Combining Texts

All the ideas for 'Parmenides', 'Second Treatise of Government' and 'Intermediate Logic'

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158 ideas

2. Reason / A. Nature of Reason / 1. On Reason
224 When questions are doubtful we should concentrate not on objects but on ideas of the intellect [Plato]
     Full Idea: Doubtful questions should not be discussed in terms of visible objects or in relation to them, but only with reference to ideas conceived by the intellect.
     From: Plato (Parmenides [c.364 BCE], 135e)
2. Reason / B. Laws of Thought / 5. Opposites
232 Opposites are as unlike as possible [Plato]
     Full Idea: Opposites are as unlike as possible.
     From: Plato (Parmenides [c.364 BCE], 159a)
2. Reason / C. Styles of Reason / 1. Dialectic
8937 Plato's 'Parmenides' is the greatest artistic achievement of the ancient dialectic [Hegel on Plato]
     Full Idea: Plato's 'Parmenides' is the greatest artistic achievement of the ancient dialectic.
     From: comment on Plato (Parmenides [c.364 BCE]) by Georg W.F.Hegel - Phenomenology of Spirit Pref 71
     A reaction: It is a long way from the analytic tradition of philosophy to be singling out a classic text for its 'artistic' achievement. Eventually we may even look back on, say, Kripke's 'Naming and Necessity' and see it in that light.
4. Formal Logic / A. Syllogistic Logic / 2. Syllogistic Logic
13439 Venn Diagrams map three predicates into eight compartments, then look for the conclusion [Bostock]
     Full Idea: Venn Diagrams are a traditional method to test validity of syllogisms. There are three interlocking circles, one for each predicate, thus dividing the universe into eight possible basic elementary quantifications. Is the conclusion in a compartment?
     From: David Bostock (Intermediate Logic [1997], 3.8)
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL
13421 'Disjunctive Normal Form' is ensuring that no conjunction has a disjunction within its scope [Bostock]
     Full Idea: 'Disjunctive Normal Form' (DNF) is rearranging the occurrences of ∧ and ∨ so that no conjunction sign has any disjunction in its scope. This is achieved by applying two of the distribution laws.
     From: David Bostock (Intermediate Logic [1997], 2.6)
13422 'Conjunctive Normal Form' is ensuring that no disjunction has a conjunction within its scope [Bostock]
     Full Idea: 'Conjunctive Normal Form' (CNF) is rearranging the occurrences of ∧ and ∨ so that no disjunction sign has any conjunction in its scope. This is achieved by applying two of the distribution laws.
     From: David Bostock (Intermediate Logic [1997], 2.6)
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL
13355 'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,ψ|= [Bostock]
     Full Idea: The Principle of Disjunction says that Γ,φ∨ψ |= iff Γ,φ |= and Γ,ψ |=.
     From: David Bostock (Intermediate Logic [1997], 2.5.G)
     A reaction: That is, a disjunction leads to a contradiction if they each separately lead to contradictions.
13350 'Assumptions' says that a formula entails itself (φ|=φ) [Bostock]
     Full Idea: The Principle of Assumptions says that any formula entails itself, i.e. φ |= φ. The principle depends just upon the fact that no interpretation assigns both T and F to the same formula.
     From: David Bostock (Intermediate Logic [1997], 2.5.A)
     A reaction: Thus one can introduce φ |= φ into any proof, and then use it to build more complex sequents needed to attain a particular target formula. Bostock's principle is more general than anything in Lemmon.
13351 'Thinning' allows that if premisses entail a conclusion, then adding further premisses makes no difference [Bostock]
     Full Idea: The Principle of Thinning says that if a set of premisses entails a conclusion, then adding further premisses will still entail the conclusion. It is 'thinning' because it makes a weaker claim. If γ|=φ then γ,ψ|= φ.
     From: David Bostock (Intermediate Logic [1997], 2.5.B)
     A reaction: It is also called 'premise-packing'. It is the characteristic of a 'monotonic' logic - where once something is proved, it stays proved, whatever else is introduced.
13356 The 'conditional' is that Γ|=φ→ψ iff Γ,φ|=ψ [Bostock]
     Full Idea: The Conditional Principle says that Γ |= φ→ψ iff Γ,φ |= ψ. With the addition of negation, this implies φ,φ→ψ |= ψ, which is 'modus ponens'.
     From: David Bostock (Intermediate Logic [1997], 2.5.H)
     A reaction: [Second half is in Ex. 2.5.4]
13352 'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z [Bostock]
     Full Idea: The Principle of Cutting is the general point that entailment is transitive, extending this to cover entailments with more than one premiss. Thus if γ |= φ and φ,Δ |= ψ then γ,Δ |= ψ. Here φ has been 'cut out'.
     From: David Bostock (Intermediate Logic [1997], 2.5.C)
     A reaction: It might be called the Principle of Shortcutting, since you can get straight to the last conclusion, eliminating the intermediate step.
13353 'Negation' says that Γ,¬φ|= iff Γ|=φ [Bostock]
     Full Idea: The Principle of Negation says that Γ,¬φ |= iff Γ |= φ. We also say that φ,¬φ |=, and hence by 'thinning on the right' that φ,¬φ |= ψ, which is 'ex falso quodlibet'.
     From: David Bostock (Intermediate Logic [1997], 2.5.E)
     A reaction: That is, roughly, if the formula gives consistency, the negation gives contradiction. 'Ex falso' says that anything will follow from a contradiction.
13354 'Conjunction' says that Γ|=φ∧ψ iff Γ|=φ and Γ|=ψ [Bostock]
     Full Idea: The Principle of Conjunction says that Γ |= φ∧ψ iff Γ |= φ and Γ |= ψ. This implies φ,ψ |= φ∧ψ, which is ∧-introduction. It is also implies ∧-elimination.
     From: David Bostock (Intermediate Logic [1997], 2.5.F)
     A reaction: [Second half is Ex. 2.5.3] That is, if they are entailed separately, they are entailed as a unit. It is a moot point whether these principles are theorems of propositional logic, or derivation rules.
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL
13610 A logic with ¬ and → needs three axiom-schemas and one rule as foundation [Bostock]
     Full Idea: For ¬,→ Schemas: (A1) |-φ→(ψ→φ), (A2) |-(φ→(ψ→ξ)) → ((φ→ψ)→(φ→ξ)), (A3) |-(¬φ→¬ψ) → (ψ→φ), Rule:DET:|-φ,|-φ→ψ then |-ψ
     From: David Bostock (Intermediate Logic [1997], 5.2)
     A reaction: A1 says everything implies a truth, A2 is conditional proof, and A3 is contraposition. DET is modus ponens. This is Bostock's compact near-minimal axiom system for proposition logic. He adds two axioms and another rule for predicate logic.
4. Formal Logic / E. Nonclassical Logics / 6. Free Logic
13846 A 'free' logic can have empty names, and a 'universally free' logic can have empty domains [Bostock]
     Full Idea: A 'free' logic is one in which names are permitted to be empty. A 'universally free' logic is one in which the domain of an interpretation may also be empty.
     From: David Bostock (Intermediate Logic [1997], 8.6)
5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
13346 Truth is the basic notion in classical logic [Bostock]
     Full Idea: The most fundamental notion in classical logic is that of truth.
     From: David Bostock (Intermediate Logic [1997], 1.1)
     A reaction: The opening sentence of his book. Hence the first half of the book is about semantics, and only the second half deals with proof. Compare Idea 10282. The thought seems to be that you could leave out truth, but that makes logic pointless.
13545 Elementary logic cannot distinguish clearly between the finite and the infinite [Bostock]
     Full Idea: In very general terms, we cannot express the distinction between what is finite and what is infinite without moving essentially beyond the resources available in elementary logic.
     From: David Bostock (Intermediate Logic [1997], 4.8)
     A reaction: This observation concludes a discussion of Compactness in logic.
13822 Fictional characters wreck elementary logic, as they have contradictions and no excluded middle [Bostock]
     Full Idea: Discourse about fictional characters leads to a breakdown of elementary logic. We accept P or ¬P if the relevant story says so, but P∨¬P will not be true if the relevant story says nothing either way, and P∧¬P is true if the story is inconsistent.
     From: David Bostock (Intermediate Logic [1997], 8.5)
     A reaction: I really like this. Does one need to invent a completely new logic for fictional characters? Or must their logic be intuitionist, or paraconsistent, or both?
5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-
13623 The syntactic turnstile |- φ means 'there is a proof of φ' or 'φ is a theorem' [Bostock]
     Full Idea: The syntactic turnstile |- φ means 'There is a proof of φ' (in the system currently being considered). Another way of saying the same thing is 'φ is a theorem'.
     From: David Bostock (Intermediate Logic [1997], 5.1)
5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
13347 Validity is a conclusion following for premises, even if there is no proof [Bostock]
     Full Idea: The classical definition of validity counts an argument as valid if and only if the conclusion does in fact follow from the premises, whether or not the argument contains any demonstration of this fact.
