Combining Philosophers

All the ideas for Anaxarchus, Joseph Melia and E.J. Lemmon

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70 ideas

2. Reason / A. Nature of Reason / 1. On Reason

5750	Consistency is modal, saying propositions are consistent if they could be true together [Melia]
	Full Idea: Consistency is a modal notion: a set of propositions is consistent iff all the members of the set could be true together.
	From: Joseph Melia (Modality [2003], Ch.6)
	A reaction: This shows why Kantian ethics, for example, needs a metaphysical underpinning. Maybe Kant should have believed in the reality of Leibnizian possible worlds? An account of reason requires an account of necessity and possibility.

4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic

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)

4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / a. Symbols of PL

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).

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).

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)

4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL

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'.

4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL

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)

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)

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)

4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL

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)

4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables

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)

4. Formal Logic / B. Propositional Logic PL / 4. Soundness of PL

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.

4. Formal Logic / B. Propositional Logic PL / 5. Completeness of PL

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]

4. Formal Logic / C. Predicate Calculus PC / 1. Predicate Calculus PC

5737	Predicate logic has connectives, quantifiers, variables, predicates, equality, names and brackets [Melia]
	Full Idea: First-order predicate language has four connectives, two quantifiers, variables, predicates, equality, names, and brackets.
	From: Joseph Melia (Modality [2003], Ch.2)
	A reaction: Look up the reference for the details! The spirit of logic is seen in this basic framework, and the main interest is in the ontological commitment of the items on the list. The list is either known a priori, or it is merely conventional.

4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / a. Symbols of PC

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)

4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / b. Terminology of PC

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)

4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / c. Derivations rules of PC

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.

4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / d. Universal quantifier ∀

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)

4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / e. Existential quantifier ∃

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.

4. Formal Logic / D. Modal Logic ML / 1. Modal Logic

5744	First-order predicate calculus is extensional logic, but quantified modal logic is intensional (hence dubious) [Melia]
	Full Idea: First-order predicate calculus is an extensional logic, while quantified modal logic is intensional (which has grave problems of interpretation, according to Quine).
	From: Joseph Melia (Modality [2003], Ch.3)
	A reaction: The battle is over ontology. Quine wants the ontology to stick with the values of the variables (i.e. the items in the real world that are quantified over in the extension). The rival view arises from attempts to explain necessity and counterfactuals.

5. Theory of Logic / B. Logical Consequence / 8. Material Implication

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.

5. Theory of Logic / G. Quantification / 5. Second-Order Quantification

5740	Second-order logic needs second-order variables and quantification into predicate position [Melia]
	Full Idea: Permitting quantification into predicate position and adding second-order variables leads to second-order logic.
	From: Joseph Melia (Modality [2003], Ch.2)
	A reaction: Often expressed by saying that we now quantify over predicates and relations, rather than just objects. Depends on your metaphysical commitments.

5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models

5741	If every model that makes premises true also makes conclusion true, the argument is valid [Melia]
	Full Idea: In first-order predicate calculus validity is defined thus: an argument is valid iff every model that makes the premises of the argument true also makes the conclusion of the argument true.
	From: Joseph Melia (Modality [2003], Ch.2)
	A reaction: See Melia Ch. 2 for an explanation of a 'model'. Traditional views of validity tend to say that if the premises are true the conclusion has to be true (necessarily), but this introduces the modal term 'necessarily', which is controversial.

7. Existence / D. Theories of Reality / 8. Facts / a. Facts

5735	Maybe names and predicates can capture any fact [Melia]
	Full Idea: Some philosophers think that any fact can be captured in a language containing only names and predicates.
	From: Joseph Melia (Modality [2003], Ch.2)
	A reaction: The problem case Melia is discussing is modal facts, such as 'x is possible'. It is hard to see how 'possible' could be an ordinary predicate, but then McGinn claims that 'existence' is, and that there are some predicates with unusual characters.

5736	No sort of plain language or levels of logic can express modal facts properly [Melia]
	Full Idea: Some philosophers say that modal facts cannot be expressed either by name/predicate language, or by first-order predicate calculus, or even by second-order logic.
	From: Joseph Melia (Modality [2003], Ch.2)
	A reaction: If 'possible' were a predicate, none of this paraphernalia would be needed. If possible worlds are accepted, then the quantifiers of first-order predicate calculus will do the job. If neither of these will do, there seems to be a problem.

9. Objects / F. Identity among Objects / 7. Indiscernible Objects

5746	The Identity of Indiscernibles is contentious for qualities, and trivial for non-qualities [Melia]
	Full Idea: If the Identity of Indiscernibles is referring to qualitative properties, such as 'being red' or 'having mass', it is contentious; if it is referring to non-qualitative properties, such as 'member of set s' or 'brother of a', it is true but trivial.
	From: Joseph Melia (Modality [2003], Ch.3 n 11)
	A reaction: I would say 'false' rather than 'contentious'. No one has ever offered a way of distinguishing two electrons, but that doesn't mean there is just one (very busy) electron. The problem is that 'indiscernible' is only an epistemological concept.

