150 ideas
12865 | Analytic philosophers may prefer formal systems because natural language is such mess [Simons] |
Full Idea: The untidiness of natural language in its use of 'part' is perhaps one of the chief reasons why mereolologists have preferred to investigate formal systems with nice algebraic properties rather than get out and mix it with reality in all its messiness. | |
From: Peter Simons (Parts [1987], 6.4) | |
A reaction: [See Idea 12864 for the uses of 'part'] I am in the unhappy (and probably doomed) position of wanting to avoid both approaches. I try to operate as if the English language were transparent and we can just discuss the world. Very naïve. |
13439 | Venn Diagrams map three predicates into eight compartments, then look for the conclusion [Bostock] |
Full Idea: Venn Diagrams are a traditional method to test validity of syllogisms. There are three interlocking circles, one for each predicate, thus dividing the universe into eight possible basic elementary quantifications. Is the conclusion in a compartment? | |
From: David Bostock (Intermediate Logic [1997], 3.8) |
13421 | 'Disjunctive Normal Form' is ensuring that no conjunction has a disjunction within its scope [Bostock] |
Full Idea: 'Disjunctive Normal Form' (DNF) is rearranging the occurrences of ∧ and ∨ so that no conjunction sign has any disjunction in its scope. This is achieved by applying two of the distribution laws. | |
From: David Bostock (Intermediate Logic [1997], 2.6) |
13422 | 'Conjunctive Normal Form' is ensuring that no disjunction has a conjunction within its scope [Bostock] |
Full Idea: 'Conjunctive Normal Form' (CNF) is rearranging the occurrences of ∧ and ∨ so that no disjunction sign has any conjunction in its scope. This is achieved by applying two of the distribution laws. | |
From: David Bostock (Intermediate Logic [1997], 2.6) |
13355 | 'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,ψ|= [Bostock] |
Full Idea: The Principle of Disjunction says that Γ,φ∨ψ |= iff Γ,φ |= and Γ,ψ |=. | |
From: David Bostock (Intermediate Logic [1997], 2.5.G) | |
A reaction: That is, a disjunction leads to a contradiction if they each separately lead to contradictions. |
13350 | 'Assumptions' says that a formula entails itself (φ|=φ) [Bostock] |
Full Idea: The Principle of Assumptions says that any formula entails itself, i.e. φ |= φ. The principle depends just upon the fact that no interpretation assigns both T and F to the same formula. | |
From: David Bostock (Intermediate Logic [1997], 2.5.A) | |
A reaction: Thus one can introduce φ |= φ into any proof, and then use it to build more complex sequents needed to attain a particular target formula. Bostock's principle is more general than anything in Lemmon. |
13351 | 'Thinning' allows that if premisses entail a conclusion, then adding further premisses makes no difference [Bostock] |
Full Idea: The Principle of Thinning says that if a set of premisses entails a conclusion, then adding further premisses will still entail the conclusion. It is 'thinning' because it makes a weaker claim. If γ|=φ then γ,ψ|= φ. | |
From: David Bostock (Intermediate Logic [1997], 2.5.B) | |
A reaction: It is also called 'premise-packing'. It is the characteristic of a 'monotonic' logic - where once something is proved, it stays proved, whatever else is introduced. |
13356 | The 'conditional' is that Γ|=φ→ψ iff Γ,φ|=ψ [Bostock] |
Full Idea: The Conditional Principle says that Γ |= φ→ψ iff Γ,φ |= ψ. With the addition of negation, this implies φ,φ→ψ |= ψ, which is 'modus ponens'. | |
From: David Bostock (Intermediate Logic [1997], 2.5.H) | |
A reaction: [Second half is in Ex. 2.5.4] |
13352 | 'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z [Bostock] |
Full Idea: The Principle of Cutting is the general point that entailment is transitive, extending this to cover entailments with more than one premiss. Thus if γ |= φ and φ,Δ |= ψ then γ,Δ |= ψ. Here φ has been 'cut out'. | |
From: David Bostock (Intermediate Logic [1997], 2.5.C) | |
A reaction: It might be called the Principle of Shortcutting, since you can get straight to the last conclusion, eliminating the intermediate step. |
13353 | 'Negation' says that Γ,¬φ|= iff Γ|=φ [Bostock] |
Full Idea: The Principle of Negation says that Γ,¬φ |= iff Γ |= φ. We also say that φ,¬φ |=, and hence by 'thinning on the right' that φ,¬φ |= ψ, which is 'ex falso quodlibet'. | |
From: David Bostock (Intermediate Logic [1997], 2.5.E) | |
A reaction: That is, roughly, if the formula gives consistency, the negation gives contradiction. 'Ex falso' says that anything will follow from a contradiction. |
13354 | 'Conjunction' says that Γ|=φ∧ψ iff Γ|=φ and Γ|=ψ [Bostock] |
Full Idea: The Principle of Conjunction says that Γ |= φ∧ψ iff Γ |= φ and Γ |= ψ. This implies φ,ψ |= φ∧ψ, which is ∧-introduction. It is also implies ∧-elimination. | |
From: David Bostock (Intermediate Logic [1997], 2.5.F) | |
A reaction: [Second half is Ex. 2.5.3] That is, if they are entailed separately, they are entailed as a unit. It is a moot point whether these principles are theorems of propositional logic, or derivation rules. |
13610 | A logic with ¬ and → needs three axiom-schemas and one rule as foundation [Bostock] |
Full Idea: For ¬,→ Schemas: (A1) |-φ→(ψ→φ), (A2) |-(φ→(ψ→ξ)) → ((φ→ψ)→(φ→ξ)), (A3) |-(¬φ→¬ψ) → (ψ→φ), Rule:DET:|-φ,|-φ→ψ then |-ψ | |
From: David Bostock (Intermediate Logic [1997], 5.2) | |
A reaction: A1 says everything implies a truth, A2 is conditional proof, and A3 is contraposition. DET is modus ponens. This is Bostock's compact near-minimal axiom system for proposition logic. He adds two axioms and another rule for predicate logic. |
13846 | A 'free' logic can have empty names, and a 'universally free' logic can have empty domains [Bostock] |
Full Idea: A 'free' logic is one in which names are permitted to be empty. A 'universally free' logic is one in which the domain of an interpretation may also be empty. | |
From: David Bostock (Intermediate Logic [1997], 8.6) |
12815 | Classical mereology doesn't apply well to the objects around us [Simons] |
Full Idea: The most fundamental criticism of classical mereology is that the theory is not applicable to most of the objects around us, and is accordingly of little use as a formal reconstruction of the concepts of part and whole which we actually employ. | |
From: Peter Simons (Parts [1987], Intro) | |
A reaction: This sounds splendidly dismissive, but one might compare it with possible worlds semantics for modal logic, which most people take with a pinch of salt as an actual commitment, but find wonderfully clarifying in modal reasoning. |
12832 | Complement: the rest of the Universe apart from some individual, written x-bar [Simons] |
Full Idea: The 'complement' of each individual in mereology is the rest of the Universe outside it, that is U - x, but written as x-bar [x with a horizontal bar above it]. | |
From: Peter Simons (Parts [1987], 1.1.10) | |
A reaction: [Don't have a font for x-bar] See Idea 12831 for the 'Universe'. Simons suggest that the interest of this term is mainly historical and algebraic. |
12834 | Criticisms of mereology: parts? transitivity? sums? identity? four-dimensional? [Simons] |
Full Idea: Main criticisms of mereology: we don't mean 'part' as improper; transitivity of 'part' is sometimes not transitive; no guarantee that there are 'sums'; the identity criteria for individuals are false; we are forced into materialistic four-dimensionalism. | |
From: Peter Simons (Parts [1987], 3.2) | |
A reaction: [Compressed summary; for four-dimensionalism see under 'Identity over Time'] Simons says these are in ascending order of importance. |
12819 | A 'part' has different meanings for individuals, classes, and masses [Simons] |
Full Idea: It emerges that 'part', like other formal concepts, is not univocal, but has analogous meanings according to whether we talk of individuals, classes, or masses. | |
From: Peter Simons (Parts [1987], Intro) | |
A reaction: He suggests that unrestricted sums are appropriate for the last two, but not for individuals. There must be something univocal about the word - some awareness of a possible whole or larger entity to which the thing could belong. |
12822 | Proper or improper part: x < y, 'x is (a) part of y' [Simons] |
Full Idea: A 'proper or improper part' is expressed by 'x < y', read as 'x is (a) part of y'. The relatively minor deviation from normal usage (of including an improper part, i.e. the whole thing) is warranted by its algebraical convenience. | |
From: Peter Simons (Parts [1987], 1.1.02) | |
A reaction: Including an improper part (i.e. the whole thing) is not, Simons points out, uncontroversial, because the part being 'equal' to the whole is read as being 'identical' to the whole, which Simons is unwilling to accept. |
12824 | Disjoint: two individuals are disjoint iff they do not overlap, written 'x | y' [Simons] |
Full Idea: Two individuals are 'disjoint' mereologically if and only if they do not overlap, expressed by 'x | y', read as 'x is disjoint from y'. Disjointedness is symmetric. | |
From: Peter Simons (Parts [1987], 1.1.04) |
12827 | Difference: the difference of individuals is the remainder of an overlap, written 'x - y' [Simons] |
Full Idea: The 'difference' of two individuals is the largest individual contained in x which has no part in common with y, expressed by 'x - y', read as 'the difference of x and y'. | |
From: Peter Simons (Parts [1987], 1.1.07) |
12823 | Overlap: two parts overlap iff they have a part in common, expressed as 'x o y' [Simons] |
Full Idea: Two parts 'overlap' mereologically if and only if they have a part in common, expressed by 'x o y', read as 'x overlaps y'. Overlapping is reflexive and symmetric but not transitive. | |