     From: David Bostock (Intermediate Logic [1997], 1.2)
     A reaction: Hence validity is given by |= rather than by |-. A common example is 'it is red so it is coloured', which seems true but beyond proof. In the absence of formal proof, you wonder whether validity is merely a psychological notion.
13348 It seems more natural to express |= as 'therefore', rather than 'entails' [Bostock]
     Full Idea: In practice we avoid quotation marks and explicitly set-theoretic notation that explaining |= as 'entails' appears to demand. Hence it seems more natural to explain |= as simply representing the word 'therefore'.
     From: David Bostock (Intermediate Logic [1997], 1.3)
     A reaction: Not sure I quite understand that, but I have trained myself to say 'therefore' for the generic use of |=. In other consequences it seems better to read it as 'semantic consequence', to distinguish it from |-.
13349 Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid' [Bostock]
     Full Idea: If we write Γ |= φ, with one formula to the right, then the turnstile abbreviates 'entails'. For a sequent of the form Γ |= it can be read as 'is inconsistent'. For |= φ we read it as 'valid'.
     From: David Bostock (Intermediate Logic [1997], 1.3)
5. Theory of Logic / B. Logical Consequence / 5. Modus Ponens
13614 MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment) [Bostock]
     Full Idea: The Rule of Detachment is a version of Modus Ponens, and says 'If |=φ and |=φ→ψ then |=ψ'. This has no assumptions. Modus Ponens is the more general rule that 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ'.
     From: David Bostock (Intermediate Logic [1997], 5.3)
     A reaction: Modus Ponens is actually designed for use in proof based on assumptions (which isn't always the case). In Detachment the formulae are just valid, without dependence on assumptions to support them.
13617 MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ [Bostock]
     Full Idea: Modus Ponens is equivalent to the converse of the Deduction Theorem, namely 'If Γ |- φ→ψ then Γ,φ|-ψ'.
     From: David Bostock (Intermediate Logic [1997], 5.3)
     A reaction: See 13615 for details of the Deduction Theorem. See 13614 for Modus Ponens.
5. Theory of Logic / D. Assumptions for Logic / 4. Identity in Logic
13799 The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b) [Bostock]
     Full Idea: We shall use 'a=b' as short for 'a is the same thing as b'. The sign '=' thus expresses a particular two-place predicate. Officially we will use 'I' as the identity predicate, so that 'Iab' is as formula, but we normally 'abbreviate' this to 'a=b'.
     From: David Bostock (Intermediate Logic [1997], 8.1)
13800 |= α=α and α=β |= φ(α/ξ ↔ φ(β/ξ) fix identity [Bostock]
     Full Idea: We usually take these two principles together as the basic principles of identity: |= α=α and α=β |= φ(α/ξ) ↔ φ(β/ξ). The second (with scant regard for history) is known as Leibniz's Law.
     From: David Bostock (Intermediate Logic [1997], 8.1)
13803 If we are to express that there at least two things, we need identity [Bostock]
     Full Idea: To say that there is at least one thing x such that Fx we need only use an existential quantifier, but to say that there are at least two things we need identity as well.
     From: David Bostock (Intermediate Logic [1997], 8.1)
     A reaction: The only clear account I've found of why logic may need to be 'with identity'. Without it, you can only reason about one thing or all things. Presumably plural quantification no longer requires '='?
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
13357 Truth-functors are usually held to be defined by their truth-tables [Bostock]
     Full Idea: The usual view of the meaning of truth-functors is that each is defined by its own truth-table, independently of any other truth-functor.
     From: David Bostock (Intermediate Logic [1997], 2.7)
5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
13812 A 'zero-place' function just has a single value, so it is a name [Bostock]
     Full Idea: We can talk of a 'zero-place' function, which is a new-fangled name for a familiar item; it just has a single value, and so it has the same role as a name.
     From: David Bostock (Intermediate Logic [1997], 8.2)
13811 A 'total' function ranges over the whole domain, a 'partial' function over appropriate inputs [Bostock]
     Full Idea: Usually we allow that a function is defined for arguments of a suitable kind (a 'partial' function), but we can say that each function has one value for any object whatever, from the whole domain that our quantifiers range over (a 'total' function).
     From: David Bostock (Intermediate Logic [1997], 8.2)
     A reaction: He points out (p.338) that 'the father of..' is a functional expression, but it wouldn't normally take stones as input, so seems to be a partial function. But then it doesn't even take all male humans either. It only takes fathers!
5. Theory of Logic / F. Referring in Logic / 1. Naming / a. Names
13360 In logic, a name is just any expression which refers to a particular single object [Bostock]
     Full Idea: The important thing about a name, for logical purposes, is that it is used to make a singular reference to a particular object; ..we say that any expression too may be counted as a name, for our purposes, it it too performs the same job.
     From: David Bostock (Intermediate Logic [1997], 3.1)
     A reaction: He cites definite descriptions as the most notoriously difficult case, in deciding whether or not they function as names. I takes it as pretty obvious that sometimes they do and sometimes they don't (in ordinary usage).
5. Theory of Logic / F. Referring in Logic / 1. Naming / e. Empty names
13361 An expression is only a name if it succeeds in referring to a real object [Bostock]
     Full Idea: An expression is not counted as a name unless it succeeds in referring to an object, i.e. unless there really is an object to which it refers.
     From: David Bostock (Intermediate Logic [1997], 3.1)
     A reaction: His 'i.e.' makes the existence condition sound sufficient, but in ordinary language you don't succeed in referring to 'that man over there' just because he exists. In modal contexts we presumably refer to hypothetical objects (pace Lewis).
5. Theory of Logic / F. Referring in Logic / 2. Descriptions / b. Definite descriptions
13813 Definite descriptions don't always pick out one thing, as in denials of existence, or errors [Bostock]
     Full Idea: It is natural to suppose one only uses a definite description when one believes it describes only one thing, but exceptions are 'there is no such thing as the greatest prime number', or saying something false where the reference doesn't occur.
     From: David Bostock (Intermediate Logic [1997], 8.3)
13814 Definite desciptions resemble names, but can't actually be names, if they don't always refer [Bostock]
     Full Idea: Although a definite description looks like a complex name, and in many ways behaves like a name, still it cannot be a name if names must always refer to objects. Russell gave the first proposal for handling such expressions.
     From: David Bostock (Intermediate Logic [1997], 8.3)
     A reaction: I take the simple solution to be a pragmatic one, as roughly shown by Donnellan, that sometimes they are used exactly like names, and sometimes as something else. The same phrase can have both roles. Confusing for logicians. Tough.
13816 Because of scope problems, definite descriptions are best treated as quantifiers [Bostock]
     Full Idea: Because of the scope problem, it now seems better to 'parse' definition descriptions not as names but as quantifiers. 'The' is to be treated in the same category as acknowledged quantifiers like 'all' and 'some'. We write Ix - 'for the x such that..'.
     From: David Bostock (Intermediate Logic [1997], 8.3)
     A reaction: This seems intuitively rather good, since quantification in normal speech is much more sophisticated than the crude quantification of classical logic. But the fact is that they often function as names (but see Idea 13817).
13817 Definite descriptions are usually treated like names, and are just like them if they uniquely refer [Bostock]
     Full Idea: In practice, definite descriptions are for the most part treated as names, since this is by far the most convenient notation (even though they have scope). ..When a description is uniquely satisfied then it does behave like a name.
     From: David Bostock (Intermediate Logic [1997], 8.3)
     A reaction: Apparent names themselves have problems when they wander away from uniquely picking out one thing, as in 'John Doe'.
13848 We are only obliged to treat definite descriptions as non-names if only the former have scope [Bostock]
     Full Idea: If it is really true that definite descriptions have scopes whereas names do not, then Russell must be right to claim that definite descriptions are not names. If, however, this is not true, then it does no harm to treat descriptions as complex names.
     From: David Bostock (Intermediate Logic [1997], 8.8)
5. Theory of Logic / F. Referring in Logic / 2. Descriptions / c. Theory of definite descriptions
13815 Names do not have scope problems (e.g. in placing negation), but Russell's account does have that problem [Bostock]
     Full Idea: In orthodox logic names are not regarded as having scope (for example, in where a negation is placed), whereas on Russell's theory definite descriptions certainly do. Russell had his own way of dealing with this.
     From: David Bostock (Intermediate Logic [1997], 8.3)
5. Theory of Logic / G. Quantification / 1. Quantification
13438 'Prenex normal form' is all quantifiers at the beginning, out of the scope of truth-functors [Bostock]
     Full Idea: A formula is said to be in 'prenex normal form' (PNF) iff all its quantifiers occur in a block at the beginning, so that no quantifier is in the scope of any truth-functor.
     From: David Bostock (Intermediate Logic [1997], 3.7)
     A reaction: Bostock provides six equivalences which can be applied to manouevre any formula into prenex normal form. He proves that every formula can be arranged in PNF.
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
13818 If we allow empty domains, we must allow empty names [Bostock]
     Full Idea: We can show that if empty domains are permitted, then empty names must be permitted too.
