67 ideas
| 10688 | 'Equivocation' is when terms do not mean the same thing in premises and conclusion [Beall/Restall] |
| Full Idea: 'Equivocation' is when the terms do not mean the same thing in the premises and in the conclusion. | |
| From: JC Beall / G Restall (Logical Consequence [2005], Intro) |
| 9535 | 'Contradictory' propositions always differ in truth-value [Lemmon] |
| Full Idea: Two propositions are 'contradictory' if they are never both true and never both false either, which means that ¬(A↔B) is a tautology. | |
| From: E.J. Lemmon (Beginning Logic [1965], 2.3) |
| 9511 | We write the conditional 'if P (antecedent) then Q (consequent)' as P→Q [Lemmon] |
| Full Idea: We write 'if P then Q' as P→Q. This is called a 'conditional', with P as its 'antecedent', and Q as its 'consequent'. | |
| From: E.J. Lemmon (Beginning Logic [1965], 1.2) | |
| A reaction: P→Q can also be written as ¬P∨Q. |
| 9510 | That proposition that either P or Q is their 'disjunction', written P∨Q [Lemmon] |
| Full Idea: If P and Q are any two propositions, the proposition that either P or Q is called the 'disjunction' of P and Q, and is written P∨Q. | |
| From: E.J. Lemmon (Beginning Logic [1965], 1.3) | |
| A reaction: This is inclusive-or (meaning 'P, or Q, or both'), and not exlusive-or (Boolean XOR), which means 'P, or Q, but not both'. The ∨ sign is sometimes called 'vel' (Latin). |
| 9512 | We write the 'negation' of P (not-P) as ¬ [Lemmon] |
| Full Idea: We write 'not-P' as ¬P. This is called the 'negation' of P. The 'double negation' of P (not not-P) would be written as ¬¬P. | |
| From: E.J. Lemmon (Beginning Logic [1965], 1.2) | |
| A reaction: Lemmons use of -P is no longer in use for 'not'. A tilde sign (squiggle) is also used for 'not', but some interpreters give that a subtly different meaning (involving vagueness). The sign ¬ is sometimes called 'hook' or 'corner'. |
| 9513 | We write 'P if and only if Q' as P↔Q; it is also P iff Q, or (P→Q)∧(Q→P) [Lemmon] |
| Full Idea: We write 'P if and only if Q' as P↔Q. It is called the 'biconditional', often abbreviate in writing as 'iff'. It also says that P is both sufficient and necessary for Q, and may be written out in full as (P→Q)∧(Q→P). | |
| From: E.J. Lemmon (Beginning Logic [1965], 1.4) | |
| A reaction: If this symbol is found in a sequence, the first move in a proof is to expand it to the full version. |
| 9514 | If A and B are 'interderivable' from one another we may write A -||- B [Lemmon] |
| Full Idea: If we say that A and B are 'interderivable' from one another (that is, A |- B and B |- A), then we may write A -||- B. | |
| From: E.J. Lemmon (Beginning Logic [1965], 1.5) |
| 9509 | That proposition that both P and Q is their 'conjunction', written P∧Q [Lemmon] |
| Full Idea: If P and Q are any two propositions, the proposition that both P and Q is called the 'conjunction' of P and Q, and is written P∧Q. | |
| From: E.J. Lemmon (Beginning Logic [1965], 1.3) | |
| A reaction: [I use the more fashionable inverted-v '∧', rather than Lemmon's '&', which no longer seems to be used] P∧Q can also be defined as ¬(¬P∨¬Q) |
| 9508 | The sign |- may be read as 'therefore' [Lemmon] |
| Full Idea: I introduce the sign |- to mean 'we may validly conclude'. To call it the 'assertion sign' is misleading. It may conveniently be read as 'therefore'. | |
| From: E.J. Lemmon (Beginning Logic [1965], 1.2) | |
| A reaction: [Actually no gap between the vertical and horizontal strokes of the sign] As well as meaning 'assertion', it may also mean 'it is a theorem that' (with no proof shown). |
| 9516 | A 'well-formed formula' follows the rules for variables, ¬, →, ∧, ∨, and ↔ [Lemmon] |
| Full Idea: A 'well-formed formula' of the propositional calculus is a sequence of symbols which follows the rules for variables, ¬, →, ∧, ∨, and ↔. | |