From: Peter Simons (Parts [1987], 1.1.03) | |
A reaction: Simons points out that we are uncomfortable with overlapping (as in overlapping national boundaries), because we seem to like conceptual boundaries. We avoid overlap even in ordering primary colour terms, by having a no-man's-land. |
12825 | Product: the product of two individuals is the sum of all of their overlaps, written 'x · y' [Simons] |
Full Idea: For two overlapping individuals their 'product' is the individual which is part of both and such that any common part of both is part of it, expressed by 'x · y', read as 'the product of x and y'. | |
From: Peter Simons (Parts [1987], 1.1.05) | |
A reaction: That is, the 'product' is the sum of any common parts between two individuals. In set theory all sets intersect at the null set, but mereology usually avoids the 'null individual'. |
12826 | Sum: the sum of individuals is what is overlapped if either of them are, written 'x + y' [Simons] |
Full Idea: The 'sum' of two individuals is that individual which something overlaps iff it overlaps at least one of x and y, expressed by 'x + y', read as 'the sum of x and y'. It is central to classical extensional mereologies that any two individuals have a sum. | |
From: Peter Simons (Parts [1987], 1.1.06) | |
A reaction: This rather technical definition (defining an individual by the possibility of it being overlapped) does not always coincide with the smallest individual containing them both. |
12828 | General sum: the sum of objects satisfying some predicate, written σx(Fx) [Simons] |
Full Idea: The 'general sum' of all objects satisfying a certain predicate is denoted by a variable-binding operator, expressed by 'σx(Fx)', read as 'the sum of objects satisfying F'. | |
From: Peter Simons (Parts [1987], 1.1.08) | |
A reaction: This, it seems, is introduced to restrict some infinite classes which aspire to be sums. |
12829 | General product: the nucleus of all objects satisfying a predicate, written πx(Fx) [Simons] |
Full Idea: The 'general product' or 'nucleus' of all objects satisfying a certain predicate is denoted by a variable-binding operator, expressed by 'πx(Fx)', read as 'the product of objects satisfying F'. | |
From: Peter Simons (Parts [1987], 1.1.08) | |
A reaction: See Idea 12825 for 'product'. 'Nucleus' is a helpful word here. Thought: is the general product a candidate for a formal definition of essence? It would be a sortal essence - roughly, what all beetles have in common, just by being beetles. |
12830 | Universe: the mereological sum of all objects whatever, written 'U' [Simons] |
Full Idea: The 'Universe' in mereology is the sum of all objects whatever, a unique individual of which all individuals are part. This is denoted by 'U'. Strictly, there can be no 'empty Universe', since the Universe is not a container, but the whole filling. | |
From: Peter Simons (Parts [1987], 1.1.09) | |
A reaction: This, of course, contrasts with set theory, which cannot have a set of all sets. At the lower end, set theory does have a null set, while mereology has no null individual. See David Lewis on combining the two theories. |
12831 | Atom: an individual with no proper parts, written 'At x' [Simons] |
Full Idea: An 'atom' in mereology is an individual with no proper parts. We shall use the expression 'At x' to mean 'x is an atom'. | |
From: Peter Simons (Parts [1987], 1.1.11) | |
A reaction: Note that 'part' in standard mereology includes improper parts, so every object has at least one part, namely itself. |
12844 | Dissective: stuff is dissective if parts of the stuff are always the stuff [Simons] |
Full Idea: Water is said not to be 'dissective', since there are parts of any quantity of water which are not water. | |
From: Peter Simons (Parts [1987], 4.2) | |
A reaction: This won't seem to do for any physical matter, but presumably parts of numbers are always numbers. |
12813 | Two standard formalisations of part-whole theory are the Calculus of Individuals, and Mereology [Simons] |
Full Idea: The standardly accepted formal theory of part-whole is classical extensional mereology, which is known in two logical guises, the Calculus of Individuals of Leonard and Goodman, and the Mereology of Lesniewski. | |
From: Peter Simons (Parts [1987], Intro) | |
A reaction: Simons catalogues several other modern attempts at axiomatisation in his chapter 2. |
12821 | The part-relation is transitive and asymmetric (and thus irreflexive) [Simons] |
Full Idea: Formally, the part-relation is transitive and asymmetric (and thus irreflexive). Hence nothing is a proper part of itself, things aren't proper parts of one another, and if one is part of two which is part of three then one is part of three. | |
From: Peter Simons (Parts [1987], 1.1.1) |
18847 | Each wheel is part of a car, but the four wheels are not a further part [Simons] |
Full Idea: The four wheels of a car are parts of it (each is part of it), but there is not a fifth part consisting of the four wheels. | |
From: Peter Simons (Parts [1987], 4.6) | |
A reaction: This raises questions about the transitivity of parthood. If there are parts of parts of wholes, the basic parts are OK, and the whole is OK, but how can there also be an intermediate part? Try counting the parts of this whole! |
12816 | Classical mereology doesn't handle temporal or modal notions very well [Simons] |
Full Idea: The underlying logic of classical extensional mereology does not have the resources to deal with temporal and modal notions such as temporary part, temporal part, essential part, or essential permanent part. | |
From: Peter Simons (Parts [1987], Intro) | |
A reaction: Simons tries to rectify this in the later chapters of his book, with modifications rather than extensions. Since everyone struggles with temporal and modal issues of identity, we shouldn't judge too harshly. |
12846 | A 'group' is a collection with a condition which constitutes their being united [Simons] |
Full Idea: We call a 'collection' of jewels a 'group' term. Several random musicians are unlikely to be an orchestra. If they come together regularly in a room to play, such conditions are constitutive of an orchestra. | |
From: Peter Simons (Parts [1987], 4.4) | |
A reaction: Clearly this invites lots of borderline cases. Eleven footballers don't immediately make a team, as followers of the game know well. |
12848 | The same members may form two groups [Simons] |
Full Idea: Groups may coincide in membership without being identical - extensionality goes. | |
From: Peter Simons (Parts [1987], 4.9) | |
A reaction: Thus an eleven-person orchestra may also constitute a football team. What if a pile of stones is an impediment to you, and useful to me? Is it then two groups? Suppose they hum while playing football? (Don't you just love philosophy?) |
12861 | 'The wolves' are the matter of 'the pack'; the latter is a group, with different identity conditions [Simons] |
Full Idea: 'The wolves' is a plural term referring to just these animals, whereas 'the pack' of wolves refers to a group, and the group and plurality, while they may coincide in membership, have different identity conditions. The wolves are the matter of the pack. | |
From: Peter Simons (Parts [1987], 6.4) | |
A reaction: Even a cautious philosopher like Simons is ready to make bold ontological commitment to 'packs', on the basis of something called 'identity conditions'. I think it is just verbal. You can qualify 'the wolves' and 'the pack' to make them identical. |
13010 | In order to select the logic justified by experience, we would need to use a lot of logic [Boghossian on Quine] |
Full Idea: Quine ends up with the logic that is maximally justified by experience, ...but a large number of the core principles of logic will have to be used to select the logic that is maximally justified by experience. | |
From: comment on Willard Quine (Carnap and Logical Truth [1954]) by Paul Boghossian - Knowledge of Logic p.233 | |
A reaction: In order to grasp some core principles of logic, you will probably need a certain amount of experience. I take logic to be an abstracted feature of reality (unless it is extended by pure fictions). Some basic logic may be hard wired in us. |
9002 | Elementary logic requires truth-functions, quantifiers (and variables), identity, and also sets of variables [Quine] |
Full Idea: Elementary logic, as commonly systematized nowadays, comprises truth-function theory (involving 'or', 'and', 'not' etc.), quantifiers (and their variables), and identity theory ('='). In addition, set theory requires classes among values of variables. | |
From: Willard Quine (Carnap and Logical Truth [1954], II) | |
A reaction: Quine is famous for trying to squeeze properties out of the picture, which would then block higher-order logics (which quantify over properties). Quine's list gives a nice programme for a student of the philosophy of logic to understand. |
13346 | Truth is the basic notion in classical logic [Bostock] |
Full Idea: The most fundamental notion in classical logic is that of truth. | |
From: David Bostock (Intermediate Logic [1997], 1.1) | |
A reaction: The opening sentence of his book. Hence the first half of the book is about semantics, and only the second half deals with proof. Compare Idea 10282. The thought seems to be that you could leave out truth, but that makes logic pointless. |
13545 | Elementary logic cannot distinguish clearly between the finite and the infinite [Bostock] |
Full Idea: In very general terms, we cannot express the distinction between what is finite and what is infinite without moving essentially beyond the resources available in elementary logic. | |
From: David Bostock (Intermediate Logic [1997], 4.8) | |