     From: David Bostock (Intermediate Logic [1997], 8.4)
5. Theory of Logic / H. Proof Systems / 1. Proof Systems
13801 An 'informal proof' is in no particular system, and uses obvious steps and some ordinary English [Bostock]
     Full Idea: An 'informal proof' is not in any particular proof system. One may use any rule of proof that is 'sufficiently obvious', and there is quite a lot of ordinary English in the proof, explaining what is going on at each step.
     From: David Bostock (Intermediate Logic [1997], 8.1)
5. Theory of Logic / H. Proof Systems / 2. Axiomatic Proof
13619 Quantification adds two axiom-schemas and a new rule [Bostock]
     Full Idea: New axiom-schemas for quantifiers: (A4) |-∀ξφ → φ(α/ξ), (A5) |-∀ξ(ψ→φ) → (ψ→∀ξφ), plus the rule GEN: If |-φ the |-∀ξφ(ξ/α).
     From: David Bostock (Intermediate Logic [1997], 5.6)
     A reaction: This follows on from Idea 13610, where he laid out his three axioms and one rule for propositional (truth-functional) logic. This Idea plus 13610 make Bostock's proposed axiomatisation of first-order logic.
13622 Axiom systems from Frege, Russell, Church, Lukasiewicz, Tarski, Nicod, Kleene, Quine... [Bostock]
     Full Idea: Notably axiomatisations of first-order logic are by Frege (1879), Russell and Whitehead (1910), Church (1956), Lukasiewicz and Tarski (1930), Lukasiewicz (1936), Nicod (1917), Kleene (1952) and Quine (1951). Also Bostock (1997).
     From: David Bostock (Intermediate Logic [1997], 5.8)
     A reaction: My summary, from Bostock's appendix 5.8, which gives details of all of these nine systems. This nicely illustrates the status and nature of axiom systems, which have lost the absolute status they seemed to have in Euclid.
5. Theory of Logic / H. Proof Systems / 3. Proof from Assumptions
13615 'Conditonalised' inferences point to the Deduction Theorem: If Γ,φ|-ψ then Γ|-φ→ψ [Bostock]
     Full Idea: If a group of formulae prove a conclusion, we can 'conditionalize' this into a chain of separate inferences, which leads to the Deduction Theorem (or Conditional Proof), that 'If Γ,φ|-ψ then Γ|-φ→ψ'.
     From: David Bostock (Intermediate Logic [1997], 5.3)
     A reaction: This is the rule CP (Conditional Proof) which can be found in the rules for propositional logic I transcribed from Lemmon's book.
13620 Proof by Assumptions can always be reduced to Proof by Axioms, using the Deduction Theorem [Bostock]
     Full Idea: By repeated transformations using the Deduction Theorem, any proof from assumptions can be transformed into a fully conditionalized proof, which is then an axiomatic proof.
     From: David Bostock (Intermediate Logic [1997], 5.6)
     A reaction: Since proof using assumptions is perhaps the most standard proof system (e.g. used in Lemmon, for many years the standard book at Oxford University), the Deduction Theorem is crucial for giving it solid foundations.
13621 The Deduction Theorem and Reductio can 'discharge' assumptions - they aren't needed for the new truth [Bostock]
     Full Idea: Like the Deduction Theorem, one form of Reductio ad Absurdum (If Γ,φ|-[absurdity] then Γ|-¬φ) 'discharges' an assumption. Assume φ and obtain a contradiction, then we know ¬&phi, without assuming φ.
     From: David Bostock (Intermediate Logic [1997], 5.7)
     A reaction: Thus proofs from assumption either arrive at conditional truths, or at truths that are true irrespective of what was initially assumed.
13616 The Deduction Theorem greatly simplifies the search for proof [Bostock]
     Full Idea: Use of the Deduction Theorem greatly simplifies the search for proof (or more strictly, the task of showing that there is a proof).
     From: David Bostock (Intermediate Logic [1997], 5.3)
     A reaction: See 13615 for details of the Deduction Theorem. Bostock is referring to axiomatic proof, where it can be quite hard to decide which axioms are relevant. The Deduction Theorem enables the making of assumptions.
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
13753 Natural deduction takes proof from assumptions (with its rules) as basic, and axioms play no part [Bostock]
     Full Idea: Natural deduction takes the notion of proof from assumptions as a basic notion, ...so it will use rules for use in proofs from assumptions, and axioms (as traditionally understood) will have no role to play.
     From: David Bostock (Intermediate Logic [1997], 6.1)
     A reaction: The main rules are those for introduction and elimination of truth functors.
13755 Excluded middle is an introduction rule for negation, and ex falso quodlibet will eliminate it [Bostock]
     Full Idea: Many books take RAA (reductio) and DNE (double neg) as the natural deduction introduction- and elimination-rules for negation, but RAA is not a natural introduction rule. I prefer TND (tertium) and EFQ (ex falso) for ¬-introduction and -elimination.
     From: David Bostock (Intermediate Logic [1997], 6.2)
13758 In natural deduction we work from the premisses and the conclusion, hoping to meet in the middle [Bostock]
     Full Idea: When looking for a proof of a sequent, the best we can do in natural deduction is to work simultaneously in both directions, forward from the premisses, and back from the conclusion, and hope they will meet in the middle.
     From: David Bostock (Intermediate Logic [1997], 6.5)
13754 Natural deduction rules for → are the Deduction Theorem (→I) and Modus Ponens (→E) [Bostock]
     Full Idea: Natural deduction adopts for → as rules the Deduction Theorem and Modus Ponens, here called →I and →E. If ψ follows φ in the proof, we can write φ→ψ (→I). φ and φ→ψ permit ψ (→E).
     From: David Bostock (Intermediate Logic [1997], 6.2)
     A reaction: Natural deduction has this neat and appealing way of formally introducing or eliminating each connective, so that you know where you are, and you know what each one means.
5. Theory of Logic / H. Proof Systems / 5. Tableau Proof
13757 Unlike natural deduction, semantic tableaux have recipes for proving things [Bostock]
     Full Idea: With semantic tableaux there are recipes for proof-construction that we can operate, whereas with natural deduction there are not.
     From: David Bostock (Intermediate Logic [1997], 6.5)
13611 Tableau proofs use reduction - seeking an impossible consequence from an assumption [Bostock]
     Full Idea: A tableau proof is a proof by reduction ad absurdum. One begins with an assumption, and one develops the consequences of that assumption, seeking to derive an impossible consequence.
     From: David Bostock (Intermediate Logic [1997], 4.1)
13613 A completed open branch gives an interpretation which verifies those formulae [Bostock]
     Full Idea: An open branch in a completed tableau will always yield an interpretation that verifies every formula on the branch.
     From: David Bostock (Intermediate Logic [1997], 4.7)
     A reaction: In other words the open branch shows a model which seems to work (on the available information). Similarly a closed branch gives a model which won't work - a counterexample.
13612 Non-branching rules add lines, and branching rules need a split; a branch with a contradiction is 'closed' [Bostock]
     Full Idea: Rules for semantic tableaus are of two kinds - non-branching rules and branching rules. The first allow the addition of further lines, and the second requires splitting the branch. A branch which assigns contradictory values to a formula is 'closed'.
     From: David Bostock (Intermediate Logic [1997], 4.1)
     A reaction: [compressed] Thus 'and' stays on one branch, asserting both formulae, but 'or' splits, checking first one and then the other. A proof succeeds when all the branches are closed, showing that the initial assumption leads only to contradictions.
13761 In a tableau proof no sequence is established until the final branch is closed; hypotheses are explored [Bostock]
     Full Idea: In a tableau system no sequent is established until the final step of the proof, when the last branch closes, and until then we are simply exploring a hypothesis.
     From: David Bostock (Intermediate Logic [1997], 7.3)
     A reaction: This compares sharply with a sequence calculus, where every single step is a conclusive proof of something. So use tableaux for exploring proofs, and then sequence calculi for writing them up?
13756 A tree proof becomes too broad if its only rule is Modus Ponens [Bostock]
     Full Idea: When the only rule of inference is Modus Ponens, the branches of a tree proof soon spread too wide for comfort.
     From: David Bostock (Intermediate Logic [1997], 6.4)
13762 Tableau rules are all elimination rules, gradually shortening formulae [Bostock]
     Full Idea: In their original setting, all the tableau rules are elimination rules, allowing us to replace a longer formula by its shorter components.
     From: David Bostock (Intermediate Logic [1997], 7.3)
5. Theory of Logic / H. Proof Systems / 6. Sequent Calculi
13759 Each line of a sequent calculus is a conclusion of previous lines, each one explicitly recorded [Bostock]
     Full Idea: A sequent calculus keeps an explicit record of just what sequent is established at each point in a proof. Every line is itself the sequent proved at that point. It is not a linear sequence or array of formulae, but a matching array of whole sequents.