10. Modality / A. Necessity / 2. Nature of Necessity

5738	We may be sure that P is necessary, but is it necessarily necessary? [Melia]
	Full Idea: We may have fairly firm beliefs as to whether or not P is necessary, but many of us find ourselves at a complete loss when wondering whether or not P is necessarily necessary.
	From: Joseph Melia (Modality [2003], Ch.2)
	A reaction: I think it is questions like this which are pushing philosophy back towards some sort of rationalism. See Idea 3651, for example. A regress of necessities would be mad, so necessity must be taken as self-evident (in itself, though maybe not to us).

10. Modality / A. Necessity / 4. De re / De dicto modality

5732	'De re' modality is about things themselves, 'de dicto' modality is about propositions [Melia]
	Full Idea: In cases of 'de re' modality, it is a particular thing that has the property essentially or accidentally; where the modality attaches to the proposition, it is 'de dicto' - it is the whole truth that all bachelors are unmarried that is necessary.
	From: Joseph Melia (Modality [2003], Ch.1)
	A reaction: This seems to me one of the most important distinctions in metaphysics (as practised by analytical philosophers, who like distinctions). The first type leads off into the ontology, the second type veers towards epistemology.

10. Modality / B. Possibility / 1. Possibility

5739	Sometimes we want to specify in what ways a thing is possible [Melia]
	Full Idea: Sometimes we want to count the ways in which something is possible, or say that there are many ways in which a certain thing is possible.
	From: Joseph Melia (Modality [2003], Ch.2)
	A reaction: This is a basic fact about talk of 'possibility'. It is not an all-or-nothing property of a situation. There can be 'faint' possibilities of things. The proximity of some possible worlds, especially those sharing our natural laws, is one answer.

10. Modality / E. Possible worlds / 1. Possible Worlds / a. Possible worlds

5734	Possible worlds make it possible to define necessity and counterfactuals without new primitives [Melia]
	Full Idea: In modal logic the concepts of necessity and counterfactuals are not interdefinable, so the language needs two primitives to represent them, but with the machinery of possible worlds they are defined by what is the case in all worlds, or close worlds.
	From: Joseph Melia (Modality [2003], Ch.1)
	A reaction: If your motivation is to reduce ontology to the barest of minimums (which it was for David Lewis) then it is paradoxical that the existence of possible worlds may be the way to achieve it. I doubt, though, whether a commitment to their reality is needed.

5742	In possible worlds semantics the modal operators are treated as quantifiers [Melia]
	Full Idea: The central idea in possible worlds semantics is that the modal operators are treated as quantifiers.
	From: Joseph Melia (Modality [2003], Ch.2)
	A reaction: It seems an essential requirement of metaphysics that an account be given of possibility and necessity, and it is also a good dream to keep the ontology simple. Commitment to possible worlds is the bizarre outcome of this dream.

5743	If possible worlds semantics is not realist about possible worlds, logic becomes merely formal [Melia]
	Full Idea: It has proved difficult to justify possible worlds semantics without accepting possible worlds. Without a secure metaphysical underpinning, the results in logic are in danger of having nothing more than a formal significance.
	From: Joseph Melia (Modality [2003], Ch.2)
	A reaction: This makes nicely clear why Lewis's controversial modal realism has to be taken seriously. It appears that the key problem is truth, because that is needed to define validity, but you can't have truth without some sort of metaphysics.

5749	Possible worlds could be real as mathematics, propositions, properties, or like books [Melia]
	Full Idea: One can be a realist about possible worlds without adopting Lewis's extreme views; they might be abstract or mathematical entities; they might be sets of propositions or maximal uninstantiated properties; they might be like books or pictures.
	From: Joseph Melia (Modality [2003], Ch.6)
	A reaction: My intuition is that once you go down the road of realism about possible worlds, Lewis's full concrete realism looks at least as attractive as any of these options. You can discuss the 'average man' in an economic theory without realism.

10. Modality / E. Possible worlds / 2. Nature of Possible Worlds / b. Worlds as fictions

5751	The truth of propositions at possible worlds are implied by the world, just as in books [Melia]
	Full Idea: Propositions are true at possible worlds in much the same way as they are true at books: by being implied by the book.
	From: Joseph Melia (Modality [2003], Ch.7)
	A reaction: An intriguing way to introduce the view that possible worlds should be seen as like books. The truth-makers of propositions about the actual world are items in it, but the truth-makers in novels (say) are the conditions of the whole work as united.

13. Knowledge Criteria / D. Scepticism / 1. Scepticism

3061	Anaxarchus said that he was not even sure that he knew nothing [Anaxarchus, by Diog. Laertius]
	Full Idea: Anaxarchus said that he was not even sure that he knew nothing.
	From: report of Anaxarchus (fragments/reports [c.340 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 09.10.1

19. Language / A. Nature of Meaning / 5. Meaning as Verification

5748	We accept unverifiable propositions because of simplicity, utility, explanation and plausibility [Melia]
	Full Idea: Many philosophers now concede that it is rational to accept a proposition not because we can directly verify it but because it is supported by considerations of simplicity, theoretical utility, explanatory power and/or intuitive plausibility.
	From: Joseph Melia (Modality [2003], Ch.5)
	A reaction: This suggests how the weakness of logical positivism may have led us to the concept of epistemic virtues (such as those listed), which are, of course, largely a matter of community consensus, just as the moral virtues are.