A reaction: This observation concludes a discussion of Compactness in logic. |
13822 | Fictional characters wreck elementary logic, as they have contradictions and no excluded middle [Bostock] |
Full Idea: Discourse about fictional characters leads to a breakdown of elementary logic. We accept P or ¬P if the relevant story says so, but P∨¬P will not be true if the relevant story says nothing either way, and P∧¬P is true if the story is inconsistent. | |
From: David Bostock (Intermediate Logic [1997], 8.5) | |
A reaction: I really like this. Does one need to invent a completely new logic for fictional characters? Or must their logic be intuitionist, or paraconsistent, or both? |
13681 | Logical consequence is marked by being preserved under all nonlogical substitutions [Quine, by Sider] |
Full Idea: Quine's view of logical consequence is that it is when there is no way of uniformly substituting nonlogical expressions in the premises and consequences so that the premises all remain true but the consequence now becomes false. | |
From: report of Willard Quine (Carnap and Logical Truth [1954], p.103) by Theodore Sider - Logic for Philosophy 1.5 | |
A reaction: One might just say that the consequence holds if you insert consistent variables for the nonlogical terms, which looks like Aristotle's view of the matter. |
13623 | The syntactic turnstile |- φ means 'there is a proof of φ' or 'φ is a theorem' [Bostock] |
Full Idea: The syntactic turnstile |- φ means 'There is a proof of φ' (in the system currently being considered). Another way of saying the same thing is 'φ is a theorem'. | |
From: David Bostock (Intermediate Logic [1997], 5.1) |
13347 | Validity is a conclusion following for premises, even if there is no proof [Bostock] |
Full Idea: The classical definition of validity counts an argument as valid if and only if the conclusion does in fact follow from the premises, whether or not the argument contains any demonstration of this fact. | |
From: David Bostock (Intermediate Logic [1997], 1.2) | |
A reaction: Hence validity is given by |= rather than by |-. A common example is 'it is red so it is coloured', which seems true but beyond proof. In the absence of formal proof, you wonder whether validity is merely a psychological notion. |
13348 | It seems more natural to express |= as 'therefore', rather than 'entails' [Bostock] |
Full Idea: In practice we avoid quotation marks and explicitly set-theoretic notation that explaining |= as 'entails' appears to demand. Hence it seems more natural to explain |= as simply representing the word 'therefore'. | |
From: David Bostock (Intermediate Logic [1997], 1.3) | |
A reaction: Not sure I quite understand that, but I have trained myself to say 'therefore' for the generic use of |=. In other consequences it seems better to read it as 'semantic consequence', to distinguish it from |-. |
13349 | Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid' [Bostock] |
Full Idea: If we write Γ |= φ, with one formula to the right, then the turnstile abbreviates 'entails'. For a sequent of the form Γ |= it can be read as 'is inconsistent'. For |= φ we read it as 'valid'. | |
From: David Bostock (Intermediate Logic [1997], 1.3) |
13614 | MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment) [Bostock] |
Full Idea: The Rule of Detachment is a version of Modus Ponens, and says 'If |=φ and |=φ→ψ then |=ψ'. This has no assumptions. Modus Ponens is the more general rule that 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ'. | |
From: David Bostock (Intermediate Logic [1997], 5.3) | |
A reaction: Modus Ponens is actually designed for use in proof based on assumptions (which isn't always the case). In Detachment the formulae are just valid, without dependence on assumptions to support them. |
13617 | MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ [Bostock] |
Full Idea: Modus Ponens is equivalent to the converse of the Deduction Theorem, namely 'If Γ |- φ→ψ then Γ,φ|-ψ'. | |
From: David Bostock (Intermediate Logic [1997], 5.3) | |
A reaction: See 13615 for details of the Deduction Theorem. See 13614 for Modus Ponens. |
13799 | The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b) [Bostock] |
Full Idea: We shall use 'a=b' as short for 'a is the same thing as b'. The sign '=' thus expresses a particular two-place predicate. Officially we will use 'I' as the identity predicate, so that 'Iab' is as formula, but we normally 'abbreviate' this to 'a=b'. | |
From: David Bostock (Intermediate Logic [1997], 8.1) |
13800 | |= α=α and α=β |= φ(α/ξ ↔ φ(β/ξ) fix identity [Bostock] |
Full Idea: We usually take these two principles together as the basic principles of identity: |= α=α and α=β |= φ(α/ξ) ↔ φ(β/ξ). The second (with scant regard for history) is known as Leibniz's Law. | |
From: David Bostock (Intermediate Logic [1997], 8.1) |
13803 | If we are to express that there at least two things, we need identity [Bostock] |
Full Idea: To say that there is at least one thing x such that Fx we need only use an existential quantifier, but to say that there are at least two things we need identity as well. | |
From: David Bostock (Intermediate Logic [1997], 8.1) | |
A reaction: The only clear account I've found of why logic may need to be 'with identity'. Without it, you can only reason about one thing or all things. Presumably plural quantification no longer requires '='? |
13829 | If logical truths essentially depend on logical constants, we had better define the latter [Hacking on Quine] |
Full Idea: Quine said a logical truth is a truth in which only logical constants occur essentially, ...but then a fruitful definition of 'logical constant' is called for. | |
From: comment on Willard Quine (Carnap and Logical Truth [1954]) by Ian Hacking - What is Logic? §02 |
13357 | Truth-functors are usually held to be defined by their truth-tables [Bostock] |
Full Idea: The usual view of the meaning of truth-functors is that each is defined by its own truth-table, independently of any other truth-functor. | |
From: David Bostock (Intermediate Logic [1997], 2.7) |
13812 | A 'zero-place' function just has a single value, so it is a name [Bostock] |
Full Idea: We can talk of a 'zero-place' function, which is a new-fangled name for a familiar item; it just has a single value, and so it has the same role as a name. | |
From: David Bostock (Intermediate Logic [1997], 8.2) |
13811 | A 'total' function ranges over the whole domain, a 'partial' function over appropriate inputs [Bostock] |
Full Idea: Usually we allow that a function is defined for arguments of a suitable kind (a 'partial' function), but we can say that each function has one value for any object whatever, from the whole domain that our quantifiers range over (a 'total' function). | |
From: David Bostock (Intermediate Logic [1997], 8.2) | |
A reaction: He points out (p.338) that 'the father of..' is a functional expression, but it wouldn't normally take stones as input, so seems to be a partial function. But then it doesn't even take all male humans either. It only takes fathers! |
13360 | In logic, a name is just any expression which refers to a particular single object [Bostock] |
Full Idea: The important thing about a name, for logical purposes, is that it is used to make a singular reference to a particular object; ..we say that any expression too may be counted as a name, for our purposes, it it too performs the same job. | |
From: David Bostock (Intermediate Logic [1997], 3.1) | |
A reaction: He cites definite descriptions as the most notoriously difficult case, in deciding whether or not they function as names. I takes it as pretty obvious that sometimes they do and sometimes they don't (in ordinary usage). |
12876 | Philosophy is stuck on the Fregean view that an individual is anything with a proper name [Simons] |
Full Idea: Modern philosophy is still under the spell of Frege's view that an individual is anything that has a proper name. (Note: But not only are empty names now recognised, but some are aware of the existence of plural reference). | |
From: Peter Simons (Parts [1987], 8.1) | |
A reaction: Presumably every electron in the universe is an individual, and every (finite) number which has never been named has a pretty clear identity. Presumably Pegasus, John Doe, and 'the person in the kitchen' have to be accommodated. |
13361 | An expression is only a name if it succeeds in referring to a real object [Bostock] |
Full Idea: An expression is not counted as a name unless it succeeds in referring to an object, i.e. unless there really is an object to which it refers. | |
From: David Bostock (Intermediate Logic [1997], 3.1) | |
A reaction: His 'i.e.' makes the existence condition sound sufficient, but in ordinary language you don't succeed in referring to 'that man over there' just because he exists. In modal contexts we presumably refer to hypothetical objects (pace Lewis). |
13814 | Definite desciptions resemble names, but can't actually be names, if they don't always refer [Bostock] |
Full Idea: Although a definite description looks like a complex name, and in many ways behaves like a name, still it cannot be a name if names must always refer to objects. Russell gave the first proposal for handling such expressions. | |
From: David Bostock (Intermediate Logic [1997], 8.3) | |
A reaction: I take the simple solution to be a pragmatic one, as roughly shown by Donnellan, that sometimes they are used exactly like names, and sometimes as something else. The same phrase can have both roles. Confusing for logicians. Tough. |
13816 | Because of scope problems, definite descriptions are best treated as quantifiers [Bostock] |