     From: David Bostock (Intermediate Logic [1997], 7.1)
13760 A sequent calculus is good for comparing proof systems [Bostock]
     Full Idea: A sequent calculus is a useful tool for comparing two systems that at first look utterly different (such as natural deduction and semantic tableaux).
     From: David Bostock (Intermediate Logic [1997], 7.2)
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
13364 Interpretation by assigning objects to names, or assigning them to variables first [Bostock, by PG]
     Full Idea: There are two approaches to an 'interpretation' of a logic: the first method assigns objects to names, and then defines connectives and quantifiers, focusing on truth; the second assigns objects to variables, then variables to names, using satisfaction.
     From: report of David Bostock (Intermediate Logic [1997], 3.4) by PG - Db (lexicon)
     A reaction: [a summary of nine elusive pages in Bostock] He says he prefers the first method, but the second method is more popular because it handles open formulas, by treating free variables as if they were names.
5. Theory of Logic / I. Semantics of Logic / 5. Extensionalism
13821 Extensionality is built into ordinary logic semantics; names have objects, predicates have sets of objects [Bostock]
     Full Idea: Extensionality is built into the semantics of ordinary logic. When a name-letter is interpreted as denoting something, we just provide the object denoted. All that we provide for a one-place predicate-letter is the set of objects that it is true of..
     From: David Bostock (Intermediate Logic [1997])
     A reaction: Could we keep the syntax of ordinary logic, and provide a wildly different semantics, much closer to real life? We could give up these dreadful 'objects' that Frege lumbered us with. Logic for processes, etc.
13362 If an object has two names, truth is undisturbed if the names are swapped; this is Extensionality [Bostock]
     Full Idea: If two names refer to the same object, then in any proposition which contains either of them the other may be substituted in its place, and the truth-value of the proposition of the proposition will be unaltered. This is the Principle of Extensionality.
     From: David Bostock (Intermediate Logic [1997], 3.1)
     A reaction: He acknowledges that ordinary language is full of counterexamples, such as 'he doesn't know the Morning Star and the Evening Star are the same body' (when he presumably knows that the Morning Star is the Morning Star). This is logic. Like maths.
5. Theory of Logic / K. Features of Logics / 2. Consistency
13541 For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ [Bostock]
     Full Idea: Any system of proof S is said to be 'negation-consistent' iff there is no formula such that |-(S)φ and |-(S)¬φ.
     From: David Bostock (Intermediate Logic [1997], 4.5)
     A reaction: Compare Idea 13542. This version seems to be a 'strong' version, as it demands a higher standard than 'absolute consistency'. Both halves of the condition would have to be established.
13542 A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula [Bostock]
     Full Idea: Any system of proof S is said to be 'absolutely consistent' iff it is not the case that for every formula we have |-(S)φ.
     From: David Bostock (Intermediate Logic [1997], 4.5)
     A reaction: Bostock notes that a sound system will be both 'negation-consistent' (Idea 13541) and absolutely consistent. 'Tonk' systems can be shown to be unsound because the two come apart.
13540 A set of formulae is 'inconsistent' when there is no interpretation which can make them all true [Bostock]
     Full Idea: 'Γ |=' means 'Γ is a set of closed formulae, and there is no (standard) interpretation in which all of the formulae in Γ are true'. We abbreviate this last to 'Γ is inconsistent'.
     From: David Bostock (Intermediate Logic [1997], 4.5)
     A reaction: This is a semantic approach to inconsistency, in terms of truth, as opposed to saying that we cannot prove both p and ¬p. I take this to be closer to the true concept, since you need never have heard of 'proof' to understand 'inconsistent'.
5. Theory of Logic / K. Features of Logics / 6. Compactness
13544 Inconsistency or entailment just from functors and quantifiers is finitely based, if compact [Bostock]
     Full Idea: Being 'compact' means that if we have an inconsistency or an entailment which holds just because of the truth-functors and quantifiers involved, then it is always due to a finite number of the propositions in question.
     From: David Bostock (Intermediate Logic [1997], 4.8)
     A reaction: Bostock says this is surprising, given the examples 'a is not a parent of a parent of b...' etc, where an infinity seems to establish 'a is not an ancestor of b'. The point, though, is that this truth doesn't just depend on truth-functors and quantifiers.
13618 Compactness means an infinity of sequents on the left will add nothing new [Bostock]
     Full Idea: The logic of truth-functions is compact, which means that sequents with infinitely many formulae on the left introduce nothing new. Hence we can confine our attention to finite sequents.
     From: David Bostock (Intermediate Logic [1997], 5.5)
     A reaction: This makes it clear why compactness is a limitation in logic. If you want the logic to be unlimited in scope, it isn't; it only proves things from finite numbers of sequents. This makes it easier to prove completeness for the system.
5. Theory of Logic / L. Paradox / 3. Antinomies
13986 Plato found antinomies in ideas, Kant in space and time, and Bradley in relations [Plato, by Ryle]
     Full Idea: Plato (in 'Parmenides') shows that the theory that 'Eide' are substances, and Kant that space and time are substances, and Bradley that relations are substances, all lead to aninomies.
     From: report of Plato (Parmenides [c.364 BCE]) by Gilbert Ryle - Are there propositions? 'Objections'
14150 Plato's 'Parmenides' is perhaps the best collection of antinomies ever made [Russell on Plato]
     Full Idea: Plato's 'Parmenides' is perhaps the best collection of antinomies ever made.
     From: comment on Plato (Parmenides [c.364 BCE]) by Bertrand Russell - The Principles of Mathematics §337
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
13358 Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all [Bostock]
     Full Idea: The principle of mathematical (or ordinary) induction says suppose the first number, 0, has a property; suppose that if any number has that property, then so does the next; then it follows that all numbers have the property.
     From: David Bostock (Intermediate Logic [1997], 2.8)
     A reaction: Ordinary induction is also known as 'weak' induction. Compare Idea 13359 for 'strong' or complete induction. The number sequence must have a first element, so this doesn't work for the integers.
13359 Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers [Bostock]
     Full Idea: The principle of complete induction says suppose that for every number, if all the numbers less than it have a property, then so does it; it then follows that every number has the property.
     From: David Bostock (Intermediate Logic [1997], 2.8)
     A reaction: Complete induction is also known as 'strong' induction. Compare Idea 13358 for 'weak' or mathematical induction. The number sequence need have no first element.
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
16150 One is, so numbers exist, so endless numbers exist, and each one must partake of being [Plato]
     Full Idea: If one is, there must also necessarily be number - Necessarily - But if there is number, there would be many, and an unlimited multitude of beings. ..So if all partakes of being, each part of number would also partake of it.
     From: Plato (Parmenides [c.364 BCE], 144a)
     A reaction: This seems to commit to numbers having being, then to too many numbers, and hence to too much being - but without backing down and wondering whether numbers had being after all. Aristotle disagreed.
7. Existence / A. Nature of Existence / 3. Being / c. Becoming
229 The one was and is and will be and was becoming and is becoming and will become [Plato]
     Full Idea: The one was and is and will be and was becoming and is becoming and will become.
     From: Plato (Parmenides [c.364 BCE], 155d)
7. Existence / A. Nature of Existence / 3. Being / f. Primary being
21821 Plato's Parmenides has a three-part theory, of Primal One, a One-Many, and a One-and-Many [Plato, by Plotinus]
     Full Idea: The Platonic Parmenides is more exact [than Parmenides himself]; the distinction is made between the Primal One, a strictly pure Unity, and a secondary One which is a One-Many, and a third which is a One-and-Many.
     From: report of Plato (Parmenides [c.364 BCE]) by Plotinus - The Enneads 5.1.08
     A reaction: Plotinus approves of this three-part theory. Parmenides has the problem that the highest Being contains no movement. By placing the One outside Being you can give it powers which an existent thing cannot have. Cf the concept of God.
7. Existence / D. Theories of Reality / 3. Reality
221 Absolute ideas, such as the Good and the Beautiful, cannot be known by us [Plato]
     Full Idea: The absolute good and the beautiful and all which we conceive to be absolute ideas are unknown to us.
     From: Plato (Parmenides [c.364 BCE], 134c)
8. Modes of Existence / A. Relations / 4. Formal Relations / a. Types of relation
13802 Relations can be one-many (at most one on the left) or many-one (at most one on the right) [Bostock]
     Full Idea: A relation is 'one-many' if for anything on the right there is at most one on the left (∀xyz(Rxz∧Ryz→x=y), and is 'many-one' if for anything on the left there is at most one on the right (∀xyz(Rzx∧Rzy→x=y).
     From: David Bostock (Intermediate Logic [1997], 8.1)
13543 A relation is not reflexive, just because it is transitive and symmetrical [Bostock]
     Full Idea: It is easy to fall into the error of supposing that a relation which is both transitive and symmetrical must also be reflexive.
     From: David Bostock (Intermediate Logic [1997], 4.7)
     A reaction: Compare Idea 14430! Transivity will take you there, and symmetricality will get you back, but that doesn't entitle you to take the shortcut?
