101 ideas
21634 | Metaphysics is (supposedly) first the ontology, then in general what things are like [Hofweber] |
Full Idea: Metaphysics can be divided into two parts: first ontology, which is supposed to tell us what there is in general. The second part is the rest of metaphysics, which is supposed to tell us what these things are like, in various general ways. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 01.1) | |
A reaction: Hofweber is a fairly sceptical guide to metaphysics, but this has been the standard view for the last decade. Before that, Quine had set an agenda of mere ontology. |
16415 | Esoteric metaphysics aims to be top science, investigating ultimate reality [Hofweber] |
Full Idea: Esoteric metaphysics appeals to those, I conjecture, who deep down hold that philosophy is the queen of sciences after all, since it investigates what the world is REALLY like. | |
From: Thomas Hofweber (Ambitious, yet modest, Metaphysics [2009], 2) | |
A reaction: He mentions Kit Fine and Jonathan Schaffer as esoteric metaphysicians. I see a pyramid of increasing generality and abstraction, with metaphysics at the top. This doesn't make it 'queen', though, because uncertainties multiply higher up. |
16413 | Science has discovered properties of things, so there are properties - so who needs metaphysics? [Hofweber] |
Full Idea: Material science has found that some features of metals make them more susceptible to corrosion but more resistant to fracture. Thus this immediately implies that there are features, i.e. properties. What is left for metaphysics to do? | |
From: Thomas Hofweber (Ambitious, yet modest, Metaphysics [2009], 1.1) | |
A reaction: Presumably economists have discovered 'features' of economies that cause unemployment, and literary critics have discovered 'features' of novels that make them good. |
21666 | 'Fundamentality' is either a superficial idea, or much too obscure [Hofweber] |
Full Idea: The dilemma of neo-Aristotelian metaphysics is that on an ordinary reading of prioriy, 'fundamentality' won't give the intended results, and on a metaphysical reading it turns into esoteric metaphysics. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 13.4.2) | |
A reaction: Hofweber is hostile to 'esoteric' metaphysics, but sympathetic to 'egalitarian' metaphysics, which anyone can understand (with a bit of effort). |
21640 | 'It's true that Fido is a dog' conjures up a contrast class, of 'it's false' or 'it's unlikely' [Hofweber] |
Full Idea: 'It's true that Fido is a dog' gives rise to a contrastive focus on 'true', with the contrast class probably depending on members like 'it's false that...' or 'it's unlikely that...'. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 02.6.3) | |
A reaction: If we introduce (from linguistics) the idea of a 'contrast class', then Ramsey's famous example begins to sound meaningful. It might occur in a discussion of 'did Antony actually say 'Friends, Romans. countrymen'?' |
17990 | Instances of minimal truth miss out propositions inexpressible in current English [Hofweber] |
Full Idea: A standard objection to minimalist truth is the 'incompleteness objection'. Since there are propositions inexpressible in present English the concept of truth isn't captured by all the instances of the Tarski biconditional. | |
From: Thomas Hofweber (Inexpressible Properties and Propositions [2006], 5.3) | |
A reaction: Sounds like a good objection. |
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. |
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) |
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) |
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). |
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. |
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) |
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'. |
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) |
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) |
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) |
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) |
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) |
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) |
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) |
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) |
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) |
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) |
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) |
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) |
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) |
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. |
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) |
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) |
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) |
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) |
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) |
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] |
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) |
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] |
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) |
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. |
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. |
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. |
21657 | Since properties can have properties, some theorists rank them in 'types' [Hofweber] |
Full Idea: Since properties themselves can have properties there is a well-known division in the theory of properties between those who take a typed and those who take a type-free approach. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 08.5) | |
A reaction: I take this idea to be about linguistic predicates, and about semantics which draws on model theory. To see it as about actual 'properties' in the physical world makes no sense. |
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. |
21653 | Maybe not even names are referential, but are just by used by speakers to refer [Hofweber] |