| From: E.J. Lemmon (Beginning Logic [1965], 2.1) |
| 9517 | The 'scope' of a connective is the connective, the linked formulae, and the brackets [Lemmon] |
| Full Idea: The 'scope' of a connective in a certain formula is the formulae linked by the connective, together with the connective itself and the (theoretically) encircling brackets | |
| From: E.J. Lemmon (Beginning Logic [1965], 2.1) |
| 9519 | A 'substitution-instance' is a wff formed by consistent replacing variables with wffs [Lemmon] |
| Full Idea: A 'substitution-instance' is a wff which results by replacing one or more variables throughout with the same wffs (the same wff replacing each variable). | |
| From: E.J. Lemmon (Beginning Logic [1965], 2.2) |
| 9529 | A wff is 'inconsistent' if all assignments to variables result in the value F [Lemmon] |
| Full Idea: If a well-formed formula of propositional calculus takes the value F for all possible assignments of truth-values to its variables, it is said to be 'inconsistent'. | |
| From: E.J. Lemmon (Beginning Logic [1965], 2.3) |
| 9531 | 'Contrary' propositions are never both true, so that ¬(A∧B) is a tautology [Lemmon] |
| Full Idea: If A and B are expressible in propositional calculus notation, they are 'contrary' if they are never both true, which may be tested by the truth-table for ¬(A∧B), which is a tautology if they are contrary. | |
| From: E.J. Lemmon (Beginning Logic [1965], 2.3) |
| 9534 | Two propositions are 'equivalent' if they mirror one another's truth-value [Lemmon] |
| Full Idea: Two propositions are 'equivalent' if whenever A is true B is true, and whenever B is true A is true, in which case A↔B is a tautology. | |
| From: E.J. Lemmon (Beginning Logic [1965], 2.3) |
| 9530 | A wff is 'contingent' if produces at least one T and at least one F [Lemmon] |
| Full Idea: If a well-formed formula of propositional calculus takes at least one T and at least one F for all the assignments of truth-values to its variables, it is said to be 'contingent'. | |
| From: E.J. Lemmon (Beginning Logic [1965], 2.3) |
| 9532 | 'Subcontrary' propositions are never both false, so that A∨B is a tautology [Lemmon] |
| Full Idea: If A and B are expressible in propositional calculus notation, they are 'subcontrary' if they are never both false, which may be tested by the truth-table for A∨B, which is a tautology if they are subcontrary. | |
| From: E.J. Lemmon (Beginning Logic [1965], 2.3) |
| 9533 | A 'implies' B if B is true whenever A is true (so that A→B is tautologous) [Lemmon] |
| Full Idea: One proposition A 'implies' a proposition B if whenever A is true B is true (but not necessarily conversely), which is only the case if A→B is tautologous. Hence B 'is implied' by A. | |
| From: E.J. Lemmon (Beginning Logic [1965], 2.3) |
| 9528 | A wff is a 'tautology' if all assignments to variables result in the value T [Lemmon] |
| Full Idea: If a well-formed formula of propositional calculus takes the value T for all possible assignments of truth-values to its variables, it is said to be a 'tautology'. | |
| From: E.J. Lemmon (Beginning Logic [1965], 2.3) |
| 9518 | A 'theorem' is the conclusion of a provable sequent with zero assumptions [Lemmon] |
| Full Idea: A 'theorem' of logic is the conclusion of a provable sequent in which the number of assumptions is zero. | |
| From: E.J. Lemmon (Beginning Logic [1965], 2.2) | |
| A reaction: This is what Quine and others call a 'logical truth'. |
| 9398 | ∧I: Given A and B, we may derive A∧B [Lemmon] |
| Full Idea: And-Introduction (&I): Given A and B, we may derive A∧B as conclusion. This depends on their previous assumptions. | |
| From: E.J. Lemmon (Beginning Logic [1965], 1.5) |
| 9397 | CP: Given a proof of B from A as assumption, we may derive A→B [Lemmon] |
