is what most people believe - that if the good are justified, the bad are permitted to fight back. He rejects them all.
Fighting unjustly under duress does not justify it, or permit it, but it may excuse it [McMahan]
     Full Idea: It is said that combatants are compelled to fight; they have no choice. But duress is not a justification; nor does it ground a permission - not even a subjective permission. It is, instead, an excusing condition.
     From: Jeff McMahan (Killing in War [2009], 3.1.2)
     A reaction: The 'subjective' permission is believing you are just, even if you aren't. A nice, accurate and true distinction made by McMahan, I think. It is roughly our postwar attitude to the Nazi army.
Soldiers cannot know enough facts to evaluate the justice of their war [McMahan]
     Full Idea: When soldiers are commanded to fight, they cannot reasonably be expected to have the factual knowledge necessary to evaluate the war as just or unjust.
     From: Jeff McMahan (Killing in War [2009], 2.3)
     A reaction: This is part of the 'epistemic' justification for a soldier to fight in an unjust war. Sometimes soldiers do have enoough knowledge, especially if they join up late on in a war, when they have studied and observed its progress.
25. Social Practice / E. Policies / 1. War / d. Non-combatants
Innocence implies not being morally responsible, rather than merely being guiltless [McMahan]
     Full Idea: My alternative conception is that one is 'innocent' if one is neither morally responsible for nor guilty of a wrong. Classical theory focused on guilt, but I think we should focus on moral responsibility (which is something less).
     From: Jeff McMahan (Killing in War [2009], 1.4)
     A reaction: This seems to make the supporters of evil equally liable to attack with its perpetrators. But you can observe perpetration a lot more easily than you can observe support.
25. Social Practice / E. Policies / 1. War / e. Peace
Unconditional surrender can't be demanded, since evil losers still have legitimate conditions [McMahan]
     Full Idea: Achieving unconditional surrender can never be a justification for the continuation of war, since there are always conditions that a vanquished adversary, no matter how evil, can be justified in demanding.
     From: Jeff McMahan (Killing in War [2009], 3.3.1)
     A reaction: McMahan is particularly discussing Hiroshima, but this also applies to the European war in 1945. Presumably a civilised victor will grant the conditions which the losers would have demanded, and that probably happened in 1945. It's about power.
29. Religion / B. Monotheistic Religion / 4. Christianity / d. Heresy
Philosophers are the forefathers of heretics [Tertullian]
     Full Idea: Philosophers are the forefathers of heretics.
     From: Tertullian (works [c.200]), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 20.2
29. Religion / D. Religious Issues / 1. Religious Commitment / e. Fideism
I believe because it is absurd [Tertullian]
     Full Idea: I believe because it is absurd ('Credo quia absurdum est').
     From: Tertullian (works [c.200]), quoted by Robert Fogelin - Walking the Tightrope of Reason n4.2
     A reaction: This seems to be a rather desperate remark, in response to what must have been rather good hostile arguments. No one would abandon the support of reason if it was easy to acquire. You can't deny its engaging romantic defiance, though.