Full Idea: A more radical alternative which takes names not to be referring even in the broader sense, but only takes speakers to refer with uses of names. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 08.1) | |
A reaction: Given that you can make up nicknames and silly nonce names for people, this seems plausible. I may say a name in a crowded room and three people look up. |
10001 | An adjective contributes semantically to a noun phrase [Hofweber] |
Full Idea: The semantic value of a determiner (an adjective) is a function from semantic values to nouns to semantic values of full noun phrases. | |
From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §3.1) | |
A reaction: This kind of states the obvious (assuming one has a compositional view of sentences), but his point is that you can't just eliminate adjectival uses of numbers by analysing them away, as if they didn't do anything. |
21636 | 'Singular terms' are not found in modern linguistics, and are not the same as noun phrases [Hofweber] |
Full Idea: Being a 'singular term' is not a category in contemporary syntactic theory and it doesn't correspond to any of the notions employed there like that of a singular noun phrase or the like. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 02.3) | |
A reaction: Hofweber has researched such things. This is an important objection to the reliance of modern Fregeans on the ontological commitments of singular terms (as proof that there are 'mathematical objects'). |
21637 | If two processes are said to be identical, that doesn't make their terms refer to entities [Hofweber] |
Full Idea: Identity between objects occurs in 'How Mary makes a chocolate cake is identical to how my grandfather used to make it', but does this show that 'how Mary makes a chocolate cake' aims to pick out an entity? | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 02.3) | |
A reaction: This is a counterexample to the Fregean thought that the criterion for the existence of the referent of a singular term is its capacity to participate in an identity relation. Defenders of the Fregean view are aware of such examples. |
16416 | The quantifier in logic is not like the ordinary English one (which has empty names, non-denoting terms etc) [Hofweber] |
Full Idea: The inferential role of the existential quantifier in first order logic does not carry over to the existential quantifier in English (we have empty names, singular terms that are not even in the business of denoting, and so on). | |
From: Thomas Hofweber (Ambitious, yet modest, Metaphysics [2009], 2) |
21643 | The inferential quantifier focuses on truth; the domain quantifier focuses on reality [Hofweber] |
Full Idea: When we ask 'is there a number?' in its inferential role (or internalist) reading, then we ask whether or not there is a true instance of 't is a number'. When we ask in its domain conditions (externalist) reading, we ask if the world contains a number. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 03.6) | |
A reaction: Hofweber's key distinction. The distinction between making truth prior and making reference prior is intriguing and important. The internalist version is close to substitutional quantification. Only the externalist view needs robust reference. |
10007 | Quantifiers for domains and for inference come apart if there are no entities [Hofweber] |
Full Idea: Quantifiers have two functions in communication - to range over a domain of entities, and to have an inferential role (e.g. F(t)→'something is F'). In ordinary language these two come apart for singular terms not standing for any entities. | |
From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §6.3) | |
A reaction: This simple observations seems to me to be wonderfully illuminating of a whole raft of problems, the sort which logicians get steamed up about, and ordinary speakers don't. Context is the key to 90% of philosophical difficulties (?). See Idea 10008. |
17988 | Quantification can't all be substitutional; some reference is obviously to objects [Hofweber] |
Full Idea: The view that all quantification is substitutional is not very plausible in general. Some uses of quantifiers clearly seem to have the function to make a claim about a domain of objects out there, no matter how they relate to the terms in our language. | |
From: Thomas Hofweber (Inexpressible Properties and Propositions [2006], 2.1) | |
A reaction: Robust realists like myself are hardly going to say that quantification is just an internal language game. |
21644 | Numbers are used as singular terms, as adjectives, and as symbols [Hofweber] |
Full Idea: Number words have a singular term use, and adjectival (or determiner) use, and the symbolic use. The main question is how they relate to each other. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 05.1) | |
A reaction: Thus 'the number four is even', 'there are four moons', and '4 comes after 3'. |
9998 | What is the relation of number words as singular-terms, adjectives/determiners, and symbols? [Hofweber] |
Full Idea: There are three different uses of the number words: the singular-term use (as in 'the number of moons of Jupiter is four'), the adjectival (or determiner) use (as in 'Jupiter has four moons'), and the symbolic use (as in '4'). How are they related? | |
From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §1) | |
A reaction: A classic philosophy of language approach to the problem - try to give the truth-conditions for all three types. The main problem is that the first one implies that numbers are objects, whereas the others do not. Why did Frege give priority to the first? |
10002 | '2 + 2 = 4' can be read as either singular or plural [Hofweber] |
Full Idea: There are two ways to read to read '2 + 2 = 4', as singular ('two and two is four'), and as plural ('two and two are four'). | |
From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §4.1) | |