| Full Idea: Conditional Proof (CP): Given a proof of B from A as assumption, we may derive A→B as conclusion, on the remaining assumptions (if any). | |
| From: E.J. Lemmon (Beginning Logic [1965], 1.5) |
| 9396 | DN: Given A, we may derive ¬¬A [Lemmon] |
| Full Idea: Double Negation (DN): Given A, we may derive ¬¬A as a conclusion, and vice versa. The conclusion depends on the assumptions of the premiss. | |
| From: E.J. Lemmon (Beginning Logic [1965], 1.5) |
| 9393 | A: we may assume any proposition at any stage [Lemmon] |
| Full Idea: Assumptions (A): any proposition may be introduced at any stage of a proof. | |
| From: E.J. Lemmon (Beginning Logic [1965], 1.5) |
| 9399 | ∧E: Given A∧B, we may derive either A or B separately [Lemmon] |
| Full Idea: And-Elimination (∧E): Given A∧B, we may derive either A or B separately. The conclusions will depend on the assumptions of the premiss. | |
| From: E.J. Lemmon (Beginning Logic [1965], 1.5) |
| 9402 | RAA: If assuming A will prove B∧¬B, then derive ¬A [Lemmon] |
| Full Idea: Reduction ad Absurdum (RAA): Given a proof of B∧¬B from A as assumption, we may derive ¬A as conclusion, depending on the remaining assumptions (if any). | |
| From: E.J. Lemmon (Beginning Logic [1965], 1.5) |
| 9395 | MTT: Given ¬B and A→B, we derive ¬A [Lemmon] |
| Full Idea: Modus Tollendo Tollens (MTT): Given ¬B and A→B, we derive ¬A as a conclusion. ¬A depends on any assumptions that have been made | |
| From: E.J. Lemmon (Beginning Logic [1965], 1.5) |
| 9400 | ∨I: Given either A or B separately, we may derive A∨B [Lemmon] |
| Full Idea: Or-Introduction (∨I): Given either A or B separately, we may derive A∨B as conclusion. This depends on the assumption of the premisses. | |
| From: E.J. Lemmon (Beginning Logic [1965], 1.5) |
| 9401 | ∨E: Derive C from A∨B, if C can be derived both from A and from B [Lemmon] |
| Full Idea: Or-Elimination (∨E): Given A∨B, we may derive C if it is proved from A as assumption and from B as assumption. This will also depend on prior assumptions. | |
| From: E.J. Lemmon (Beginning Logic [1965], 1.5) |
| 9394 | MPP: Given A and A→B, we may derive B [Lemmon] |
| Full Idea: Modus Ponendo Ponens (MPP): Given A and A→B, we may derive B as a conclusion. B will rest on any assumptions that have been made. | |
| From: E.J. Lemmon (Beginning Logic [1965], 1.5) |
| 9521 | 'Modus tollendo ponens' (MTP) says ¬P, P ∨ Q |- Q [Lemmon] |
| Full Idea: 'Modus tollendo ponens' (MTP) says that if a disjunction holds and also the negation of one of its disjuncts, then the other disjunct holds. Thus ¬P, P ∨ Q |- Q may be introduced as a theorem. | |
| From: E.J. Lemmon (Beginning Logic [1965], 2.2) | |
| A reaction: Unlike Modus Ponens and Modus Tollens, this is a derived rule. |
| 9522 | 'Modus ponendo tollens' (MPT) says P, ¬(P ∧ Q) |- ¬Q [Lemmon] |
| Full Idea: 'Modus ponendo tollens' (MPT) says that if the negation of a conjunction holds and also one of its conjuncts, then the negation of the other conjunct holds. Thus P, ¬(P ∧ Q) |- ¬Q may be introduced as a theorem. | |
| From: E.J. Lemmon (Beginning Logic [1965], 2.2) | |
| A reaction: Unlike Modus Ponens and Modus Tollens, this is a derived rule. |
| 9525 | We can change conditionals into negated conjunctions with P→Q -||- ¬(P ∧ ¬Q) [Lemmon] |
| Full Idea: The proof that P→Q -||- ¬(P ∧ ¬Q) is useful for enabling us to change conditionals into negated conjunctions | |
| From: E.J. Lemmon (Beginning Logic [1965], 2.2) |
| 9524 | We can change conditionals into disjunctions with P→Q -||- ¬P ∨ Q [Lemmon] |
| Full Idea: The proof that P→Q -||- ¬P ∨ Q is useful for enabling us to change conditionals into disjunctions. | |
| From: E.J. Lemmon (Beginning Logic [1965], 2.2) |
| 9523 | De Morgan's Laws make negated conjunctions/disjunctions into non-negated disjunctions/conjunctions [Lemmon] |