Full Idea: Because of the scope problem, it now seems better to 'parse' definition descriptions not as names but as quantifiers. 'The' is to be treated in the same category as acknowledged quantifiers like 'all' and 'some'. We write Ix - 'for the x such that..'. | |
From: David Bostock (Intermediate Logic [1997], 8.3) | |
A reaction: This seems intuitively rather good, since quantification in normal speech is much more sophisticated than the crude quantification of classical logic. But the fact is that they often function as names (but see Idea 13817). |
13817 | Definite descriptions are usually treated like names, and are just like them if they uniquely refer [Bostock] |
Full Idea: In practice, definite descriptions are for the most part treated as names, since this is by far the most convenient notation (even though they have scope). ..When a description is uniquely satisfied then it does behave like a name. | |
From: David Bostock (Intermediate Logic [1997], 8.3) | |
A reaction: Apparent names themselves have problems when they wander away from uniquely picking out one thing, as in 'John Doe'. |
13848 | We are only obliged to treat definite descriptions as non-names if only the former have scope [Bostock] |
Full Idea: If it is really true that definite descriptions have scopes whereas names do not, then Russell must be right to claim that definite descriptions are not names. If, however, this is not true, then it does no harm to treat descriptions as complex names. | |
From: David Bostock (Intermediate Logic [1997], 8.8) |
13813 | Definite descriptions don't always pick out one thing, as in denials of existence, or errors [Bostock] |
Full Idea: It is natural to suppose one only uses a definite description when one believes it describes only one thing, but exceptions are 'there is no such thing as the greatest prime number', or saying something false where the reference doesn't occur. | |
From: David Bostock (Intermediate Logic [1997], 8.3) |
13815 | Names do not have scope problems (e.g. in placing negation), but Russell's account does have that problem [Bostock] |
Full Idea: In orthodox logic names are not regarded as having scope (for example, in where a negation is placed), whereas on Russell's theory definite descriptions certainly do. Russell had his own way of dealing with this. | |
From: David Bostock (Intermediate Logic [1997], 8.3) |
13438 | 'Prenex normal form' is all quantifiers at the beginning, out of the scope of truth-functors [Bostock] |
Full Idea: A formula is said to be in 'prenex normal form' (PNF) iff all its quantifiers occur in a block at the beginning, so that no quantifier is in the scope of any truth-functor. | |
From: David Bostock (Intermediate Logic [1997], 3.7) | |
A reaction: Bostock provides six equivalences which can be applied to manouevre any formula into prenex normal form. He proves that every formula can be arranged in PNF. |
13818 | If we allow empty domains, we must allow empty names [Bostock] |
Full Idea: We can show that if empty domains are permitted, then empty names must be permitted too. | |
From: David Bostock (Intermediate Logic [1997], 8.4) |
12845 | Some natural languages don't distinguish between singular and plural [Simons] |
Full Idea: The syntactic distinction between singular and plural is not a universal feature of natural languages. Chinese manages nicely without it, and Sanskrit makes a tripartite distinction between singular, dual, and plural (more than two). | |
From: Peter Simons (Parts [1987], 4.3) | |
A reaction: Simons is mounting an attack on the way in which modern philosophy and logic has been mesmerised by singular terms and individuated objects. Most people seem now to agree with Simons. There is stuff, as well as plurals. |
13801 | An 'informal proof' is in no particular system, and uses obvious steps and some ordinary English [Bostock] |
Full Idea: An 'informal proof' is not in any particular proof system. One may use any rule of proof that is 'sufficiently obvious', and there is quite a lot of ordinary English in the proof, explaining what is going on at each step. | |
From: David Bostock (Intermediate Logic [1997], 8.1) |
13619 | Quantification adds two axiom-schemas and a new rule [Bostock] |
Full Idea: New axiom-schemas for quantifiers: (A4) |-∀ξφ → φ(α/ξ), (A5) |-∀ξ(ψ→φ) → (ψ→∀ξφ), plus the rule GEN: If |-φ the |-∀ξφ(ξ/α). | |
From: David Bostock (Intermediate Logic [1997], 5.6) | |
A reaction: This follows on from Idea 13610, where he laid out his three axioms and one rule for propositional (truth-functional) logic. This Idea plus 13610 make Bostock's proposed axiomatisation of first-order logic. |
13622 | Axiom systems from Frege, Russell, Church, Lukasiewicz, Tarski, Nicod, Kleene, Quine... [Bostock] |
Full Idea: Notably axiomatisations of first-order logic are by Frege (1879), Russell and Whitehead (1910), Church (1956), Lukasiewicz and Tarski (1930), Lukasiewicz (1936), Nicod (1917), Kleene (1952) and Quine (1951). Also Bostock (1997). | |
From: David Bostock (Intermediate Logic [1997], 5.8) | |
A reaction: My summary, from Bostock's appendix 5.8, which gives details of all of these nine systems. This nicely illustrates the status and nature of axiom systems, which have lost the absolute status they seemed to have in Euclid. |
13615 | 'Conditonalised' inferences point to the Deduction Theorem: If Γ,φ|-ψ then Γ|-φ→ψ [Bostock] |
Full Idea: If a group of formulae prove a conclusion, we can 'conditionalize' this into a chain of separate inferences, which leads to the Deduction Theorem (or Conditional Proof), that 'If Γ,φ|-ψ then Γ|-φ→ψ'. | |
From: David Bostock (Intermediate Logic [1997], 5.3) | |
A reaction: This is the rule CP (Conditional Proof) which can be found in the rules for propositional logic I transcribed from Lemmon's book. |
13616 | The Deduction Theorem greatly simplifies the search for proof [Bostock] |
Full Idea: Use of the Deduction Theorem greatly simplifies the search for proof (or more strictly, the task of showing that there is a proof). | |
From: David Bostock (Intermediate Logic [1997], 5.3) | |
A reaction: See 13615 for details of the Deduction Theorem. Bostock is referring to axiomatic proof, where it can be quite hard to decide which axioms are relevant. The Deduction Theorem enables the making of assumptions. |
13620 | Proof by Assumptions can always be reduced to Proof by Axioms, using the Deduction Theorem [Bostock] |
Full Idea: By repeated transformations using the Deduction Theorem, any proof from assumptions can be transformed into a fully conditionalized proof, which is then an axiomatic proof. | |
From: David Bostock (Intermediate Logic [1997], 5.6) | |
A reaction: Since proof using assumptions is perhaps the most standard proof system (e.g. used in Lemmon, for many years the standard book at Oxford University), the Deduction Theorem is crucial for giving it solid foundations. |
13621 | The Deduction Theorem and Reductio can 'discharge' assumptions - they aren't needed for the new truth [Bostock] |
Full Idea: Like the Deduction Theorem, one form of Reductio ad Absurdum (If Γ,φ|-[absurdity] then Γ|-¬φ) 'discharges' an assumption. Assume φ and obtain a contradiction, then we know ¬&phi, without assuming φ. | |
From: David Bostock (Intermediate Logic [1997], 5.7) | |
A reaction: Thus proofs from assumption either arrive at conditional truths, or at truths that are true irrespective of what was initially assumed. |
13753 | Natural deduction takes proof from assumptions (with its rules) as basic, and axioms play no part [Bostock] |
Full Idea: Natural deduction takes the notion of proof from assumptions as a basic notion, ...so it will use rules for use in proofs from assumptions, and axioms (as traditionally understood) will have no role to play. | |
From: David Bostock (Intermediate Logic [1997], 6.1) | |
A reaction: The main rules are those for introduction and elimination of truth functors. |
13755 | Excluded middle is an introduction rule for negation, and ex falso quodlibet will eliminate it [Bostock] |
Full Idea: Many books take RAA (reductio) and DNE (double neg) as the natural deduction introduction- and elimination-rules for negation, but RAA is not a natural introduction rule. I prefer TND (tertium) and EFQ (ex falso) for ¬-introduction and -elimination. | |
From: David Bostock (Intermediate Logic [1997], 6.2) |
13758 | In natural deduction we work from the premisses and the conclusion, hoping to meet in the middle [Bostock] |
Full Idea: When looking for a proof of a sequent, the best we can do in natural deduction is to work simultaneously in both directions, forward from the premisses, and back from the conclusion, and hope they will meet in the middle. | |
From: David Bostock (Intermediate Logic [1997], 6.5) |
13754 | Natural deduction rules for → are the Deduction Theorem (→I) and Modus Ponens (→E) [Bostock] |
Full Idea: Natural deduction adopts for → as rules the Deduction Theorem and Modus Ponens, here called →I and →E. If ψ follows φ in the proof, we can write φ→ψ (→I). φ and φ→ψ permit ψ (→E). | |
From: David Bostock (Intermediate Logic [1997], 6.2) | |
A reaction: Natural deduction has this neat and appealing way of formally introducing or eliminating each connective, so that you know where you are, and you know what each one means. |
13611 | Tableau proofs use reduction - seeking an impossible consequence from an assumption [Bostock] |
Full Idea: A tableau proof is a proof by reduction ad absurdum. One begins with an assumption, and one develops the consequences of that assumption, seeking to derive an impossible consequence. | |
From: David Bostock (Intermediate Logic [1997], 4.1) |
13613 | A completed open branch gives an interpretation which verifies those formulae [Bostock] |
Full Idea: An open branch in a completed tableau will always yield an interpretation that verifies every formula on the branch. | |
