118 ideas
| 15447 | We shouldn't always follow where the argument leads! [Lewis on Plato] |
| Full Idea: There comes a time not to go on following where the argument leads! | |
| From: comment on Plato (The Laws [c.348 BCE], 667b) by David Lewis - Against Structural Universals 'Variant' | |
| A reaction: Lewis is a fine one to talk, since he follows argument that take him past innumerable incredulous stares of onlookers. |
| 243 | It is foolish to quarrel with the mind's own reasoning processes [Plato] |
| Full Idea: When the soul quarrels with knowledge or opinion or reason, its natural ruling principles, you have there what I call 'folly'. | |
| From: Plato (The Laws [c.348 BCE], 689b) |
| 241 | We ought to follow where the argument leads us [Plato] |
| Full Idea: We ought to follow where the argument leads us. | |
| From: Plato (The Laws [c.348 BCE], 667a) |
| 21264 | Mortals are incapable of being fully rational [Plato] |
| Full Idea: We mustn't assume that mortal eyes will ever be able to look upon reason and get to know it adequately. | |
| From: Plato (The Laws [c.348 BCE], 897d) | |
| A reaction: This is in the context of the rational control of the whole Cosmos. I presume Plato would be flabbergasted by the findings of recent physics and cosmology. Did Kant believe that he was being completely rational about ethics? |
| 251 | Truth has the supreme value, for both gods and men [Plato] |
| Full Idea: Truth heads the list of all things good, for gods and men alike. | |
| From: Plato (The Laws [c.348 BCE], 730c) |
| 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) |
| 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) |
| 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. |
| 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. |
| 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) |
| 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? |
| 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) |
| 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) |
| 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. |
| 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 '='? |
| 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) |
| 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! |
| 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). |
| 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). |
| 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) |
| 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) |
| 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. |
| 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) |
| 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) |
| 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. |
| 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. |
| 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. |
| 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. |
| 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. |
| 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) |
| 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) |
| 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) |
| 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) |
| 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. |
| 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. |
| 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'. |
| 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. |
| 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. |
| 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? |
| 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) |
| 21259 | To grasp a thing we need its name, its definition, and what it really is [Plato] |
| Full Idea: There are three elements in any given thing: the first is what the object actually is, the second is the definition of this, and the third is the name. | |
| From: Plato (The Laws [c.348 BCE], 895d) | |
| A reaction: I take the importance of this to be its distinction between what it is, and the definition of what it is. Aristotle maintains this distinction, but some modern Aristotelians seem to get the confused. Plato worried a lot more about names than we do. |
| 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) |
| 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) |
| 21260 | Soul is what is defined by 'self-generating motion' [Plato] |
| Full Idea: The entity which we call 'soul' is precisely that which is defined by the expression 'self-generating motion'. | |
| From: Plato (The Laws [c.348 BCE], 896a) | |