| Full Idea: The forms of De Morgan's Laws [P∨Q -||- ¬(¬P ∧ ¬Q); ¬(P∨Q) -||- ¬P ∧ ¬Q; ¬(P∧Q) -||- ¬P ∨ ¬Q); P∧Q -||- ¬(¬P∨¬Q)] transform negated conjunctions and disjunctions into non-negated disjunctions and conjunctions respectively. | |
| From: E.J. Lemmon (Beginning Logic [1965], 2.2) |
| 9527 | The Distributive Laws can rearrange a pair of conjunctions or disjunctions [Lemmon] |
| Full Idea: The Distributive Laws say that P ∧ (Q∨R) -||- (P∧Q) ∨ (P∧R), and that P ∨ (Q∨R) -||- (P∨Q) ∧ (P∨R) | |
| From: E.J. Lemmon (Beginning Logic [1965], 2.2) |
| 9526 | We can change conjunctions into negated conditionals with P→Q -||- ¬(P → ¬Q) [Lemmon] |
| Full Idea: The proof that P∧Q -||- ¬(P → ¬Q) is useful for enabling us to change conjunctions into negated conditionals. | |
| From: E.J. Lemmon (Beginning Logic [1965], 2.2) |
| 9537 | Truth-tables are good for showing invalidity [Lemmon] |
| Full Idea: The truth-table approach enables us to show the invalidity of argument-patterns, as well as their validity. | |
| From: E.J. Lemmon (Beginning Logic [1965], 2.4) |
| 9538 | A truth-table test is entirely mechanical, but this won't work for more complex logic [Lemmon] |
| Full Idea: A truth-table test is entirely mechanical, ..and in propositional logic we can even generate proofs mechanically for tautological sequences, ..but this mechanical approach breaks down with predicate calculus, and proof-discovery is an imaginative process. | |
| From: E.J. Lemmon (Beginning Logic [1965], 2.5) |
| 9536 | If any of the nine rules of propositional logic are applied to tautologies, the result is a tautology [Lemmon] |
| Full Idea: If any application of the nine derivation rules of propositional logic is made on tautologous sequents, we have demonstrated that the result is always a tautologous sequent. Thus the system is consistent. | |
| From: E.J. Lemmon (Beginning Logic [1965], 2.4) | |
| A reaction: The term 'sound' tends to be used now, rather than 'consistent'. See Lemmon for the proofs of each of the nine rules. |
| 9539 | Propositional logic is complete, since all of its tautologous sequents are derivable [Lemmon] |
| Full Idea: A logical system is complete is all expressions of a specified kind are derivable in it. If we specify tautologous sequent-expressions, then propositional logic is complete, because we can show that all tautologous sequents are derivable. | |
| From: E.J. Lemmon (Beginning Logic [1965], 2.5) | |
| A reaction: [See Lemmon 2.5 for details of the proofs] |
| 13909 | Write '(∀x)(...)' to mean 'take any x: then...', and '(∃x)(...)' to mean 'there is an x such that....' [Lemmon] |
| Full Idea: Just as '(∀x)(...)' is to mean 'take any x: then....', so we write '(∃x)(...)' to mean 'there is an x such that....' | |
| From: E.J. Lemmon (Beginning Logic [1965], 3.1) | |
| A reaction: [Actually Lemmon gives the universal quantifier symbol as '(x)', but the inverted A ('∀') seems to have replaced it these days] |
| 13902 | 'Gm' says m has property G, and 'Pmn' says m has relation P to n [Lemmon] |
| Full Idea: A predicate letter followed by one name expresses a property ('Gm'), and a predicate-letter followed by two names expresses a relation ('Pmn'). We could write 'Pmno' for a complex relation like betweenness. | |
| From: E.J. Lemmon (Beginning Logic [1965], 3.1) |
| 13911 | The 'symbols' are bracket, connective, term, variable, predicate letter, reverse-E [Lemmon] |
| Full Idea: I define a 'symbol' (of the predicate calculus) as either a bracket or a logical connective or a term or an individual variable or a predicate-letter or reverse-E (∃). | |
| From: E.J. Lemmon (Beginning Logic [1965], 4.1) |
| 13910 | Our notation uses 'predicate-letters' (for 'properties'), 'variables', 'proper names', 'connectives' and 'quantifiers' [Lemmon] |