From: David Bostock (Intermediate Logic [1997], 4.7) | |
A reaction: In other words the open branch shows a model which seems to work (on the available information). Similarly a closed branch gives a model which won't work - a counterexample. |
13612 | Non-branching rules add lines, and branching rules need a split; a branch with a contradiction is 'closed' [Bostock] |
Full Idea: Rules for semantic tableaus are of two kinds - non-branching rules and branching rules. The first allow the addition of further lines, and the second requires splitting the branch. A branch which assigns contradictory values to a formula is 'closed'. | |
From: David Bostock (Intermediate Logic [1997], 4.1) | |
A reaction: [compressed] Thus 'and' stays on one branch, asserting both formulae, but 'or' splits, checking first one and then the other. A proof succeeds when all the branches are closed, showing that the initial assumption leads only to contradictions. |
13761 | In a tableau proof no sequence is established until the final branch is closed; hypotheses are explored [Bostock] |
Full Idea: In a tableau system no sequent is established until the final step of the proof, when the last branch closes, and until then we are simply exploring a hypothesis. | |
From: David Bostock (Intermediate Logic [1997], 7.3) | |
A reaction: This compares sharply with a sequence calculus, where every single step is a conclusive proof of something. So use tableaux for exploring proofs, and then sequence calculi for writing them up? |
13757 | Unlike natural deduction, semantic tableaux have recipes for proving things [Bostock] |
Full Idea: With semantic tableaux there are recipes for proof-construction that we can operate, whereas with natural deduction there are not. | |
From: David Bostock (Intermediate Logic [1997], 6.5) |
13756 | A tree proof becomes too broad if its only rule is Modus Ponens [Bostock] |
Full Idea: When the only rule of inference is Modus Ponens, the branches of a tree proof soon spread too wide for comfort. | |
From: David Bostock (Intermediate Logic [1997], 6.4) |
13762 | Tableau rules are all elimination rules, gradually shortening formulae [Bostock] |
Full Idea: In their original setting, all the tableau rules are elimination rules, allowing us to replace a longer formula by its shorter components. | |
From: David Bostock (Intermediate Logic [1997], 7.3) |
13759 | Each line of a sequent calculus is a conclusion of previous lines, each one explicitly recorded [Bostock] |
Full Idea: A sequent calculus keeps an explicit record of just what sequent is established at each point in a proof. Every line is itself the sequent proved at that point. It is not a linear sequence or array of formulae, but a matching array of whole sequents. | |
From: David Bostock (Intermediate Logic [1997], 7.1) |
13760 | A sequent calculus is good for comparing proof systems [Bostock] |
Full Idea: A sequent calculus is a useful tool for comparing two systems that at first look utterly different (such as natural deduction and semantic tableaux). | |
From: David Bostock (Intermediate Logic [1997], 7.2) |
13364 | Interpretation by assigning objects to names, or assigning them to variables first [Bostock, by PG] |
Full Idea: There are two approaches to an 'interpretation' of a logic: the first method assigns objects to names, and then defines connectives and quantifiers, focusing on truth; the second assigns objects to variables, then variables to names, using satisfaction. | |
From: report of David Bostock (Intermediate Logic [1997], 3.4) by PG - Db (lexicon) | |
A reaction: [a summary of nine elusive pages in Bostock] He says he prefers the first method, but the second method is more popular because it handles open formulas, by treating free variables as if they were names. |
13821 | Extensionality is built into ordinary logic semantics; names have objects, predicates have sets of objects [Bostock] |
Full Idea: Extensionality is built into the semantics of ordinary logic. When a name-letter is interpreted as denoting something, we just provide the object denoted. All that we provide for a one-place predicate-letter is the set of objects that it is true of.. | |
From: David Bostock (Intermediate Logic [1997]) | |
A reaction: Could we keep the syntax of ordinary logic, and provide a wildly different semantics, much closer to real life? We could give up these dreadful 'objects' that Frege lumbered us with. Logic for processes, etc. |
13362 | If an object has two names, truth is undisturbed if the names are swapped; this is Extensionality [Bostock] |
Full Idea: If two names refer to the same object, then in any proposition which contains either of them the other may be substituted in its place, and the truth-value of the proposition of the proposition will be unaltered. This is the Principle of Extensionality. | |
From: David Bostock (Intermediate Logic [1997], 3.1) | |
A reaction: He acknowledges that ordinary language is full of counterexamples, such as 'he doesn't know the Morning Star and the Evening Star are the same body' (when he presumably knows that the Morning Star is the Morning Star). This is logic. Like maths. |
13541 | For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ [Bostock] |
Full Idea: Any system of proof S is said to be 'negation-consistent' iff there is no formula such that |-(S)φ and |-(S)¬φ. | |
From: David Bostock (Intermediate Logic [1997], 4.5) | |
A reaction: Compare Idea 13542. This version seems to be a 'strong' version, as it demands a higher standard than 'absolute consistency'. Both halves of the condition would have to be established. |
13542 | A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula [Bostock] |
Full Idea: Any system of proof S is said to be 'absolutely consistent' iff it is not the case that for every formula we have |-(S)φ. | |
From: David Bostock (Intermediate Logic [1997], 4.5) | |
A reaction: Bostock notes that a sound system will be both 'negation-consistent' (Idea 13541) and absolutely consistent. 'Tonk' systems can be shown to be unsound because the two come apart. |
13540 | A set of formulae is 'inconsistent' when there is no interpretation which can make them all true [Bostock] |
Full Idea: 'Γ |=' means 'Γ is a set of closed formulae, and there is no (standard) interpretation in which all of the formulae in Γ are true'. We abbreviate this last to 'Γ is inconsistent'. | |
From: David Bostock (Intermediate Logic [1997], 4.5) | |
A reaction: This is a semantic approach to inconsistency, in terms of truth, as opposed to saying that we cannot prove both p and ¬p. I take this to be closer to the true concept, since you need never have heard of 'proof' to understand 'inconsistent'. |
13544 | Inconsistency or entailment just from functors and quantifiers is finitely based, if compact [Bostock] |
Full Idea: Being 'compact' means that if we have an inconsistency or an entailment which holds just because of the truth-functors and quantifiers involved, then it is always due to a finite number of the propositions in question. | |
From: David Bostock (Intermediate Logic [1997], 4.8) | |
A reaction: Bostock says this is surprising, given the examples 'a is not a parent of a parent of b...' etc, where an infinity seems to establish 'a is not an ancestor of b'. The point, though, is that this truth doesn't just depend on truth-functors and quantifiers. |
13618 | Compactness means an infinity of sequents on the left will add nothing new [Bostock] |
Full Idea: The logic of truth-functions is compact, which means that sequents with infinitely many formulae on the left introduce nothing new. Hence we can confine our attention to finite sequents. | |
From: David Bostock (Intermediate Logic [1997], 5.5) | |
A reaction: This makes it clear why compactness is a limitation in logic. If you want the logic to be unlimited in scope, it isn't; it only proves things from finite numbers of sequents. This makes it easier to prove completeness for the system. |
9003 | Set theory was struggling with higher infinities, when new paradoxes made it baffling [Quine] |
Full Idea: Unlike elementary logic, the truths of set theory are not obvious. Set theory was straining at the leash of intuition ever since Cantor discovered higher infinites; and with the added impetus of the paradoxes of set theory the leash snapped. | |
From: Willard Quine (Carnap and Logical Truth [1954], II) | |
A reaction: This problem seems to have forced Quine into platonism about sets, because he felt they were essential for mathematics and science, but couldn't be constructed with precision. So they must be real, but we don't quite understand them. |
13358 | Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all [Bostock] |
Full Idea: The principle of mathematical (or ordinary) induction says suppose the first number, 0, has a property; suppose that if any number has that property, then so does the next; then it follows that all numbers have the property. | |
From: David Bostock (Intermediate Logic [1997], 2.8) | |
A reaction: Ordinary induction is also known as 'weak' induction. Compare Idea 13359 for 'strong' or complete induction. The number sequence must have a first element, so this doesn't work for the integers. |
13359 | Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers [Bostock] |
Full Idea: The principle of complete induction says suppose that for every number, if all the numbers less than it have a property, then so does it; it then follows that every number has the property. | |
From: David Bostock (Intermediate Logic [1997], 2.8) | |
A reaction: Complete induction is also known as 'strong' induction. Compare Idea 13358 for 'weak' or mathematical induction. The number sequence need have no first element. |