| A reaction: We may suspect that he defines soul in this way for a particular context, aimed at proving the existence of a First Mover. He must think there is more to soul than the generation of movement. |
| 276 | My individuality is my soul, which carries my body around [Plato] |
| Full Idea: While I am alive I have nothing to thank for my individuality except my soul, whereas my body is just the likeness that I carry around with me. | |
| From: Plato (The Laws [c.348 BCE], 959a) |
| 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'. |
| 249 | People who value beauty above virtue insult the soul by placing the body above it [Plato] |
| Full Idea: When a man values beauty above virtue, the disrespect he shows his soul is total and fundamental, because he argues that the body is more to be honoured than the soul. | |
| From: Plato (The Laws [c.348 BCE], 727e) |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |
| 265 | An action is only just if it is performed by someone with a just character and outlook [Plato] |
| Full Idea: The description 'just' is applicable only to the benefit conferred or injury inflicted by someone with a just character and outlook. | |
| From: Plato (The Laws [c.348 BCE], 862b) | |
| A reaction: How should we describe the occasional administering of good justice by a generally wicked judge. Greeks focus on character, but moderns focus on actions. |
| 269 | Attempted murder is like real murder, but we should respect the luck which avoided total ruin [Plato] |
| Full Idea: An attempted murder should be treated like a successful one, but with respect shown for the luck which saved him from total ruin. | |
| From: Plato (The Laws [c.348 BCE], 877a) | |
| A reaction: The earliest reference to moral luck, I think. 'Repect' sounds vague, but it is asking judges to 'take it into consideration', which is quite practical. Attempted murderers are just as dangerous. |
| 240 | It would be strange if the gods rewarded those who experienced the most pleasure in life [Plato] |
| Full Idea: It would be strange if the gods gave the greatest rewards in heaven to those who led the most pleasant life, rather than the most just. | |
| From: Plato (The Laws [c.348 BCE], 662c) | |
| A reaction: All of philosophy is just footnotes to Plato.... See Idea 1454. |
| 264 | The conquest of pleasure is the noblest victory of all [Plato] |
| Full Idea: The conquest of pleasure is the noblest victory of all. | |
| From: Plato (The Laws [c.348 BCE], 840c) | |
| A reaction: Plato's puritanical streak. Even Aristotle doesn't agree with this. Self-control does not imply conquest of pleasure. Has a good professional wine taster conquered pleasure? |
| 4332 | Virtue is a concord of reason and emotion, with pleasure and pain trained to correct ends [Plato] |
| Full Idea: Virtue is the general concord of reason and emotion, but there is one key element, which is the correct formation of our feelings of pleasure and pain, which makes us hate what we ought to hate, and love what we ought to love. | |
| From: Plato (The Laws [c.348 BCE], 653c) | |
| A reaction: An important truth, taken up by Aristotle. To see another person humiliated gives some people pleasure and other people pain. |
| 248 | A serious desire for moral excellence is very rare indeed [Plato] |
| Full Idea: People who are anxious to attain moral excellence with all possible speed are pretty thin on the ground. | |
| From: Plato (The Laws [c.348 BCE], 718e) |
| 253 | Every crime is the result of excessive self-love [Plato] |
| Full Idea: The cause of each and every crime we commit is precisely this excessive love of ourselves. | |
| From: Plato (The Laws [c.348 BCE], 731e) |
| 263 | The only worthwhile life is one devoted to physical and moral perfection [Plato] |
| Full Idea: A life devoted to every physical perfection and every moral virtue is the only life worth the name. | |
| From: Plato (The Laws [c.348 BCE], 807c) |
| 235 | Virtue is the aim of all laws [Plato] |
| Full Idea: Virtue is the aim of the laws the legislator lays down. | |
| From: Plato (The Laws [c.348 BCE], 631a) |
| 277 | The Guardians must aim to discover the common element in the four cardinal virtues [Plato] |
| Full Idea: The guardians of the state should aim to get an exact idea of the common element in all the four virtues. | |
| From: Plato (The Laws [c.348 BCE], 965d) |
| 254 | Excessive laughter and tears must be avoided [Plato] |
| Full Idea: Excessive laughter and tears must be avoided. | |
| From: Plato (The Laws [c.348 BCE], 732c) |
| 266 | Injustice is the mastery of the soul by bad feelings, even if they do not lead to harm [Plato] |