| Full Idea: Quantifier-notation might be thus: first, render into sentences about 'properties', and use 'predicate-letters' for them; second, introduce 'variables'; third, introduce propositional logic 'connectives' and 'quantifiers'. Plus letters for 'proper names'. | |
| From: E.J. Lemmon (Beginning Logic [1965], 3.1) |
| 13904 | Universal Elimination (UE) lets us infer that an object has F, from all things having F [Lemmon] |
| Full Idea: Our rule of universal quantifier elimination (UE) lets us infer that any particular object has F from the premiss that all things have F. It is a natural extension of &E (and-elimination), as universal propositions generally affirm a complex conjunction. | |
| From: E.J. Lemmon (Beginning Logic [1965], 3.2) |
| 13906 | With finite named objects, we can generalise with &-Intro, but otherwise we need ∀-Intro [Lemmon] |
| Full Idea: If there are just three objects and each has F, then by an extension of &I we are sure everything has F. This is of no avail, however, if our universe is infinitely large or if not all objects have names. We need a new device, Universal Introduction, UI. | |
| From: E.J. Lemmon (Beginning Logic [1965], 3.2) |
| 13908 | UE all-to-one; UI one-to-all; EI arbitrary-to-one; EE proof-to-one [Lemmon] |
| Full Idea: Univ Elim UE - if everything is F, then something is F; Univ Intro UI - if an arbitrary thing is F, everything is F; Exist Intro EI - if an arbitrary thing is F, something is F; Exist Elim EE - if a proof needed an object, there is one. | |
| From: E.J. Lemmon (Beginning Logic [1965], 3.3) | |
| A reaction: [My summary of Lemmon's four main rules for predicate calculus] This is the natural deduction approach, of trying to present the logic entirely in terms of introduction and elimination rules. See Bostock on that. |
| 13901 | Predicate logic uses propositional connectives and variables, plus new introduction and elimination rules [Lemmon] |
| Full Idea: In predicate calculus we take over the propositional connectives and propositional variables - but we need additional rules for handling quantifiers: four rules, an introduction and elimination rule for the universal and existential quantifiers. | |
| From: E.J. Lemmon (Beginning Logic [1965]) | |
| A reaction: This is Lemmon's natural deduction approach (invented by Gentzen), which is largely built on introduction and elimination rules. |
| 13903 | Universal elimination if you start with the universal, introduction if you want to end with it [Lemmon] |
| Full Idea: The elimination rule for the universal quantifier concerns the use of a universal proposition as a premiss to establish some conclusion, whilst the introduction rule concerns what is required by way of a premiss for a universal proposition as conclusion. | |
| From: E.J. Lemmon (Beginning Logic [1965], 3.2) | |
| A reaction: So if you start with the universal, you need to eliminate it, and if you start without it you need to introduce it. |
| 13905 | If there is a finite domain and all objects have names, complex conjunctions can replace universal quantifiers [Lemmon] |
| Full Idea: If all objects in a given universe had names which we knew and there were only finitely many of them, then we could always replace a universal proposition about that universe by a complex conjunction. | |
| From: E.J. Lemmon (Beginning Logic [1965], 3.2) |
| 13900 | 'Some Frenchmen are generous' is rendered by (∃x)(Fx→Gx), and not with the conditional → [Lemmon] |
| Full Idea: It is a common mistake to render 'some Frenchmen are generous' by (∃x)(Fx→Gx) rather than the correct (∃x)(Fx&Gx). 'All Frenchmen are generous' is properly rendered by a conditional, and true if there are no Frenchmen. | |
| From: E.J. Lemmon (Beginning Logic [1965], 3.1) | |
| A reaction: The existential quantifier implies the existence of an x, but the universal quantifier does not. |