9004 | If set theory is not actually a branch of logic, then Frege's derivation of arithmetic would not be from logic [Quine] |
Full Idea: We might say that set theory is not really logic, but a branch of mathematics. This would deprive 'includes' of the status of a logical word. Frege's derivation of arithmetic would then cease to count as a derivation from logic: for he used set theory. | |
From: Willard Quine (Carnap and Logical Truth [1954], II) | |
A reaction: Quine has been making the point that higher infinities and the paradoxes undermine the status of set theory as logic, but he decides to continue thinking of set theory as logic. Critics of logicism frequently ask whether the reduction is to logic. |
12838 | Four-dimensional ontology has no change, since that needs an object, and time to pass [Simons] |
Full Idea: In the four-dimensional ontology there may be timeless variation, but there is no change. Change consists in an object having first one property and then another contrary one. But processes all have their properties timelessly. | |
From: Peter Simons (Parts [1987], 3.4) | |
A reaction: Possibly Simons is begging the question here. The phenomena which are traditionally labelled as 'change' are all nicely covered in the four-D account. Change is, we might say, subsumed in the shape of the space-time 'worm'. |
12842 | There are real relational changes, as well as bogus 'Cambridge changes' [Simons] |
Full Idea: It is a mistake to call bogus Cambridge changes 'relational changes', since there are real relational changes, such as the changes in the relative positions and distances of several bodies. | |
From: Peter Simons (Parts [1987], 4.1) | |
A reaction: I'm not sure how you distinguish the two. If we swap seats, that is a real change. If everyone moves away from where I am sitting, is that real or Cambridge? If I notice, I might be upset, but suppose I don't notice? Nothing about me changes. |
12841 | I don't believe in processes [Simons] |
Full Idea: I have been unable to see that there are processes. | |
From: Peter Simons (Parts [1987], 4.1 n4) | |
A reaction: My problem here is that I am inclined to think of the mind as a process of the brain. The fact that a reductive account can be given of a process doesn't mean that we can deny there existence. Is there no such thing as decay, or erosion? |
12836 | Fans of process ontology cheat, since river-stages refer to 'rivers' [Simons] |
Full Idea: Proponents of process ontology (except perhaps Whitehead, who is obscure) indulge in double-talk with concrete examples. It is cheating to talk of 'cat-processes', or 'bathing in river-stages'. You can't change the subject and leave the predicate alone. | |
From: Peter Simons (Parts [1987], 3.4) | |
A reaction: It is one thing to admit processes into one's ontology, and another to have a 'process ontology', which presumably reduces objects to processes. I suppose the interest of continuant objects is precisely the aspect of them that is above any process. |
12880 | Moments are things like smiles or skids, which are founded on other things [Simons] |
Full Idea: A 'moment' is something which is founded on something else. Examples are legion: smiles, headaches, gestures, skids, collisions, fights, thought, all founded on their participants, the continuants involved in them. | |
From: Peter Simons (Parts [1987], 8.4) | |
A reaction: The idea of a 'moment' and 'foundation' come from Husserl Log. Inv. 3. Simons says moments 'have a bright future in ontology'. It would be better if fewer of his examples involved human beings and their perceptions. |
12883 | Moving disturbances are are moments which continuously change their basis [Simons] |
Full Idea: Moving disturbances are a special and interesting kind of continuant: moments which continuously change their fundaments. | |
From: Peter Simons (Parts [1987], 8.5) | |
A reaction: [a smile is a moment, and the face its fundament] I'm thinking he's got this wrong. Compare Idea 12882. Disturbances can't be continuants, because the passing of time is essential to them, but not to a continuant. |
12881 | A smiling is an event with causes, but the smile is a continuant without causes [Simons] |
Full Idea: A smiling, being an event, has causes and effects, whereas the smile thereby produced is a continuant, and has itself neither causes nor effects. | |
From: Peter Simons (Parts [1987], 8.5) | |
A reaction: This is dogmatic, hopeful and a bit dubious. Simons is very scathing about processes in ontology. There seem to be two descriptions, with distinctive syntax, but it is hard to believe that in reality we have two types of thing present. |
12882 | A wave is maintained by a process, but it isn't a process [Simons] |
Full Idea: A wave is maintained by a process transferring motion from particle to particle of the medium, but it is not identical with this process. | |
From: Peter Simons (Parts [1987], 8.5) | |
A reaction: I'm inclined to think of the mind as a process. There are some 'things' which only seem to exist if they have a duration. Bricks can be instantaneous, but minds and waves can't. A wave isn't a continuant. A hill isn't a wave. |
12840 | I do not think there is a general identity condition for events [Simons] |
Full Idea: Like Anscombe (1979) I do not think there is such a creature as a general identity condition for events. | |
From: Peter Simons (Parts [1987], 4.1 n1) | |
A reaction: My working definition of an event is 'any part of a process which can be individuated'. This leaves you trying to define a process, and define individuate, and then to realise that individuation is not an objective matter. |
12839 | Relativity has an ontology of things and events, not on space-time diagrams [Simons] |
Full Idea: A closer examination of the concepts and principles of relativity shows that they rest squarely on an ontology of things and events (not on convenient 'space-time diagrams'). Acceleration concerns non-zero mass, but only continuants can have a mass. | |
From: Peter Simons (Parts [1987], 3.4) | |
A reaction: The point here is that fans of four-dimensionalism like to claim that they are more in touch with modern physics, because 'time is just another dimension, like space, so objects are spread across it'. Simons sounds right about this. |
12879 | Independent objects can exist apart, and maybe even entirely alone [Simons] |
Full Idea: An object a is ontologically independent of b if a can exist without b, if there is a possible world in which in which a exists and b does not. In the strongest sense, an object is independent if it could be all there is. | |
From: Peter Simons (Parts [1987], 8.4) | |
A reaction: Simons calls the strongest version a 'startling' one which maybe not even God could achieve. |
12847 | Mass nouns admit 'much' and 'a little', and resist 'many' and 'few'. [Simons] |
Full Idea: Syntactic criteria for mass nouns include that they admit 'much' and 'a little', and resist 'many' and 'few'. | |
From: Peter Simons (Parts [1987], 4.6) | |
A reaction: That is, they don't seem to be countable. Sortal terms are those which pick out countables. |
12863 | Mass terms (unlike plurals) are used with indifference to whether they can exist in units [Simons] |
Full Idea: Mass terms and plural terms differ principally in the indifference of mass terms to matters of division. A mass term can be used irrespective of how, indeed whether, the denotatum comes parcelled in units. | |
From: Peter Simons (Parts [1987], 6.4) | |
A reaction: It seems more to the point to say that mass terms (stuff) don't need units to exist, and you can disperse the units (the cups of water) without affecting the identity of the stuff. You can't pulverise a pile of stones and retain the stones. |
12862 | Gold is not its atoms, because the atoms must be all gold, but gold contains neutrons [Simons] |
Full Idea: The mass of gold cannot be identified with the gold atoms, because whatever is part of the gold atoms is gold, whereas not every part of the gold is gold (for example, the neutrons in it are not gold). | |
From: Peter Simons (Parts [1987], 6.4) | |
A reaction: There is something too quick about arguments like this. It comes back to nominal v real essence. We apply 'gold' to the superficial features of the stuff, but deep down we may actually mean the atomic structure. See Idea 12812. |
12859 | A mixture can have different qualities from its ingredients. [Simons] |
Full Idea: The qualities of a mixture need not be those of its ingredients in isolation. | |
From: Peter Simons (Parts [1987], 6.2) | |
A reaction: It depends on what you mean by a quality. Presumably we can give a reductive account of the qualities of the mixture, as long as no reaction has taken place. The taste of a salad is just the sum of its parts. |
12858 | Mixtures disappear if nearly all of the mixture is one ingredient [Simons] |
Full Idea: If a cupful of dirty water is mixed evenly with a ton of earth, no dirty water remains, and the same goes if we mix it evenly with a lake of clean water. | |
From: Peter Simons (Parts [1987], 6.2) | |
A reaction: This means that a mixture is a vague entity, subject to the sorites paradox. If the dirt was cyanide, we would consider the water to be polluted by it down to a much lower level. |
13543 | A relation is not reflexive, just because it is transitive and symmetrical [Bostock] |
Full Idea: It is easy to fall into the error of supposing that a relation which is both transitive and symmetrical must also be reflexive. | |
From: David Bostock (Intermediate Logic [1997], 4.7) | |
A reaction: Compare Idea 14430! Transivity will take you there, and symmetricality will get you back, but that doesn't entitle you to take the shortcut? |