8. Modes of Existence / D. Universals / 2. Need for Universals
227 You must always mean the same thing when you utter the same name [Plato]
     Full Idea: You must always mean the same thing when you utter the same name.
     From: Plato (Parmenides [c.364 BCE], 147d)
223 If you deny that each thing always stays the same, you destroy the possibility of discussion [Plato]
     Full Idea: If a person denies that the idea of each thing is always the same, he will utterly destroy the power of carrying on discussion.
     From: Plato (Parmenides [c.364 BCE], 135c)
8. Modes of Existence / D. Universals / 6. Platonic Forms / a. Platonic Forms
211 If admirable things have Forms, maybe everything else does as well [Plato]
     Full Idea: It is troubling that if admirable things have abstract ideas, then perhaps everything else must have ideas as well.
     From: Plato (Parmenides [c.364 BCE], 130d)
219 If absolute ideas existed in us, they would cease to be absolute [Plato]
     Full Idea: None of the absolute ideas exists in us, because then it would no longer be absolute.
     From: Plato (Parmenides [c.364 BCE], 133c)
228 Greatness and smallness must exist, to be opposed to one another, and come into being in things [Plato]
     Full Idea: These two ideas, greatness and smallness, exist, do they not? For if they did not exist, they could not be opposites of one another, and could not come into being in things.
     From: Plato (Parmenides [c.364 BCE], 149e)
16151 Plato moves from Forms to a theory of genera and principles in his later work [Plato, by Frede,M]
     Full Idea: It seems to me that Plato in the later dialogues, beginning with the second half of 'Parmenides', wants to substitute a theory of genera and theory of principles that constitute these genera for the earlier theory of forms.
     From: report of Plato (Parmenides [c.364 BCE]) by Michael Frede - Title, Unity, Authenticity of the 'Categories' V
     A reaction: My theory is that the later Plato came under the influence of the brilliant young Aristotle, and this idea is a symptom of it. The theory of 'principles' sounds like hylomorphism to me.
210 It would be absurd to think there were abstract Forms for vile things like hair, mud and dirt [Plato]
     Full Idea: Are there abstract ideas for such things as hair, mud and dirt, which are particularly vile and worthless? That would be quite absurd.
     From: Plato (Parmenides [c.364 BCE], 130d)
220 The concept of a master includes the concept of a slave [Plato]
     Full Idea: Mastership in the abstract is mastership of slavery in the abstract.
     From: Plato (Parmenides [c.364 BCE], 133e)
8. Modes of Existence / D. Universals / 6. Platonic Forms / b. Partaking
218 Participation is not by means of similarity, so we are looking for some other method of participation [Plato]
     Full Idea: Participation is not by means of likeness, so we must seek some other method of participation.
     From: Plato (Parmenides [c.364 BCE], 133a)
213 Each idea is in all its participants at once, just as daytime is a unity but in many separate places at once [Plato]
     Full Idea: Just as day is in many places at once, but not separated from itself, so each idea might be in all its participants at once.
     From: Plato (Parmenides [c.364 BCE], 131b)
216 If things are made alike by participating in something, that thing will be the absolute idea [Plato]
     Full Idea: That by participation in which like things are made like, will be the absolute idea, will it not?
     From: Plato (Parmenides [c.364 BCE], 132e)
212 The whole idea of each Form must be found in each thing which participates in it [Plato]
     Full Idea: The whole idea of each form (of beauty, justice etc) must be found in each thing which participates in it.
     From: Plato (Parmenides [c.364 BCE], 131a)
215 If things partake of ideas, this implies either that everything thinks, or that everything actually is thought [Plato]
     Full Idea: If all things partake of ideas, must either everything be made of thoughts and everything thinks, or everything is thought, and so can't think?
     From: Plato (Parmenides [c.364 BCE], 132c)
8. Modes of Existence / D. Universals / 6. Platonic Forms / c. Self-predication
214 If absolute greatness and great things are seen as the same, another thing appears which makes them seem great [Plato]
     Full Idea: If you regard the absolute great and the many great things in the same way, will not another appear beyond, by which all these must appear to be great?
     From: Plato (Parmenides [c.364 BCE], 132a)
217 Nothing can be like an absolute idea, because a third idea intervenes to make them alike (leading to a regress) [Plato]
     Full Idea: It is impossible for anything to be like an absolute idea, because a third idea will appear to make them alike, and if that is like anything, it will lead to another idea, and so on.
     From: Plato (Parmenides [c.364 BCE], 133a)
9. Objects / B. Unity of Objects / 1. Unifying an Object / b. Unifying aggregates
15851 Parts must belong to a created thing with a distinct form [Plato]
     Full Idea: The part would not be the part of many things or all, but of some one character ['ideas'] and of some one thing, which we call a 'whole', since it has come to be one complete [perfected] thing composed [created] of all.
     From: Plato (Parmenides [c.364 BCE], 157d)
     A reaction: A serious shot by Plato at what identity is. Harte quotes it (125) and shows that 'character' is Gk 'idea', and 'composed' will translate as 'created'. 'Form' links this Platonic passage to Aristotle's hylomorphism.
9. Objects / C. Structure of Objects / 5. Composition of an Object
15846 In Parmenides, if composition is identity, a whole is nothing more than its parts [Plato, by Harte,V]
     Full Idea: At the heart of the 'Parmenides' puzzles about composition is the thesis that composition is identity. Considered thus, a whole adds nothing to an ontology that already includes its parts
     From: report of Plato (Parmenides [c.364 BCE]) by Verity Harte - Plato on Parts and Wholes 2.5
     A reaction: There has to be more to a unified identity that mere proximity of the parts. When do parts come together, and when do they actually 'compose' something?
9. Objects / C. Structure of Objects / 8. Parts of Objects / a. Parts of objects
15849 Plato says only a one has parts, and a many does not [Plato, by Harte,V]
     Full Idea: In 'Parmenides' it is argued that a part cannot be part of a many, but must be part of something one.
     From: report of Plato (Parmenides [c.364 BCE], 157c) by Verity Harte - Plato on Parts and Wholes 3.2
     A reaction: This looks like the right way to go with the term 'part'. We presuppose a unity before we even talk of its parts, so we can't get into contradictions and paradoxes about their relationships.
15850 Anything which has parts must be one thing, and parts are of a one, not of a many [Plato]
     Full Idea: The whole of which the parts are parts must be one thing composed of many; for each of the parts must be part, not of a many, but of a whole.
     From: Plato (Parmenides [c.364 BCE], 157c)
     A reaction: This is a key move of metaphysics, and we should hang on to it. The other way madness lies.
9. Objects / C. Structure of Objects / 8. Parts of Objects / c. Wholes from parts
13259 It seems that the One must be composed of parts, which contradicts its being one [Plato]
     Full Idea: The One must be composed of parts, both being a whole and having parts. So on both grounds the One would thus be many and not one. But it must be not many, but one. So if the One will be one, it will neither be a whole, nor have parts.
     From: Plato (Parmenides [c.364 BCE], 137c09), quoted by Kathrin Koslicki - The Structure of Objects 5.2
     A reaction: This is the starting point for Plato's metaphysical discussion of objects. It seems to begin a line of thought which is completed by Aristotle, surmising that only an essential structure can bestow identity on a bunch of parts.
9. Objects / F. Identity among Objects / 5. Self-Identity
13847 If non-existent things are self-identical, they are just one thing - so call it the 'null object' [Bostock]
     Full Idea: If even non-existent things are still counted as self-identical, then all non-existent things must be counted as identical with one another, so there is at most one non-existent thing. We might arbitrarily choose zero, or invent 'the null object'.
     From: David Bostock (Intermediate Logic [1997], 8.6)
9. Objects / F. Identity among Objects / 6. Identity between Objects
15847 Two things relate either as same or different, or part of a whole, or the whole of the part [Plato]
     Full Idea: Everything is surely related to everything as follows: either it is the same or different; or, if it is not the same or different, it would be related as part to whole or as whole to part.
     From: Plato (Parmenides [c.364 BCE], 146b)
     A reaction: This strikes me as a really helpful first step in trying to analyse the nature of identity. Two things are either two or (actually) one, or related mereologically.
10. Modality / A. Necessity / 6. Logical Necessity
13820 The idea that anything which can be proved is necessary has a problem with empty names [Bostock]
     Full Idea: The common Rule of Necessitation says that what can be proved is necessary, but this is incorrect if we do not permit empty names. The most straightforward answer is to modify elementary logic so that only necessary truths can be proved.
     From: David Bostock (Intermediate Logic [1997], 8.4)
19. Language / C. Assigning Meanings / 3. Predicates
13363 A (modern) predicate is the result of leaving a gap for the name in a sentence [Bostock]
     Full Idea: A simple way of approaching the modern notion of a predicate is this: given any sentence which contains a name, the result of dropping that name and leaving a gap in its place is a predicate. Very different from predicates in Aristotle and Kant.