A reaction: Hofweber doesn't notice that this phenomenon occurs elsewhere in English. 'The team is playing well', or 'the team are splitting up'; it simply depends whether you are holding the group in though as an entity, or as individuals. Important for numbers. |
21646 | The Amazonian Piraha language is said to have no number words [Hofweber] |
Full Idea: The now famous Piraha language, of the Amazon region in Brazil, allegedly has no number words. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 05.6) | |
A reaction: Two groups can be shown to be of equal cardinality, by one-to-one matching rather than by counting. They could get by using 'equals' (and maybe unequally bigger and unequally smaller), and intuitive feelings for sizes of groups. |
21665 | The fundamental theorem of arithmetic is that all numbers are composed uniquely of primes [Hofweber] |
Full Idea: The prime numbers are more fundamental than the even numbers, and than the composite non-prime numbers. The result that all numbers uniquely decompose into a product of prime numbers is called the 'Fundamental Theorem of Arithmetic'. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 13.4.2) | |
A reaction: I could have used this example in my thesis, which defended the view that essences are the fundamentals of explanation, even in abstract theoretical realms. |
21649 | How can words be used for counting if they are objects? [Hofweber] |
Full Idea: Number words as singular terms seem to refer to objects; numbers words in determiner or adjectival position are tied to counting. How these objects are related to counting is what the application problem is about. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 06.1.3) | |
A reaction: You can't use stones for counting, so there must be more to numbers than the announcement that they are 'objects'. They seem to have internal relations, which makes them unusual objects. |
10003 | Why is arithmetic hard to learn, but then becomes easy? [Hofweber] |
Full Idea: Why is arithmetic so hard to learn, and why does it seem so easy to us now? For example, subtracting 789 from 26,789. | |
From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §4.2) | |
A reaction: His answer that we find thinking about objects very easy, but as children we have to learn with difficulty the conversion of the determiner/adjectival number words, so that we come to think of them as objects. |
10008 | Arithmetic is not about a domain of entities, as the quantifiers are purely inferential [Hofweber] |
Full Idea: I argue for an internalist conception of arithmetic. Arithmetic is not about a domain of entities, not even quantified entities. Quantifiers over natural numbers occur in their inferential-role reading in which they merely generalize over the instances. | |
From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §6.3) | |
A reaction: Hofweber offers the hope that modern semantics can disentangle the confusions in platonist arithmetic. Very interesting. The fear is that after digging into the semantics for twenty years, you find the same old problems re-emerging at a lower level. |
10005 | Arithmetic doesn’t simply depend on objects, since it is true of fictional objects [Hofweber] |
Full Idea: That 'two dogs are more than one' is clearly true, but its truth doesn't depend on the existence of dogs, as is seen if we consider 'two unicorns are more than one', which is true even though there are no unicorns. | |
From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §6.2) | |
A reaction: This is an objection to crude empirical accounts of arithmetic, but the idea would be that there is a generalisation drawn from objects (dogs will do nicely), which then apply to any entities. If unicorns are entities, it will be true of them. |
10000 | We might eliminate adjectival numbers by analysing them into blocks of quantifiers [Hofweber] |
Full Idea: Determiner uses of number words may disappear on analysis. This is inspired by Russell's elimination of the word 'the'. The number becomes blocks of first-order quantifiers at the level of semantic representation. | |
From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §2) | |
A reaction: [compressed] The proposal comes from platonists, who argue that numbers cannot be analysed away if they are objects. Hofweber says the analogy with Russell is wrong, as 'the' can't occur in different syntactic positions, the way number words can. |
21647 | Logicism makes sense of our ability to know arithmetic just by thought [Hofweber] |
Full Idea: Frege's tying the objectivity of arithmetic to the objectivity of logic makes sense of the fact that can find out about arithmetic by thinking alone. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 06.1.1) | |
A reaction: This assumes that logic is entirely a priori. We might compare the geometry of land surfaces with 'pure' geometry. If numbers are independent objects, it is unclear how we could have any a priori knowledge of them. |
21648 | Neo-Fregeans are dazzled by a technical result, and ignore practicalities [Hofweber] |
Full Idea: A major flaw of the neo-Fregean program is that it is more impressed by the technical result that Peano Arithmetic can be interpreted by second-order logic plus Hume's Principle, than empirical considerations about how numbers come about. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 06.1.3) | |
A reaction: This doesn't sound like a problem that would bother Fregeans or neo-Fregeans much. Deriving the Peano Axioms from various beginnings has become a parlour game for modern philosophers of mathematics. |
10006 | First-order logic captures the inferential relations of numbers, but not the semantics [Hofweber] |
Full Idea: Representing arithmetic formally we do not primarily care about semantic features of number words. We are interested in capturing the inferential relations of arithmetical statements to one another, which can be done elegantly in first-order logic. | |
From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §6.3) | |
