52 ideas
| 12027 | There must be a plausible epistemological theory alongside any metaphysical theory [Forbes,G] |
| Full Idea: No metaphysical account which renders it impossible to give a plausible epistemological theory is to be countenanced. | |
| From: Graeme Forbes (The Metaphysics of Modality [1985], 9.1) | |
| A reaction: It is hard to object to this principle, though we certainly don't want to go verificationist, and thus rule out speculations about metaphysics which are beyond any possible knowledge. Some have tried to prove that something must exist (e.g. Jacquette). |
| 17749 | Post proved the consistency of propositional logic in 1921 [Walicki] |
| Full Idea: A proof of the consistency of propositional logic was given by Emil Post in 1921. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.2.1) |
| 17765 | Propositional language can only relate statements as the same or as different [Walicki] |
| Full Idea: Propositional language is very rudimentary and has limited powers of expression. The only relation between various statements it can handle is that of identity and difference. As are all the same, but Bs can be different from As. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 7 Intro) | |
| A reaction: [second sentence a paraphrase] In predicate logic you could represent two statements as being the same except for one element (an object or predicate or relation or quantifier). |
| 12005 | The symbol 'ι' forms definite descriptions; (ιx)F(x) says 'the x which is such that F(x)' [Forbes,G] |
| Full Idea: We use the symbol 'ι' (Greek 'iota') to form definite descriptions, reading (ιx)F(x) as 'the x which is such that F(x)', or simply as 'the F'. | |
| From: Graeme Forbes (The Metaphysics of Modality [1985], 4.1) | |
| A reaction: Compare the lambda operator in modal logic, which picks out predicates from similar formulae. |
| 17764 | Boolean connectives are interpreted as functions on the set {1,0} [Walicki] |
| Full Idea: Boolean connectives are interpreted as functions on the set {1,0}. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 5.1) | |
| A reaction: 1 and 0 are normally taken to be true (T) and false (F). Thus the functions output various combinations of true and false, which are truth tables. |
| 17752 | The empty set is useful for defining sets by properties, when the members are not yet known [Walicki] |
| Full Idea: The empty set is mainly a mathematical convenience - defining a set by describing the properties of its members in an involved way, we may not know from the very beginning what its members are. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 1.1) |
| 17753 | The empty set avoids having to take special precautions in case members vanish [Walicki] |
| Full Idea: Without the assumption of the empty set, one would often have to take special precautions for the case where a set happened to contain no elements. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 1.1) | |
| A reaction: Compare the introduction of the concept 'zero', where special precautions are therefore required. ...But other special precautions are needed without zero. Either he pays us, or we pay him, or ...er. Intersecting sets need the empty set. |
| 17759 | Ordinals play the central role in set theory, providing the model of well-ordering [Walicki] |
| Full Idea: Ordinals play the central role in set theory, providing the paradigmatic well-orderings. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) | |
| A reaction: When you draw the big V of the iterative hierarchy of sets (built from successive power sets), the ordinals are marked as a single line up the middle, one ordinal for each level. |
| 17741 | To determine the patterns in logic, one must identify its 'building blocks' [Walicki] |
| Full Idea: In order to construct precise and valid patterns of arguments one has to determine their 'building blocks'. One has to identify the basic terms, their kinds and means of combination. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History Intro) | |
| A reaction: A deceptively simple and important idea. All explanation requires patterns and levels, and it is the idea of building blocks which makes such things possible. It is right at the centre of our grasp of everything. |
| 12010 | Is the meaning of 'and' given by its truth table, or by its introduction and elimination rules? [Forbes,G] |
| Full Idea: The typical semantic account of validity for propositional connectives like 'and' presupposes that meaning is given by truth-tables. On the natural deduction view, the meaning of 'and' is given by its introduction and elimination rules. | |
| From: Graeme Forbes (The Metaphysics of Modality [1985], 4.4) |
| 17747 | A 'model' of a theory specifies interpreting a language in a domain to make all theorems true [Walicki] |
| Full Idea: A specification of a domain of objects, and of the rules for interpreting the symbols of a logical language in this domain such that all the theorems of the logical theory are true is said to be a 'model' of the theory. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.1.3) | |
| A reaction: The basic ideas of this emerged 1915-30, but it needed Tarski's account of truth to really get it going. |
| 17748 | The L-S Theorem says no theory (even of reals) says more than a natural number theory [Walicki] |
| Full Idea: The L-S Theorem is ...a shocking result, since it implies that any consistent formal theory of everything - even about biology, physics, sets or the real numbers - can just as well be understood as being about natural numbers. It says nothing more. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.2) | |
| A reaction: Illuminating. Particularly the point that no theory about the real numbers can say anything more than a theory about the natural numbers. So the natural numbers contain all the truths we can ever express? Eh????? |
| 17761 | A compact axiomatisation makes it possible to understand a field as a whole [Walicki] |
| Full Idea: Having such a compact [axiomatic] presentation of a complicated field [such as Euclid's], makes it possible to relate not only to particular theorems but also to the whole field as such. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1) |
| 17763 | Axiomatic systems are purely syntactic, and do not presuppose any interpretation [Walicki] |
| Full Idea: Axiomatic systems, their primitive terms and proofs, are purely syntactic, that is, do not presuppose any interpretation. ...[142] They never address the world directly, but address a possible semantic model which formally represents the world. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1) |
| 17758 | Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [Walicki] |