     From: David Bostock (Intermediate Logic [1997], 3.2)
     A reaction: This concept derives from Frege. To get to grips with contemporary philosophy you have to relearn all sorts of basic words like 'predicate' and 'object'.
24. Political Theory / A. Basis of a State / 1. A People / b. The natural life
19906 All countries are in a mutual state of nature [Locke]
     Full Idea: All commonwealths are in a state of Nature one with another.
     From: John Locke (Second Treatise of Government [1690], 153)
     A reaction: A striking remark. It is easy to think that the state of nature no longer exists. International law attempts to rectify this, but diplomacy is much more like negotiations in nature than it is like obedience to laws.
19882 We are not created for solitude, but are driven into society by our needs [Locke]
     Full Idea: God, having made man such a creature that, in His own judgement, it was not good for him to be alone, put him under strong obligations of necessity, convenience, and inclination, to drive him into society.
     From: John Locke (Second Treatise of Government [1690], 077)
     A reaction: This is almost Aristotelian, apart from the individualistic assumption that we are 'driven' into society. The only time I see other people looking generally happy is when they are sitting around at leisure and talking to other people.
24. Political Theory / A. Basis of a State / 3. Natural Values / a. Natural freedom
19864 In nature men can dispose of possessions and their persons in any way that is possible [Locke]
     Full Idea: The estate all men are naturally in is perfect freedom to order their actions, and dispose of their possessions and persons as they think fit, within the bounds of the laws of nature.
     From: John Locke (Second Treatise of Government [1690], 004)
     A reaction: Note that they have possessions, so property is not an invention of society, but something which society should protect. Presumably Locke thinks they could sell themselves into slavery, which Rousseau rejects.
24. Political Theory / A. Basis of a State / 3. Natural Values / b. Natural equality
19865 There is no subjection in nature, and all creatures of the same species are equal [Locke]
     Full Idea: Creatures of the same species and rank, promiscuously born to all the same advantages of Nature, are also equal one among another, without subordination or subjection.
     From: John Locke (Second Treatise of Government [1690], 004)
     A reaction: The birds in my garden don't behave as if that were true. Physical strength is surely a natural inequality.
24. Political Theory / A. Basis of a State / 3. Natural Values / c. Natural rights
19866 The rational law of nature says we are all equal and independent, and should show mutual respect [Locke]
     Full Idea: The state of Nature has a law of Nature to govern it, which obliges everyone, and reason, which is that law, teaches mankind that all being equal and independent, no one ought to harm another in his life, health, liberty or possessions.
     From: John Locke (Second Treatise of Government [1690], 006)
     A reaction: He adds that this is because we are all the property of God. Locke is more optimistic than Hobbes or Rousseau about this, since he thinks we have a natural obligation to be nice.
19872 The animals and fruits of the earth belong to mankind [Locke]
     Full Idea: All the fruits the earth naturally produces, and beasts it feeds, belong to mankind in common, as they are produced by the spontaneous hand of Nature.
     From: John Locke (Second Treatise of Government [1690], 026)
     A reaction: Not a popular view among 21st century ecologists, I guess, but this remains the implicit belief of anyone who goes hunting in the woods, and our enclosed gardens seem to endorse the idea.
19907 There is a natural right to inheritance within a family [Locke]
     Full Idea: Every man is born with a right before any other man, to inherit, with his brethren, his father's goods.
     From: John Locke (Second Treatise of Government [1690], 190)
     A reaction: If a child is fully grown, they may well have drifted into a state of partial ownership of the goods of the parent, of which it would be hard then to deprive them. It is hard to see this as a natural right of tiny orphaned infants.
24. Political Theory / B. Nature of a State / 1. Purpose of a State
19863 Politics is the right to make enforceable laws to protect property and the state, for the common good [Locke]
     Full Idea: Political power is the right of making laws, with penalties up to death, for the preserving of property, employing the force of community in the execution of such laws, in defence of the commonwealth, and only for the common good.
     From: John Locke (Second Treatise of Government [1690], 003)
     A reaction: Since political power can be used for selfish corruption and genocide, this isn't very accurate, so I take it this is how power ought to be exercised! Notice that defence gets equal billing with his famous defence of property.
24. Political Theory / B. Nature of a State / 2. State Legitimacy / c. Social contract
5654 The Second Treatise explores the consequences of the contractual view of the state [Locke, by Scruton]
     Full Idea: In his second Treatise, Locke gave us perhaps the first extended account of the true logical consequences of Hobbes's contractual view of the state.
     From: report of John Locke (Second Treatise of Government [1690]) by Roger Scruton - Short History of Modern Philosophy Ch.14
     A reaction: The issue seems to boil down to an opposition between the Cartesian and the Aristotelian view of the individual, with Locke following Descartes. The alternative, endorsed by Hegel, which I prefer, is that the state is part of human nature.
19888 A society only begins if there is consent of all the individuals to join it [Locke]
     Full Idea: The beginning of politic society depends upon the consent of the individuals to join into and make one society.
     From: John Locke (Second Treatise of Government [1690], 106)
     A reaction: This is the dramatic new political idea (originating with Hobbes), that all of the members must (at some point) consent to the state. In practice we are all born into a state, so it is not clear what this means in real life.
6702 If anyone enjoys the benefits of government (even using a road) they give tacit assent to its laws [Locke]
     Full Idea: Every man, that hath an possession, or enjoyment, of any part of the dominions of any government, doth thereby give his tacit consent, and is obliged to obedience to the laws, ..whether it be barely travelling on the highway.
     From: John Locke (Second Treatise of Government [1690], 119), quoted by Gordon Graham - Eight Theories of Ethics Ch.8
     A reaction: Locke's famous assertion of an unspoken and inescapable contract, to which we are all subject. Hume gave an effective reply (Idea 6703). Locke has a point though. The more you accept, the more obliged you are. I accept the law more as I get older.
19909 A politic society is created from a state of nature by a unanimous agreement [Locke]
     Full Idea: That which makes the community, and brings men out of the loose state of Nature into one politic society, is the agreement that everyone has with the rest to incorporate and act as one body.
     From: John Locke (Second Treatise of Government [1690], 211)
     A reaction: Geography usually keeps commonwealths in place once they have been established, but some of them become disfunctional hell holes because they are trapped in perpetual disagreement.
24. Political Theory / B. Nature of a State / 2. State Legitimacy / d. General will
19910 A single will creates the legislature, which is duty-bound to preserve that will [Locke]
     Full Idea: The essence and union of the society consisting in having one will; the legislative, when once established by the majority, has the declaring and, as it were, keeping of that will.
     From: John Locke (Second Treatise of Government [1690], 212)
     A reaction: Not far from Rousseau's big idea, apart from the emphasis on the 'majority'. Rousseau reduced the role of the general will to preliminaries and basics, but wanted close to unanimity, so that everyone accepts being a subject, to government and law.
24. Political Theory / B. Nature of a State / 4. Citizenship
19893 Anyone who enjoys the benefits of a state has given tacit consent to be part of it [Locke]
     Full Idea: Every man that hath any possession or enjoyment of any part of the dominions of any government doth thereby give his tacit consent, and is as far forth obliged to obedience to the laws of that government, during such enjoyment.
     From: John Locke (Second Treatise of Government [1690], 119)
     A reaction: I wondered at the age of about 18 whether I had given tacit consent to be a British citizen. Locke says you only have to travel freely down the highways to give consent! We are all free, of course, to apply for citizenship elsewhere. But Idea 19894.
19894 You can only become an actual member of a commonwealth by an express promise [Locke]
     Full Idea: Nothing can make any man a subject or member of a commonwealth but his actually entering into it by positive engagement, and express promise and compact.
     From: John Locke (Second Treatise of Government [1690], 122)
     A reaction: In practice the indigenous population never do this. But it a clear distinction for foreign residents in any country. States cannot induct resident foreigners into their army, or allow them to vote.
19892 Children are not born into citizenship of a state [Locke]
     Full Idea: It is plain, by the practices of governments themselves, as well as by the laws of right reason, that a child is born a subject of no country nor government.
     From: John Locke (Second Treatise of Government [1690], 118)
     A reaction: At what age do they become citizens, given that there is no induction ceremony? If a small British child were attacked overseas, we would expect the British government to defend its rights.
24. Political Theory / C. Ruling a State / 2. Leaders / b. Monarchy
19885 Absolute monarchy is inconsistent with civil society [Locke]
     Full Idea: Absolute monarchy, which by some men is counted for the only government in the world, is inconsistent with civil society, and so can be no form of civil government at all.
     From: John Locke (Second Treatise of Government [1690], 090)
     A reaction: This is because citizens do not have a 'decisive' power to appeal for redress of injuries. Rousseau thought that there could be an absolute monarchy, as long as the general will agreed it, and its term of office could be brought to an end by the assembly.
24. Political Theory / C. Ruling a State / 2. Leaders / c. Despotism
19886 The idea that absolute power improves mankind is confuted by history [Locke]
     Full Idea: He that thinks absolute power purifies men's blood, and corrects the baseness of human nature, need but read the history of this, or any other age, to be convinced to the contrary.