A reaction: This begins to pinpoint the difference between the approach of logicists like Frege, and those who are interested in the psychology of numbers, and the empirical roots of numbers in the process of counting. |
21664 | Supervenience offers little explanation for things which necessarily go together [Hofweber] |
Full Idea: The results from the use of supervenience in philosophical theorising are limited. In particular, modal notions can't distinguish between things which necessarily go together. For example, that truths about numbers are grounded in truths about sets. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 13.4.1) | |
A reaction: [compressed] |
21660 | Reality can be seen as the totality of facts, or as the totality of things [Hofweber] |
Full Idea: Reality can be seen as everything that is the case - the totality of all facts that obtain - or reality can be seen as everything there is - the totality of all things that exist. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 10) | |
A reaction: Things are a lot easier to specify than facts, but on the whole I prefer facts, just in order to affirm that there is more to reality than the mere 'things' that compose it. Our ontology must capture the dynamic and relational character of reality. |
21661 | There are probably ineffable facts, systematically hidden from us [Hofweber] |
Full Idea: We do have reason to think that there are ineffable facts, and that these facts are systematically hidden from us. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 10.2.4) | |
A reaction: [Hofweber's Ch.10 is a lengthy and interesting discussion of ineffable facts] Things which are very very small, or very very remote in space seem obvious candidates. The most obvious candidates are tiny detail about the remote past. |
17989 | Since properties have properties, there can be a typed or a type-free theory of them [Hofweber] |
Full Idea: Since properties themselves can have properties there is a well-known division in the theory of properties between those who take a typed and those who take a type-free approach. | |
From: Thomas Hofweber (Inexpressible Properties and Propositions [2006], 2.2) | |
A reaction: A typed approach would imply restrictions on what it can be a property of. 'Green' is a property of surfaces, 'dark' is a property of colours. My first reaction is to opt for type-free. |
21652 | Our perceptual beliefs are about ordinary objects, not about simples arranged chair-wise [Hofweber] |
Full Idea: The belief that there are simples arranged chair-wise is not a perceptual belief. Our perceptual beliefs have a content about ordinary objects, not simples arranged chair-wise. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 07.3.1) | |
A reaction: Hofweber gives ontological priority to 'perceptual beliefs'. I'm inclined to agree, but I hear the critical hordes swarming against the gate. |
21663 | Counterfactuals are essential for planning, and learning from mistakes [Hofweber] |
Full Idea: Counterfactuals are important for reasoning about the past and to plan for the future. If we want to learn from our mistakes, it is important to think about what would have happened if I had done things differently. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 13.4.1) | |
A reaction: A thought also found in Tim Williamson, but not the sort of thing you hear from Lewis or Stalnaker. It is a nice example of how highly abstract and theoretical problems need to be slotted into human psychology. |
10004 | Our minds are at their best when reasoning about objects [Hofweber] |
Full Idea: Our minds mainly reason about objects. Most cognitive problems we are faced with deal with particular objects, whether they are people or material things. Reasoning about them is what our minds are good at. | |
From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §4.3) | |
A reaction: Hofweber is suggesting this as an explanation of why we continually reify various concepts, especially numbers. Very plausible. It works for qualities of character, and explains our tendency to talk about universals as objects ('redness'). |
21654 | The "Fido"-Fido theory of meaning says every expression in a language has a referent [Hofweber] |
Full Idea: The picture of language often called the "Fido"-Fido theory of meaning says every expression in natural languages refers; they simply differ in what they refer to. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 08.2) | |
A reaction: It seems obvious that at least there are syncategorematic terms like 'not' and 'or' and 'maybe' that are internal to language. I'm inclining to the opposite view of Paul Pietroski. Hofweber says if all words are names, they can't add up to truth. |
21641 | Inferential role semantics is an alternative to semantics that connects to the world [Hofweber] |
Full Idea: An inferential role semantics is generally seen as a large-scale alternative to a semantics based on reference and other language-world relations. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 03.4.5) | |
A reaction: Presumably the other obvious language-world relation is truth. Being a robust realist, I take it I have to be strongly committed to semantics which connects to the world - or do I? Reality is robust, but our talk about it is evasive? |
21638 | Syntactic form concerns the focus of the sentence, as well as the truth-conditions [Hofweber] |
Full Idea: Syntactic form is not only related to the truth conditions of a sentence; it is also related to what focus an utterance of a sentence will have. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 02.5.2) | |
A reaction: Hofweber has commendably studied some linguistics. The idea of mental and linguistic 'focus' increasingly strikes me as of importance in many areas of philosophy. E.g. in the scope of ethics, on whom should you focus? |
21658 | Properties can be expressed in a language despite the absence of a single word for them [Hofweber] |