13802 | Relations can be one-many (at most one on the left) or many-one (at most one on the right) [Bostock] |
Full Idea: A relation is 'one-many' if for anything on the right there is at most one on the left (∀xyz(Rxz∧Ryz→x=y), and is 'many-one' if for anything on the left there is at most one on the right (∀xyz(Rzx∧Rzy→x=y). | |
From: David Bostock (Intermediate Logic [1997], 8.1) |
9006 | Commitment to universals is as arbitrary or pragmatic as the adoption of a new system of bookkeeping [Quine] |
Full Idea: One's hypothesis as to there being universals is at bottom just as arbitrary or pragmatic a matter as one's adoption of a new brand of set theory or even a new system of bookkeeping. | |
From: Willard Quine (Carnap and Logical Truth [1954], x) | |
A reaction: This spells out clearly the strongly pragmatist vein in Quine's thinking. |
12850 | To individuate something we must pick it out, but also know its limits of variation [Simons] |
Full Idea: We have not finished deciding what Fido is when we can pick him out from his surroundings at any one time. ...Knowing what Fido is depends on knowing roughly within what limits his flux of parts is tolerable. | |
From: Peter Simons (Parts [1987], 5.2) | |
A reaction: I like this. We don't know the world until we know its modal characteristics (its powers or dispositions). Have you 'individuated' a hand grenade if you think it is a nice ornament? |
12860 | Sortal nouns for continuants tell you their continuance- and cessation-conditions [Simons] |
Full Idea: A sortal noun for a kind of continuant tells us, among other things, under what conditions the object continues to exist and under what conditions it ceases to exist. | |
From: Peter Simons (Parts [1987], 6.3) | |
A reaction: This sounds blatantly false. If you know something is a 'snake', that doesn't tell you how hot it must get before the snakes die. Obviously if you know all about snakes (from studying individual snakes!), then you know a lot about the next snake. |
12886 | A whole requires some unique relation which binds together all of the parts [Simons] |
Full Idea: A whole must at least approximate to this condition: every member of some division of the object stands in a certain relation to every other member, and no member bears this relation to anything other than members of the division. | |
From: Peter Simons (Parts [1987], 9.2) | |
A reaction: Simons proceeds to formalise this, and I suspect that he goes for this definition because (unlike looser ones) it can be formalised. See Simons's Idea 12865. We'll need to know whether these are internal or external relations. |
12857 | Tibbles isn't Tib-plus-tail, because Tibbles can survive its loss, but the sum can't [Simons] |
Full Idea: There mere fact that Tibbles can survive the mutilation of losing a tail, whereas the sum of Tib and the tail cannot, is enough to distinguish them, even if no such mutilation ever occurs. | |
From: Peter Simons (Parts [1987], 6.1) | |
A reaction: See Idea 12835 for details of the Tibbles example. Either we go for essentialism here, or the whole notion of identity collapses. But the essential features of a person are not just those whose loss would kill them. |
12835 | Does Tibbles remain the same cat when it loses its tail? [Simons] |
Full Idea: The cat is 'Tibbles' with a tail; 'Tib' is Tibbles after the loss of the tail. 1) Tibbles isn't Tib at t; 2) Tibbles is Tib at t'; 3) Tibbles at t is Tibbles at t'; 4) Tib at t is Tib at t'; so 5) Tibbles at t is Tib at t (contradicting 1). What's wrong? | |
From: Peter Simons (Parts [1987], 3.3) | |
A reaction: [The example is in Wiggins 1979, from Geach, from William of Sherwood] Simons catalogues nine assumptions which are being made to produce the contradiction. 1) rests on Leibniz's law. Simons says two objects are occupying Tibbles. |
12820 | Without extensional mereology two objects can occupy the same position [Simons] |
Full Idea: If we reject extensionality in mereology, it has as a consequence that more than one object may have exactly the same parts at the same time, and hence occupy the same position. | |
From: Peter Simons (Parts [1987], Intro) | |
A reaction: Simons defends this claim. I'm unconvinced that we must choose between the two views. The same parts should ensure the same physical essence, which seems to guarantee the same identity. Not any old parts generate an essence. |
12866 | Composition is asymmetric and transitive [Simons] |
Full Idea: Composition is asymmetric and transitive: if a is made up of b, and b of c, then a is made up of c; and if a is made of b, then b is not made up of a. We cannot say the snow is made up of the snowball. | |
From: Peter Simons (Parts [1987], 6.5) | |
A reaction: ...And snowballs composed of snow can then compose a snowman (transitivity). |
12867 | A hand constitutes a fist (when clenched), but a fist is not composed of an augmented hand [Simons] |
Full Idea: Composition entails constitution, but does the converse hold? A hand constitutes a fist in virtue of being clenched, but it is not obvious that it composes a fist, and certainly a fist is not composed of a hand plus some additional part. | |
From: Peter Simons (Parts [1987], 6.5) | |
A reaction: There are subtleties of ordinary usage in 'compose' and 'constitute' which are worth teasing apart, but that isn't the last word on such relationships. 'Compose' seems to point towards matter, while 'constitute' seems to point towards form. |
12864 | We say 'b is part of a', 'b is a part of a', 'b are a part of a', or 'b are parts of a'. [Simons] |
Full Idea: There are four cases of possible forms of expression when a is made up of b: we say 'b is part of a', or 'b is a part of a', or 'b are a part of a', or 'b are parts of a'. | |
From: Peter Simons (Parts [1987], 6.4) | |
A reaction: Personally I don't want to make much of these observations of normal English usage, but they are still interesting, and Simons offers a nice discussion of them. |
12814 | Classical mereology says there are 'sums', for whose existence there is no other evidence [Simons] |
Full Idea: Either out of conviction or for reasons of algebraic neatness, classical extensional mereology asserts the existence of certain individuals, mereological sums, for whose existence in general we have no evidence outside the theory itself. | |
From: Peter Simons (Parts [1987], Intro) | |
A reaction: Observing that we have no evidence for sums 'outside the theory' is nice. It is a nice ontological test, with interesting implications for Quinean ontological commitment. |
12817 | 'Mereological extensionality' says objects with the same parts are identical [Simons] |
Full Idea: Classical extensional mereology won't extend well to temporal and modal facts, because of 'mereological extensionality', which is the thesis that objects with the same parts are identical (by analogy with the extensionality of sets). | |
From: Peter Simons (Parts [1987], Intro) | |
A reaction: Simons challenges this view, claiming, for example, that the Ship of Theseus is two objects rather than one. I suppose 'my building bricks' might be 'your sculpture', but this is very ontologically extravagant. This is a mereological Leibniz's Law. |
12833 | If there are c atoms, this gives 2^c - 1 individuals, so there can't be just 2 or 12 individuals [Simons] |
Full Idea: In classical mereology, if there are c atoms, where c is any cardinal number, there are 2^c - 1 individuals, so the cardinality of models is restricted. There are no models with cardinality 2, 12 or aleph-0, for example. | |
From: Peter Simons (Parts [1987], 1.2) | |
A reaction: The news that there is no possible world containing just 2 or just 12 individuals ought to worry fans of extensional mereology. A nice challenge for God - create a world containing just 12 individuals. |
12849 | Sums are more plausible for pluralities and masses than they are for individuals [Simons] |
Full Idea: We are on stronger grounds in asserting the general existence of sums when considering pluralities and masses than when considering individuals. | |
From: Peter Simons (Parts [1987], 5.2) | |
A reaction: I was thinking that the modern emphasis on referring to plurals was precisely to resist the idea that we must 'sum' them into one thing. If so, we wouldn't want to then sum several plurals. If a mass isn't a sum, how can we sum some masses? |
12877 | Sums of things in different categories are found within philosophy. [Simons] |
Full Idea: Cross-categorial sums are not unknown in philosophy. A body and the events which befall it are intimately connected, and the mysterious four-dimensional blocks might be mereological sums of the body and its life. | |
From: Peter Simons (Parts [1987], 8.1) | |
A reaction: Simons here ventures into the territory of abstracta, which he said he wouldn't touch. Presumably his first example has 'a biography' as its whole, which is not just a philosophical notion. Why will some categories sum, and others won't? |
12888 | The wholeness of a melody seems conventional, but of an explosion it seems natural [Simons] |
Full Idea: The example of a melody shows that what counts as a temporal individual is partly a matter of human stipulation. But with a natural event like an explosion there is little or no room for decision about what is a part, and whether it is a single event. | |
From: Peter Simons (Parts [1987], 9.6) | |
A reaction: You could have a go at giving a natural account of the wholeness of a melody, in terms of the little aesthetic explosion that occurs in the brain of a listener. |
12871 | Objects have their essential properties because of the kind of objects they are [Simons] |
Full Idea: An object has the essential properties it has in virtue of being the kind of object it is. | |