| Full Idea: My general description of injustice is this: the mastery of the soul by anger, fear, pleasure, pain, envy and desires, whether they lead to actual damage or not. | |
| From: Plato (The Laws [c.348 BCE], 863e) |
| 242 | The best people are produced where there is no excess of wealth or poverty [Plato] |
| Full Idea: The community in which neither wealth nor poverty exists will produce the finest characters. | |
| From: Plato (The Laws [c.348 BCE], 679b) |
| 256 | Virtue and great wealth are incompatible [Plato] |
| Full Idea: Virtue and great wealth are quite incompatible. | |
| From: Plato (The Laws [c.348 BCE], 742e) |
| 245 | Totalitarian states destroy friendships and community spirit [Plato] |
| Full Idea: Excessively authoritarian government destroys all friendship and community of spirit in the state. | |
| From: Plato (The Laws [c.348 BCE], 697d) |
| 239 | Education in virtue produces citizens who are active but obedient [Plato] |
| Full Idea: Education in virtue produces a keen desire to become a perfect citizen who knows how to rule and be ruled as justice demands. | |
| From: Plato (The Laws [c.348 BCE], 643e) |
| 1402 | Friendship is impossible between master and slave, even if they are made equal [Plato] |
| Full Idea: Even if you proclaim that a master and a slave shall have equal status, friendship between them is inherently impossible. | |
| From: Plato (The Laws [c.348 BCE], 757a) |
| 262 | Men and women should qualify equally for honours on merit [Plato] |
| Full Idea: Men and women who have shown conspicuous merit should qualify for all honours without distinction of sex. | |
| From: Plato (The Laws [c.348 BCE], 802a) |
| 236 | Sound laws achieve the happiness of those who observe them [Plato] |
| Full Idea: Sound laws achieve the happiness of those who observe them. | |
| From: Plato (The Laws [c.348 BCE], 631b) |
| 259 | Justice is granting the equality which unequals deserve [Plato] |
| Full Idea: Justice consists of granting the 'equality' which unequals deserve to get. | |
| From: Plato (The Laws [c.348 BCE], 757d) | |
| A reaction: Beautifully simple, and hard to improve on. It shows the close link between equality and justice, but shows why they are not the same. The main debate about justice concerns the criteria for 'deserving'. |
| 238 | Children's games should channel their pleasures into adult activity [Plato] |
| Full Idea: We should use children's games to channel their pleasures and desires towards activities in which they will have to engage when they are adult. | |
| From: Plato (The Laws [c.348 BCE], 643c) |
| 260 | Control of education is the key office of state, and should go to the best citizen [Plato] |
| Full Idea: The Minister of Education is by far the most important of all the supreme offices of the state; the best all-round citizen in the state should be appointed. | |
| From: Plato (The Laws [c.348 BCE], 765e) |
| 257 | Mathematics has the widest application of any subject on the curriculum [Plato] |
| Full Idea: For domestic and public purposes, and all professional skills, no branch of a child's education has as big a range of applications as mathematics. | |
| From: Plato (The Laws [c.348 BCE], 747a) |
| 4331 | Education is channelling a child's feelings into the right course before it understands why [Plato] |
| Full Idea: I call 'education' the initial acquisition of virtue by the child, when the feelings of pleasure and affection, pain and hatred, are channelled in the right courses before he can understand the reason why. | |
| From: Plato (The Laws [c.348 BCE], 653b) | |
| A reaction: A precursor of Aristotle's view (Ethics 1104b11). A profound, simple and important insight. |
| 250 | The best way to educate the young is not to rebuke them, but to set a good example [Plato] |
| Full Idea: The best way to educate the younger generation (as well as yourself) is not to rebuke them but patently to practise all your life what you preach to others. | |
| From: Plato (The Laws [c.348 BCE], 729c) |
| 275 | Creation is not for you; you exist for the sake of creation [Plato] |
| Full Idea: Creation is not for your benefit; you exist for the sake of the universe. | |
| From: Plato (The Laws [c.348 BCE], 903c) |
| 273 | Movement is transmitted through everything, and it must have started with self-generated motion [Plato] |
| Full Idea: Motion is transmitted to innumerable things, and this must spring from some initial principle, which must be the change effected by self-generated motion. | |
| From: Plato (The Laws [c.348 BCE], 895a) |
| 8004 | In 'The Laws', to obey the law is to be obey god [Plato, by MacIntyre] |
