display all the ideas for this combination of philosophers
54 ideas
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) |
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) |
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) |
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) |
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. |
10183 | An infinite set maps into its own proper subset [Dedekind, by Reck/Price] |
Full Idea: A set is 'Dedekind-infinite' iff there exists a one-to-one function that maps a set into a proper subset of itself. | |
From: report of Richard Dedekind (Nature and Meaning of Numbers [1888], §64) by E Reck / M Price - Structures and Structuralism in Phil of Maths n 7 | |
A reaction: Sounds as if it is only infinite if it is contradictory, or doesn't know how big it is! |
22288 | We have the idea of self, and an idea of that idea, and so on, so infinite ideas are available [Dedekind, by Potter] |
Full Idea: Dedekind had an interesting proof of the Axiom of Infinity. He held that I have an a priori grasp of the idea of my self, and that every idea I can form the idea of that idea. Hence there are infinitely many objects available to me a priori. | |
From: report of Richard Dedekind (Nature and Meaning of Numbers [1888], no. 66) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 12 'Numb' | |
A reaction: Who said that Descartes' Cogito was of no use? Frege endorsed this, as long as the ideas are objective and not subjective. |
10706 | Dedekind originally thought more in terms of mereology than of sets [Dedekind, by Potter] |
Full Idea: Dedekind plainly had fusions, not collections, in mind when he avoided the empty set and used the same symbol for membership and inclusion - two tell-tale signs of a mereological conception. | |
From: report of Richard Dedekind (Nature and Meaning of Numbers [1888], 2-3) by Michael Potter - Set Theory and Its Philosophy 02.1 | |
A reaction: Potter suggests that mathematicians were torn between mereology and sets, and eventually opted whole-heartedly for sets. Maybe this is only because set theory was axiomatised by Zermelo some years before Lezniewski got to mereology. |