     From: John Locke (Second Treatise of Government [1690], 092)
     A reaction: I can't imagine who proposed the view that Locke is attacking, but it will have been some real 17th century thinker. Attitudes to monarchy changed drastically in England, but Louis XIV was still ruling in France.
19903 Despotism is arbitrary power to kill, based neither on natural equality, nor any social contract [Locke]
     Full Idea: Despotical power is an absolute, arbitrary power one man over another, to take away his life whenever he pleases; and this is a power which neither Nature gives, for it has made no such distinction between one man and another, nor compact can convey.
     From: John Locke (Second Treatise of Government [1690], 172)
     A reaction: Colonies of seals, walruses and apes seem to display despotism, based on physical strength, though that is largely to do with mating. There could be such a compact, but Locke would regard it as invalid.
19905 People stripped of their property are legitimately subject to despotism [Locke]
     Full Idea: Forfeiture gives despotical power to lords for their own benefit over those who are stripped of all property. ...Despotical power is over such as have no property at all.
     From: John Locke (Second Treatise of Government [1690], 173)
     A reaction: Nasty! Shylock is stripped of his property by Venice, so these things happened. This is taking the significance of property a long way beyond its role at the beginning of Locke's book. Property is the start of society, but then becomes your passport.
19904 Legitimate prisoners of war are subject to despotism, because that continues the state of war [Locke]
     Full Idea: Captives, taken in a just and lawful war, and such only, are subject to a despotical power, which, as it arises not from compact, so neither is it capable of any, but is the state of war continued.
     From: John Locke (Second Treatise of Government [1690], 205)
     A reaction: How long after a war finishes is such despotism legitimate? What happened to the German prisoners in Russia in 1945? Locke defined despotism as the right to kill, but that is expressly contrary to the rules of war, look you.
24. Political Theory / C. Ruling a State / 3. Government / b. Legislature
19895 Even the legislature must be preceded by a law which gives it power to make laws [Locke]
     Full Idea: The first and fundamental positive law of all commonwealths is the establishing of the legislative power, as the first and fundamental natural law which is to govern even the legislative.
     From: John Locke (Second Treatise of Government [1690], 134)
     A reaction: I think Rousseau says that there cannot be a law which enables the general will to set up legislative powers. It just seems to be something which happens. Locke is threatened with an infinite regress. What legitimises the enabling law?
24. Political Theory / C. Ruling a State / 3. Government / c. Executive
19900 The executive must not be the legislature, or they may exempt themselves from laws [Locke]
     Full Idea: It may be too great temptation to human frailty, apt to grasp at power, for the same persons to have the power of making laws to also have in their hands the power to execute them, whereby they may exempt themselves.
     From: John Locke (Second Treatise of Government [1690], 143)
     A reaction: The main principles of modern constitutions are devised to avoid corruption. If people were incorruptible (yeah, right) the world would presumably be run very differently, and rather more efficiently, like a good family.
24. Political Theory / C. Ruling a State / 4. Changing the State / c. Revolution
19902 Any obstruction to the operation of the legislature can be removed forcibly by the people [Locke]
     Full Idea: Having erect a legislative with the power of making laws, when they are hindered by any force from what is so necessary to society, and wherein the safety and preservation of the people consists, the people have a right to remove it by force.
     From: John Locke (Second Treatise of Government [1690], 155)
     A reaction: I doubt if he was thinking of the French Revolution, but this will clearly have application to the English events of 1642. The Speaker of the Commons was held down in his chair in the 1620s, so that some legislation could be enacted.
19908 Rebelling against an illegitimate power is no sin [Locke]
     Full Idea: It is plain that shaking off a power which force, and not right, hath set over any one, though it have the name of rebellion, yet it is no offence against God.
     From: John Locke (Second Treatise of Government [1690], 196)
     A reaction: [He cites Hezekiah at 2 Kings 18.7] At this time the English Civil War was referred to as the 'Great Rebellion' (so this is an interesting and brave remark of Locke's), though few people would think that Charles I had illegitimate power.
19911 If legislators confiscate property, or enslave people, they are no longer owed obedience [Locke]
     Full Idea: Whenever the legislators endeavour to take away and destroy the property of the people, or reduce them to slavery under arbitrary power, they put themselves into a state of war with the people, who are thereupon absolved from any further obedience.
     From: John Locke (Second Treatise of Government [1690], 222)
     A reaction: This might fit Louis XVI in 1788. Locke was certainly not averse to consideration the situations in which revolution might be justified. He was trying to be even-handed about 1642. Locke seems to think that without property you ARE a slave.
24. Political Theory / D. Ideologies / 5. Democracy / a. Nature of democracy
19901 The people have supreme power, to depose a legislature which has breached their trust [Locke]
     Full Idea: There remains still in the people a supreme power to remove or alter the legislative, when they find the legislative act contrary to the trust reposed in them.
     From: John Locke (Second Treatise of Government [1690], 149)
     A reaction: This seems to be the most important aspect of representative democracy. It is not the power of people to make decisions, but the power to get rid of bad rulers.
19887 Unanimous consent makes a united community, which is then ruled by the majority [Locke]
     Full Idea: When any number of men have, by the consent of every individual, made a community, they have thereby made that community into one body, with a power to act as one body, which is only by the will and determination of the majority.
     From: John Locke (Second Treatise of Government [1690], 096)
     A reaction: This seems to be presume democracy without discussion, although the formation of the community is by universal consent, which is the 'general will'. Rousseau has the constitution also made almost unanimously, not by a majority.
25. Social Practice / A. Freedoms / 1. Slavery
19913 A master forfeits ownership of slaves he abandons [Locke]
     Full Idea: A master forfeits the dominion over his slaves whom he hath abandoned.
     From: John Locke (Second Treatise of Government [1690], 237)
     A reaction: How often did slave owners take a day off, I wonder? Presumably slaves will take back their freedom, even if the masters haven't 'forfeited' their ownership, so Locke's point is fairly academic.
19883 Slaves captured in a just war have no right to property, so are not part of civil society [Locke]
     Full Idea: Slave are captives taken in a just war, and by right of Nature subjected to the absolute dominion and arbitrary power of their masters. ...Being not capable of any property, they cannot in that state be considered any part of civil society.
     From: John Locke (Second Treatise of Government [1690], 085)
     A reaction: If the test of citizenship is being capable of holding property, presumably children and mentally damaged people (including the very old) will also fail to qualify. I see no principled reason why slaves should not be allowed to vote. Note 'just' war.
19870 If you try to enslave me, you have declared war on me [Locke]
     Full Idea: He who makes an attempt to enslave me thereby puts himself into a state of war with me.
     From: John Locke (Second Treatise of Government [1690], 017)
     A reaction: So presumably actual slaves are in a state of permanent war with their owners. What of a woman who is enslaved by her husband?
25. Social Practice / A. Freedoms / 6. Political freedom
19871 Freedom is not absence of laws, but living under laws arrived at by consent [Locke]
     Full Idea: Liberty of man in society is to be under no other legislative power but that established by consent in the commonwealth. Freedom is not (as Filmer suggests) doing what you please while not tied by any laws.
     From: John Locke (Second Treatise of Government [1690], 022)
     A reaction: That sounds plausible if the consent is unanimous, but a minority is not free if the laws made by a large majority are a sort of persecution.
25. Social Practice / B. Equalities / 4. Economic equality
19880 All value depends on the labour involved [Locke]
     Full Idea: It is labour that puts the difference of value on everything. ...Whatever bread is worth more than acorns, wine than water, that is wholly owing to labour and industry.
     From: John Locke (Second Treatise of Government [1690], 040)
     A reaction: In capitalism this is nonsense. Supply and demand fix all the values. Locke has slid from labour bestowing ownership to labour bestowing value. No one gets paid on the basis of how hard they work, except on piece rates.
25. Social Practice / C. Rights / 3. Alienating rights
19884 There is only a civil society if the members give up all of their natural executive rights [Locke]
     Full Idea: Wherever any number of men so unite into one society as to quite every one his executive power of the law of Nature, and to resign it to the public, there and there only is a civil society.
     From: John Locke (Second Treatise of Government [1690], 089)
     A reaction: This seems to mean that you must give up your active ('executive') natural rights, but not your passive ones (which are inviolable).
19873 We all own our bodies, and the work we do is our own [Locke]
     Full Idea: Every man has a 'property' in his own 'person'. This nobody has any right to it but himself. The 'labour' of his body and the 'work' of his hands, we may say, are properly his.
     From: John Locke (Second Treatise of Government [1690], 027)
     A reaction: He doesn't have any grounds for this claim. Why doesn't a cow own its body? He slides from my ownership of my laborious efforts to my ownership of what I have been working on. I can't acquire your car by servicing it.
25. Social Practice / C. Rights / 4. Property rights
6580 Locke (and Marx) held that ownership of objects is a natural relation, based on the labour put into it [Locke, by Fogelin]
     Full Idea: Locke thought that property ownership reflected a natural relationship; for him the primordial notion of the ownership of an object is a function of the labour that one puts into it; Marx held a similar view.