Full Idea: Simply because there is no single word in a certain language for a certain property doesn't mean that it isn't expressible in that language. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 09.1.1) | |
A reaction: Good. For example a shade of blue for which there is no label might be 'the next darkest discriminable shade of blue adjacent to the one we are looking at'. And then the one after that... But 'tastes better than Diet Pepsi' in ancient Greek? |
21659 | 'Being taller than this' is a predicate which can express many different properties [Hofweber] |
Full Idea: It is said that not every property can be expressed because there are more properties than there are predicates. ...But the same predicate can be used to express many different properties: 'being taller than this' depends on what 'this' refers to. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 09.2) | |
A reaction: A good example, but being a comparative and relying on a demonstrative indexical makes it a favourable example. 'Being made of iron' doesn't have much scope for expressing many properties. |
21655 | Compositonality is a way to build up the truth-conditions of a sentence [Hofweber] |
Full Idea: Compositional semantics assigns semantic values to various expressions in order to generate the truth conditions of the sentences in which they can occur correctly, ...thus leading to the truth-conditions of the sentence. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 08.3) | |
A reaction: I favour both the compositional and the truth-conditional accounts of semantics, but I am not sure how to fit the pragmatic and contextual ingredient into that picture. You can't leave out psychology. |
21656 | Proposition have no content, because they are content [Hofweber] |
Full Idea: If there propositions then they do not have content, because they are content. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 08.4) | |
A reaction: This sounds right. A rather obvious regress threatens if you say otherwise. |
21635 | Without propositions there can be no beliefs or desires [Hofweber] |
Full Idea: If there are no propositions, then there are no contents, and thus there are no beliefs and desires. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 01.4.3) | |
A reaction: A simple but powerful point. Those who claim that there are only sentences (and no propositions) can hardly claim that you must formulate a sentence every time you have a specific belief or desire. |
21662 | Do there exist thoughts which we are incapable of thinking? [Hofweber] |
Full Idea: Might there be some thought token that has a different content than any such token we can in principle have? | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 10.3.3) | |
A reaction: For me the idea that a thought might exist which can never be thought is an absurdity, but people who believe in the external existence of parts of reality called 'propositions' seem committed to it. A baffling view. |
21645 | 'Semantic type coercion' is selecting the reading of a word to make the best sense [Hofweber] |
Full Idea: 'Semantic type coercion' is where an expression of variable type is forced to take a particular type on a particular occasion so that the sentence as a whole in which it occurse is semantically interpretable. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 05.4.4) | |
A reaction: He compares 'and' in 'John sang and Mary danced' with 'John and Mary danced together', where 'and' can vary in type, and we adopt the reading that makes sense. Hofweber says we do this with number language. He favours 'cognitive need'. |
21639 | 'Background deletion' is appropriately omitting background from an answer [Hofweber] |
Full Idea: 'Background deletion' is the pheomenon that what isn't focused in an answer, what is the background, can be left out of the answer, with the resulting sub-sentential answer nonetheless being appropriate. | |
From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 02.6.2) | |
A reaction: [I'm struck by the verbosity of this sentence, from an over-long book] It is not unreasonable to think that each conversational exchange has an implicit and agreed domain of quantification. Well, 'focus', then. |
17991 | Holism says language can't be translated; the expressibility hypothesis says everything can [Hofweber] |
Full Idea: Holism says that nothing that can be said in one language can be said in another one. The expressibility hypothesis says that everything that can be said in one language can be said in every other one. | |
From: Thomas Hofweber (Inexpressible Properties and Propositions [2006], 6.4) | |
A reaction: Obviously expressibility would only refer to reasonably comprehensive languages (with basic logical connectives, for example). Personally I vote for the expressibility hypothesis, which Hofweber seems to favour. |
22595 | Liberty is the triumph of the individual, over both despotic government and enslaving majorities [Constant] |
Full Idea: Lliberty is the triumph of the individual, as much over a government which seeks to rule by despotic methods, as over the masses who seek to render the minority the slave of the majority. | |
From: Benjamin Constant (Principles of Politics [1806]), quoted by Ian Dunt - How to be a Liberal 4 | |
A reaction: [No page given] Dunt describes Constant's book as the first really systematic account of liberalism. Very important to have rights against the majority, as well as against government. |
22597 | Minority rights are everyone's rights, because we all have turns in the minority [Constant] |
Full Idea: To defend the rights of minorities is to defend the rights of all. Everyone in turn finds himself in the minority. | |
From: Benjamin Constant (Principles of Politics [1806]), quoted by Ian Dunt - How to be a Liberal 4 | |
A reaction: Very conformist people, who are often the most oppressive, are rarely in the minority, and are unlikely to be impressed by this idea. |