| 21695 | The set scheme discredited by paradoxes is actually the most natural one [Quine] |
| Full Idea: Each proposed revision of set theory is unnatural, because the natural scheme is the unrestricted one that the antinomies discredit. | |
| From: Willard Quine (The Ways of Paradox [1961], p.16) | |
| A reaction: You can either takes this free-far-all version of set theory, and gradually restrain it for each specific problem, or start from scratch and build up in safe steps. The latter is (I think) the 'iterated' approach. |
| 21693 | Russell's antinomy challenged the idea that any condition can produce a set [Quine] |
| Full Idea: In the case of Russell's antinomy, the tacit and trusted pattern of reasoning that is found wanting is this: for any condition you can formulate, there is a class whose members are the things meeting the condition. | |
| From: Willard Quine (The Ways of Paradox [1961], p.11) | |
| A reaction: This is why Russell's Paradox is so important for set theory, which in turn makes it important for the foundations of mathematics. |
| 10690 | Formal logic is invariant under permutations, or devoid of content, or gives the norms for thought [Beall/Restall] |
| Full Idea: Logic is purely formal either when it is invariant under permutation of object (Tarski), or when it has totally abstracted away from all contents, or it is the constitutive norms for thought. | |
| From: JC Beall / G Restall (Logical Consequence [2005], 2) | |
| A reaction: [compressed] The third account sounds rather woolly, and the second one sounds like a tricky operation, but the first one sounds clear and decisive, so I vote for Tarski. |
| 10691 | Logical consequence needs either proofs, or absence of counterexamples [Beall/Restall] |
| Full Idea: Technical work on logical consequence has either focused on proofs, where validity is the existence of a proof of the conclusions from the premises, or on models, which focus on the absence of counterexamples. | |
| From: JC Beall / G Restall (Logical Consequence [2005], 3) |
| 10695 | Logical consequence is either necessary truth preservation, or preservation based on interpretation [Beall/Restall] |
| Full Idea: Two different views of logical consequence are necessary truth-preservation (based on modelling possible worlds; favoured by Realists), or truth-preservation based on the meanings of the logical vocabulary (differing in various models; for Anti-Realists). | |
| From: JC Beall / G Restall (Logical Consequence [2005], 2) | |
| A reaction: Thus Dummett prefers the second view, because the law of excluded middle is optional. My instincts are with the first one. |
| 10689 | A step is a 'material consequence' if we need contents as well as form [Beall/Restall] |
| Full Idea: A logical step is a 'material consequence' and not a formal one, if we need the contents as well as the structure or form. | |
| From: JC Beall / G Restall (Logical Consequence [2005], 2) |
| 9520 | The paradoxes of material implication are P |- Q → P, and ¬P |- P → Q [Lemmon] |
| Full Idea: The paradoxes of material implication are P |- Q → P, and ¬P |- P → Q. That is, since Napoleon was French, then if the moon is blue then Napoleon was French; and since Napoleon was not Chinese, then if Napoleon was Chinese, the moon is blue. | |
| From: E.J. Lemmon (Beginning Logic [1965], 2.2) | |
| A reaction: This is why the symbol → does not really mean the 'if...then' of ordinary English. Russell named it 'material implication' to show that it was a distinctively logical operator. |
| 10696 | A 'logical truth' (or 'tautology', or 'theorem') follows from empty premises [Beall/Restall] |
| Full Idea: If a conclusion follows from an empty collection of premises, it is true by logic alone, and is a 'logical truth' (sometimes a 'tautology'), or, in the proof-centred approach, 'theorems'. | |