From: Peter Simons (Parts [1987], 7.1) | |
A reaction: He attributes this to Husserl and Wiggins. I just don't get it. What makes something the 'kind of object it is'? They've got it the wrong way round. Does God announce that this thing is a tiger, and is then pleasantly surprised to discover its stripes? |
12870 | We must distinguish the de dicto 'must' of propositions from the de re 'must' of essence [Simons] |
Full Idea: We must distinguish the 'must' of necessity as applied to a proposition or state of affairs (de dicto) from the 'must' of essence, concerning the way in which an object has an attribute (de re). | |
From: Peter Simons (Parts [1987], 7.1) | |
A reaction: A helfpful distinction, but a possible confusion of necessity and essentiality (Simons knows this). Modern logicians seem to run them together, because they only care about identity. I don't, because I care about explanations. |
12873 | Original parts are the best candidates for being essential to artefacts [Simons] |
Full Idea: Original parts are the best candidates for being essential to artefacts. It is hard to conceive how an object could have as essential a part which was attached at some time after the object had come into being. | |
From: Peter Simons (Parts [1987], 7.4) | |
A reaction: Without its big new memory upgrade my computer would be hopelessly out of date. Simons is awesome in some ways, but seems rather confused when it comes to discussing essence. I think Wiggins may have been a bad influence on him. |
12874 | An essential part of an essential part is an essential part of the whole [Simons] |
Full Idea: An essential part of an essential part is an essential part of the whole. | |
From: Peter Simons (Parts [1987], 7.4) | |
A reaction: Sounds beyond dispute, but worth pondering. It seems to be only type-parts, not token-parts, which are essential. Simons is thinking of identity rather than function, but he rejects Chisholm's idea that all parts are essential. So which ones are? |
12837 | Four dimensional-objects are stranger than most people think [Simons] |
Full Idea: The strangeness of four-dimensional objects is almost always underestimated in the literature. | |
From: Peter Simons (Parts [1987], 3.4) | |
A reaction: See Idea 12836, where he has criticised process ontologists for smuggling in stages and process as being OF conventional objects. |
12856 | Intermittent objects would be respectable if they occurred in nature, as well as in artefacts [Simons] |
Full Idea: If we could show that intermittence could occur not only among artefacts and higher-order objects, but also among natural things, then we should have given it a secure place on the ontological map. | |
From: Peter Simons (Parts [1987], 5.7) | |
A reaction: Interesting ontological test. Having identified fairly clear intermittent artefacts (Idea 12851), if we then fail to find any examples in nature, must we revisit the artefacts and say they are not intermittents? He suggests freezing an organ in surgery. |
12885 | Objects like chess games, with gaps in them, are thereby less unified [Simons] |
Full Idea: Temporal objects which are scattered in time - i.e. have temporal gaps in them, like interrupted discussions or chess games - are less unified than those without gaps. | |
From: Peter Simons (Parts [1987], 9.2) | |
A reaction: Is he really saying that a discussion or a chess game is less unified if there is even the slightest pause in it? Otherwise, how long must the pause be before it disturbs the unity? Do people play internet chess, as they used to play correspondence chess? |
12854 | An entrepreneur and a museum curator would each be happy with their ship at the end [Simons] |
Full Idea: At the end of the Ship of Theseus story both an entrepreneur and a museum curator can be content, each having his ship all to himself, ..because each was all along claiming a different object from the other. | |
From: Peter Simons (Parts [1987], 5.5) | |
A reaction: Simons has the entrepreneur caring about function (for cruises), and the curator caring about matter (as a relic of Theseus). It is bold of Simons to say on that basis that it starts as two objects, one 'matter-constant', the other 'form-constant'. |
12855 | The 'best candidate' theories mistakenly assume there is one answer to 'Which is the real ship?' [Simons] |
Full Idea: The 'best candidate' theories get into difficulty because it is assumed that there is a single uniquely correct answer to the question 'Which is the real ship?' | |
From: Peter Simons (Parts [1987], 5.5) | |
A reaction: My own example supports Simons. If Theseus discards the old planks as rubbish, then his smart new ship is the original. But if he steals his own ship (to evade insurance regulations) by substituting a plank at a time, the removed planks are the original. |
12872 | The zygote is an essential initial part, for a sexually reproduced organism [Simons] |
Full Idea: It is essential to an organism arising from sexual reproduction that it has its zygote as initial improper part. | |
From: Peter Simons (Parts [1987], 7.3) | |
A reaction: It can't be necessary that an organism which appears to be sexually reproduced actually is so (if you don't believe that, read more science fiction). It may well just be analytic that sexual reproduction involves a zygote. Nothing to do with essence. |
13847 | If non-existent things are self-identical, they are just one thing - so call it the 'null object' [Bostock] |
Full Idea: If even non-existent things are still counted as self-identical, then all non-existent things must be counted as identical with one another, so there is at most one non-existent thing. We might arbitrarily choose zero, or invent 'the null object'. | |
From: David Bostock (Intermediate Logic [1997], 8.6) |
9001 | Frege moved Kant's question about a priori synthetic to 'how is logical certainty possible?' [Quine] |
Full Idea: When Kant's arithmetical examples of a priori synthetic judgements were sweepingly disqualified by Frege's reduction of arithmetic to logic, attention moved to the less tendentious and logically prior question 'How is logical certainty possible?' | |
From: Willard Quine (Carnap and Logical Truth [1954], I) | |
A reaction: A nice summary of the story so far, from someone who should know. This still leaves the question open of whether any synthetic truths can be derived from the logical certainties which are available. |
13820 | The idea that anything which can be proved is necessary has a problem with empty names [Bostock] |
Full Idea: The common Rule of Necessitation says that what can be proved is necessary, but this is incorrect if we do not permit empty names. The most straightforward answer is to modify elementary logic so that only necessary truths can be proved. | |
From: David Bostock (Intermediate Logic [1997], 8.4) |
12889 | The limits of change for an individual depend on the kind of individual [Simons] |
Full Idea: What determines the limits of admissible change and secures the identity of a continuant is a matter of the kind of object in question. | |
From: Peter Simons (Parts [1987], 9.6) | |
A reaction: This gives some motivation for the sortal view of essence, which I find hard to take. However, if my statue were pulverised it would make good compost. |
9005 | Examination of convention in the a priori begins to blur the distinction with empirical knowledge [Quine] |
Full Idea: In trying to make sense of the role of convention in a priori knowledge, the very distinction between a priori and empirical begins to waver and dissolve. | |
From: Willard Quine (Carnap and Logical Truth [1954], VI) | |
A reaction: This is the next stage in the argument after Wittgenstein presents the apriori as nothing more than what arises from truth tables. The rationalists react by taking us back to the original 'natural light of reason' view. Then we go round again... |
13363 | A (modern) predicate is the result of leaving a gap for the name in a sentence [Bostock] |
Full Idea: A simple way of approaching the modern notion of a predicate is this: given any sentence which contains a name, the result of dropping that name and leaving a gap in its place is a predicate. Very different from predicates in Aristotle and Kant. | |
From: David Bostock (Intermediate Logic [1997], 3.2) | |
A reaction: This concept derives from Frege. To get to grips with contemporary philosophy you have to relearn all sorts of basic words like 'predicate' and 'object'. |
12843 | With activities if you are doing it you've done it, with performances you must finish to have done it [Simons] |
Full Idea: Action theorists distinguish between activity verbs such as 'weep' and 'talk' (where continuous entails perfect - John is weeping so John has now wept), and performance verbs like 'wash', where John is washing doesn't yet mean John has washed. | |
From: Peter Simons (Parts [1987], 4.2) | |
A reaction: How to distinguish them, bar examples? In 'has wept' and 'has washed', I'm thinking that it is the 'has' which is ambiguous, rather than the more contentful word. One is 'has participated' and the other is 'has completed'. I've participated in washing! |
12875 | One false note doesn't make it a performance of a different work [Simons] |
Full Idea: A performance of a certain work with a false note is still a performance of that work, albeit a slightly imperfect one, and not (as Goodman has argued) a performance of a different work. | |
From: Peter Simons (Parts [1987], 7.6) | |
A reaction: This is clearly right, but invites the question of how many wrong notes are permissable. One loud very wrong note could ruin a very long performance (but of that work, presumably). This is about classical music, but think about jazz. |