| Full Idea: The divine is important in 'The Laws' because it is identified with law; to be obedient before the law is to be obedient before god. | |
| From: report of Plato (The Laws [c.348 BCE]) by Alasdair MacIntyre - A Short History of Ethics Ch.6 | |
| A reaction: Christian conservativism in a nutshell. Plato is rejecting his view in Euthyphro that piety (etc.) must precede the will of the gods. The obvious problem is bad laws, made by corrupt rulers. |
| 21258 | The only possible beginning for the endless motions of reality is something self-generated [Plato] |
| Full Idea: When the motion in reality is transmitted to thousands of things one after another, the entire sequence of their movements must surely spring from some initial principle, which can hardly be anything except the change effected by self-generated motion. | |
| From: Plato (The Laws [c.348 BCE], 895a) | |
| A reaction: This gives a domino picture of reality, with all of reality responding inertly to a first kick. Much better is to see self-generated motion in the active qualities of all matter, as seen in the sea of virtual subatomic particles at the smallest level. |
| 21257 | Self-generating motion is clearly superior to all other kinds of motion [Plato] |
| Full Idea: We can't resist the conclusion that the motion which can generate itself is infinitely superior, and all the others are inferior to it. | |
| From: Plato (The Laws [c.348 BCE], 894d) | |
| A reaction: Who said you can't get values from facts! Not that the argument depends on superiority. There could be an inferior First Mover, as a bus driver is subservient to the passengers, or (my favourite) a head teacher is inferior to the pupils. |
| 21261 | Self-moving soul has to be the oldest thing there is [Plato] |
| Full Idea: Soul, being the source of motion, is the most ancient thing there is. | |
| From: Plato (The Laws [c.348 BCE], 896b) | |
| A reaction: Plato seems to assume that the First Mover must still exist, which doesn't follow from anything in the argument. The First Pusher could be dead before the last domino falls. Why can't activity be the default state of everything? |
| 274 | Soul must be the cause of all the opposites, such as good and evil or beauty and ugliness [Plato] |
| Full Idea: Soul must be cause of good and evil, beauty and ugliness, justice and injustice, and all the opposites. | |
| From: Plato (The Laws [c.348 BCE], 896d) |
| 21263 | If all the motions of nature reflect calculations of reason, then the best kind of soul must direct it [Plato] |
| Full Idea: If the movement of the heavens and all that is in them reflects the motion and revolution and calculation of reason ...then clearly we have to admit that it is the best kind of soul that cares for the entire universe and directs it along the best path. | |
| From: Plato (The Laws [c.348 BCE], 897c) | |
| A reaction: Most of this passage reflects the cosmological argument - that without some initiating source natural events could not occur - but this slides into the design argument. So who designed mud (which is too inferior to have a Form)? |
| 278 | If astronomical movements are seen as necessary instead of by divine will, this leads to atheism [Plato] |
| Full Idea: If a man studying astronomy sees events apparently happening by necessity rather than being directed by the intention of a benevolent will, he will turn into an atheist. | |
| From: Plato (The Laws [c.348 BCE], 967a) |
| 21265 | The heavens must be full of gods, controlling nature either externally or from within [Plato] |
| Full Idea: A soul or souls have been shown to be cause of all the phenomena, and whether it is by their living presence in matter that they direct all the heavens, or by some other means, we insist that these souls are gods. So 'everything is full of gods'. | |
| From: Plato (The Laws [c.348 BCE], 899b) | |
| A reaction: This seems to have little to do with the pagan gods on Olympus. It is also notably not monotheistic. It is somewhere between animism and panpsychism. Does he think the rivers and woods contain gods? Probably not. Just the orderly heavens. |
| 21262 | There must be at least two souls controlling the cosmos, one doing good, the other the opposite [Plato] |
| Full Idea: There must be more than one soul (not fewer than two) controlling movement and the heavens: that which does good, and that which has the opposite capacity. | |
| From: Plato (The Laws [c.348 BCE], 896e) | |
| A reaction: [Wording compressed - as often with the dialogues] This idea of controlling opposites is found in Empedocles. Presumably this good soul defers to the Form of the Good, as implied by the Euthyphro Question. |