| From: JC Beall / G Restall (Logical Consequence [2005], 4) | |
| A reaction: These truths are written as following from the empty set Φ. They are just implications derived from the axioms and the rules. |
| 10693 | Models are mathematical structures which interpret the non-logical primitives [Beall/Restall] |
| Full Idea: Models are abstract mathematical structures that provide possible interpretations for each of the non-logical primitives in a formal language. | |
| From: JC Beall / G Restall (Logical Consequence [2005], 3) |
| 21691 | Antinomies contradict accepted ways of reasoning, and demand revisions [Quine] |
| Full Idea: An 'antinomy' produces a self-contradiction by accepted ways of reasoning. It establishes that some tacit and trusted pattern of reasoning must be made explicit and henceforward be avoided or revised. | |
| From: Willard Quine (The Ways of Paradox [1961], p.05) | |
| A reaction: Quine treats antinomies as of much greater importance than mere paradoxes. It is often possible to give simple explanations of paradoxes, but antinomies go to the root of our belief system. This was presumably Kant's intended meaning. |
| 21690 | Whenever the pursuer reaches the spot where the pursuer has been, the pursued has moved on [Quine] |
| Full Idea: The Achilles argument is that (if the front runner keeps running) each time the pursuer reaches a spot where the pursuer has been, the pursued has moved a bit beyond. | |
| From: Willard Quine (The Ways of Paradox [1961], p.03) | |
| A reaction: Quine is always wonderfully lucid, and this is the clearest simple statement of the paradox. |
| 21689 | A barber shaves only those who do not shave themselves. So does he shave himself? [Quine] |
| Full Idea: In a certain village there is a barber, who shaves all and only those men in the village who do not shave themselves. So does the barber shave himself? The barber shaves himself if and only if he does not shave himself. | |
| From: Willard Quine (The Ways of Paradox [1961], p.02) | |
| A reaction: [Russell himself quoted this version of his paradox, from an unnamed source] Quine treats his as trivial because it only concerns barbers, but the full Russell paradox is a major 'antinomy', because it concerns sets. |
| 21694 | Membership conditions which involve membership and non-membership are paradoxical [Quine] |
| Full Idea: With Russell's antinomy, ...each tie the trouble comes of taking a membership condition that itself talks in turn of membership and non-membership. | |
| From: Willard Quine (The Ways of Paradox [1961], p.13) | |
| A reaction: Hence various stipulations to rule out vicious circles or referring to sets of the 'wrong type' are invoked to cure the problem. The big question is how strong to make the restrictions. |
| 21692 | If we write it as '"this sentence is false" is false', there is no paradox [Quine] |
| Full Idea: If we supplant the sentence 'this sentence is false' with one saying what it refers to, we get '"this sentence is false" is false'. But then the whole outside sentence attributes falsity no longer to itself but to something else, so there is no paradox. | |
| From: Willard Quine (The Ways of Paradox [1961], p.07) | |
| A reaction: Quine is pointing us towards type theory and meta-languages to solve the problem. We now have the Revenge Liar, and the problem has not been fully settled. |
| 10692 | Hilbert proofs have simple rules and complex axioms, and natural deduction is the opposite [Beall/Restall] |
| Full Idea: There are many proof-systems, the main being Hilbert proofs (with simple rules and complex axioms), or natural deduction systems (with few axioms and many rules, and the rules constitute the meaning of the connectives). | |
| From: JC Beall / G Restall (Logical Consequence [2005], 3) |