     From: report of John Locke (Second Treatise of Government [1690]) by Robert Fogelin - Walking the Tightrope of Reason Ch.3
     A reaction: Marx would have to think that, in order to believe that capitalist ownership of the means of production used by the workers was a fundamental injustice. A deeper Marxism might see the whole idea of 'ownership' as a capitalist (or feudal) conspiracy.
20520 Locke says 'mixing of labour' entitles you to land, as well as nuts and berries [Wolff,J on Locke]
     Full Idea: The great advantage of Locke's 'labour-mixing' argument is that it seems it can justify the appropriation of land, as well as nuts and berries.
     From: comment on John Locke (Second Treatise of Government [1690]) by Jonathan Wolff - An Introduction to Political Philosophy (Rev) 5 'Locke'
     A reaction: The argument is dubious at best, and plausibly downright wicked. How much labour achieves ownership? What of previous people who worked the land but never thought to claim 'ownership'? Suppose I do more labour than you on 'your' land?
19875 A man's labour gives ownership rights - as long as there are fair shares for all [Locke]
     Full Idea: The 'labour' being the unquestionable property of the labourer, no man but he can have a right to what that is once joined to, at least where there is enough, and as good left in common for others.
     From: John Locke (Second Treatise of Government [1690], 027)
     A reaction: The qualification at the end is a crucial (and problematic) addition to his theory. What is the situation when an area of wilderness is 98% owned? What of the single source of water? Who gets the best parts? Getting there first seems crucial.
19874 If a man mixes his labour with something in Nature, he thereby comes to own it [Locke]
     Full Idea: Whatever a man removes out of the state that Nature hath provided and left it in, he hath mixed his labour with it, and joined something to it that is his own, and thereby makes it his property. ...This excludes the common right of other men.
     From: John Locke (Second Treatise of Government [1690], 027)
     A reaction: This is Locke's famous Labour Theory of Value. Does picking it up count as labour? Putting a fence round it? Paying someone else to do the labour? Do bees own their honey? Settlers in the wilderness own nothing on day one?
19877 Fountain water is everyone's, but a drawn pitcher of water has an owner [Locke]
     Full Idea: Though the water running in the fountain be every one's, yet who can doubt but that in the pitcher is his only who drew it out?
     From: John Locke (Second Treatise of Government [1690], 029)
     A reaction: This would certainly be the normal consensus of a community, as long as there is plenty of water. The strong and fit gatherers get all the best firewood, so I suppose that is just tough on the others.
19876 Gathering natural fruits gives ownership; the consent of other people is irrelevant [Locke]
     Full Idea: If the first gathering of acorns and apples made them not a man's, nothing else could. ...Will anyone say he had no right to them because he had not the consent of all mankind to make them his?
     From: John Locke (Second Treatise of Government [1690], 028)
     A reaction: The ideas of Nozick are all in this sentence. Does this idea justify the enclosure of common land? The first member of the community who thought of Locke's labour theory had a huge head's start. Liberal individualism rampant.
19878 Mixing labour with a thing bestows ownership - as long as the thing is not wasted [Locke]
     Full Idea: How far has God given us all things 'to enjoy'? As much as any one can make use of to any advantage of his life before it spoils, so much he may by his labour fix a property in.
     From: John Locke (Second Treatise of Government [1690], 031)
     A reaction: This adds a very different value to Locke's theory, because the person seems to be answerable to fellow citizens if they harvest important resources and then waste them. Where do luxuries fit in?
19879 A man owns land if he cultivates it, to the limits of what he needs [Locke]
     Full Idea: As much land as a man tills, plants, improves, cultivates, and can use the product of, so much is his property.
     From: John Locke (Second Treatise of Government [1690], 032)
     A reaction: Industrial farming rather changes this picture. Does the man himself decide how much he can use the product of, or do the neighbours tell him where his boundaries must be? 'Reason not the need', as King Lear said. What if he stops cultivating it?
19898 Soldiers can be commanded to die, but not to hand over their money [Locke]
     Full Idea: The sergeant that can command a soldier to march up to the mouth of a cannon ...cannot command that soldier to give him one penny of his money.
     From: John Locke (Second Treatise of Government [1690], 139)
     A reaction: A very nice and accurate illustration of a principle which runs so deep that it does indeed look like a basis of society.
25. Social Practice / D. Justice / 2. The Law / a. Legal system
19881 The aim of law is not restraint, but to make freedom possible [Locke]
     Full Idea: The end of law is not to abolish or restrain, but to preserve and enlarge freedom, for where there is no law there is no freedom.
     From: John Locke (Second Treatise of Government [1690], 057)
     A reaction: This fits both the liberal and the communitarian view of the matter. Talk of 'freedom' is commonplace in England by this date, where it is hardly mention 60 years earler. John Lilburne almost single-handedly brought this about.
25. Social Practice / D. Justice / 2. The Law / c. Natural law
19868 It is only by a law of Nature that we can justify punishing foreigners [Locke]
     Full Idea: If by the law of Nature every man hath not a power to punish offences against [the state], as he soberly judges the case to require, I see not how the magistrates of any community can punish an alien of another country.
     From: John Locke (Second Treatise of Government [1690], 009)
     A reaction: This is a nice point. You can't expect to be above the law in a foreign country, but you have entered into no social contract, unless visiting a place is a sort of contract. Intrusions into air space are often accidental visits.
25. Social Practice / D. Justice / 3. Punishment / a. Right to punish
19867 Reparation and restraint are the only justifications for punishment [Locke]
     Full Idea: Reparation and restraint are the only two reasons why one man may lawfully do harm to another, which is that we call punishment.
     From: John Locke (Second Treatise of Government [1690], 008)
     A reaction: But by 'reparation' does be mean retribution, or compensation? He doesn't rule out capital punishment, but that may qualify as maximum restraint.
19912 Self-defence is natural, but not the punishment of superiors by inferiors [Locke]
     Full Idea: It is natural for us to defend life and limb, but that an inferior should punish a superior is against nature.
     From: John Locke (Second Treatise of Government [1690], 236)
     A reaction: He is obliquely referring to the execution of Charles I, even though he may have been legitimately overthrown. I wonder what exactly he means by 'superior' and 'inferior'. An idea from another age!
19869 Punishment should make crime a bad bargain, leading to repentance and deterrence [Locke]
     Full Idea: Each transgression may be punished to that degree, and with so much severity, as will suffice to make it an ill bargain to the offender, give him cause to repent, and terrify others from doing the like.
     From: John Locke (Second Treatise of Government [1690], 012)
     A reaction: I gather that the consensus among experts is that the biggest deterrence is a high likelihood of being caught, rather than the severity of the punishment.
25. Social Practice / E. Policies / 4. Taxation
19899 The consent of the people is essential for any tax [Locke]
     Full Idea: The legislative power must not raise taxes on the property of the people without the consent of the people given by themselves or their deputies.
     From: John Locke (Second Treatise of Government [1690], 142)
     A reaction: He will be thinking of the resistance to Ship Money in the 1630s, which was a step towards civil war. The people of Boston, Ma, may have read this sentence 80 years later!
25. Social Practice / E. Policies / 5. Education / c. Teaching
222 Only a great person can understand the essence of things, and an even greater person can teach it [Plato]
     Full Idea: Only a man of very great natural gifts will be able to understand that everything has a class and absolute essence, and an even more wonderful man can teach this.
     From: Plato (Parmenides [c.364 BCE], 135a)
26. Natural Theory / A. Speculations on Nature / 6. Early Matter Theories / d. The unlimited
225 The unlimited has no shape and is endless [Plato]
     Full Idea: The unlimited partakes neither of the round nor of the straight, because it has no ends nor edges.
     From: Plato (Parmenides [c.364 BCE], 137e)
26. Natural Theory / A. Speculations on Nature / 6. Early Matter Theories / e. The One
233 Some things do not partake of the One [Plato]
     Full Idea: The others cannot partake of the one in any way; they can neither partake of it nor of the whole.
     From: Plato (Parmenides [c.364 BCE], 159d)
     A reaction: Compare Idea 231
2062 The only movement possible for the One is in space or in alteration [Plato]
     Full Idea: If the One moves it either moves spatially or it is altered, since these are the only motions.
     From: Plato (Parmenides [c.364 BCE], 138b)
231 Everything partakes of the One in some way [Plato]
     Full Idea: The others are not altogether deprived of the one, for they partake of it in some way.
     From: Plato (Parmenides [c.364 BCE], 157c)
     A reaction: Compare Idea 233.
28. God / B. Proving God / 2. Proofs of Reason / a. Ontological Proof
234 We couldn't discuss the non-existence of the One without knowledge of it [Plato]
     Full Idea: There must be knowledge of the one, or else not even the meaning of the words 'if the one does not exist' would be known.
     From: Plato (Parmenides [c.364 BCE], 160d)