| Full Idea: An ordinal can be defined as a transitive set of transitive sets, or else, as a transitive set totally ordered by set inclusion. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) |
| 17755 | Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals [Walicki] |
| Full Idea: The collection of ordinals is defined inductively: Basis: the empty set is an ordinal; Ind: for an ordinal x, the union with its singleton is also an ordinal; and any arbitrary (possibly infinite) union of ordinals is an ordinal. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) | |
| A reaction: [symbolism translated into English] Walicki says they are called 'ordinal numbers', but are in fact a set. |
| 17756 | The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki] |
| Full Idea: We can form infinite ordinals by taking unions of ordinals. We can thus form 'limit ordinals', which have no immediate predecessor. ω is the first (the union of all finite ordinals), ω + ω = sω is second, 3ω the third.... | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) |
| 17760 | Two infinite ordinals can represent a single infinite cardinal [Walicki] |
| Full Idea: There may be several ordinals for the same cardinality. ...Two ordinals can represent different ways of well-ordering the same number (aleph-0) of elements. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) | |
| A reaction: This only applies to infinite ordinals and cardinals. For the finite, the two coincide. In infinite arithmetic the rules are different. |
| 17757 | Members of ordinals are ordinals, and also subsets of ordinals [Walicki] |
| Full Idea: Every member of an ordinal is itself an ordinal, and every ordinal is a transitive set (its members are also its subsets; a member of a member of an ordinal is also a member of the ordinal). | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) |
| 17762 | In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate [Walicki] |
| Full Idea: Since non-Euclidean geometry preserves all Euclid's postulates except the fifth one, all the theorems derived without the use of the fifth postulate remain valid. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1) |
| 17754 | Inductive proof depends on the choice of the ordering [Walicki] |
| Full Idea: Inductive proof is not guaranteed to work in all cases and, particularly, it depends heavily on the choice of the ordering. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.1.1) | |
| A reaction: There has to be an well-founded ordering for inductive proofs to be possible. |
| 12023 | Vagueness problems arise from applying sharp semantics to vague languages [Forbes,G] |
| Full Idea: It is very plausible that the sorites paradoxes arose from the application of a semantic apparatus appropriate only for sharp predicates to languages containing vague predicates (rather than from deficiency of meaning, or from incoherence). | |
| From: Graeme Forbes (The Metaphysics of Modality [1985], 7.3) | |
| A reaction: Sounds wrong. Of course, logic has been designed for sharp predicates, and natural languages are awash with vagueness. But the problems of vagueness bothered lawyers long before logicians like Russell began to worry about it. |
| 12017 | In all instances of identity, there must be some facts to ensure the identity [Forbes,G] |
| Full Idea: For each instance of identity or failure of identity, there must be facts in virtue of which that instance obtains. ..Enough has been said to lend this doctrine some plausibility. | |
| From: Graeme Forbes (The Metaphysics of Modality [1985], 5.5) | |
| A reaction: Penelope Mackie picks this out from Forbes as a key principle. It sounds to be in danger of circularity, unless the 'facts' can be cited without referring to, or implicitly making use of, identities - which seems unlikely. |
| 12024 | If we combined two clocks, it seems that two clocks may have become one clock. [Forbes,G] |
| Full Idea: If we imagine a possible world in which two clocks in a room make one clock from half the parts of each, the judgement 'these two actual clocks could have been a single clock' does not seem wholly false. | |
| From: Graeme Forbes (The Metaphysics of Modality [1985], 7.4) | |
| A reaction: You would, of course, have sufficient parts left over to make a second clock, so they look like a destroyed clock, so I don't think I find Forbes's intuition on this one very persuasive. |
| 11885 | Only individual essences will ground identities across worlds in other properties [Forbes,G, by Mackie,P] |
| Full Idea: Forbes argues that, unless we posit individual essences, we cannot guarantee that identities across possible worlds will be appropriately grounded in other properties. | |
| From: report of Graeme Forbes (The Metaphysics of Modality [1985]) by Penelope Mackie - How Things Might Have Been 2.4 | |
| A reaction: There is a confrontation between Wiggins, who says identity is primitive, and Forbes, who says identity must be grounded in other properties. I think I side with Forbes. |
| 12014 | An individual essence is a set of essential properties which only that object can have [Forbes,G] |
| Full Idea: An individual essence of an object x is a set of properties I which satisfies the following conditions: i. every property P in I is an essential property of x; ii. it is not possible that some object y distinct from x has every member of I. | |
| From: Graeme Forbes (The Metaphysics of Modality [1985], 5.1) | |
| A reaction: I am coming to the view that stable natural kinds (like electrons or gold) do not have individual essences, but complex kinds (like tigers or tables) do. The view is based on the idea that explanatory power is what individuates an essence. |
| 12015 | Non-trivial individual essence is properties other than de dicto, or universal, or relational [Forbes,G] |
| Full Idea: A non-trivial individual essence is properties other than a) those following from a de dicto truth, b) properties of existence and self-identity (or their cognates), c) properties derived from necessities in some other category. | |
| From: Graeme Forbes (The Metaphysics of Modality [1985], 5.1) | |
| A reaction: [I have compressed Forbes] Rather than adding all these qualificational clauses to our concept, we could just tighten up on the notion of a property, saying it is something which is causally efficacious, and hence explanatory. |
| 12013 | Essential properties depend on a category, and perhaps also on particular facts [Forbes,G] |
| Full Idea: The essential properties of a thing will typically depend upon what category of thing it is, and perhaps also on some more particular facts about the thing itself. | |
| From: Graeme Forbes (The Metaphysics of Modality [1985], 5.1) | |
| A reaction: I see no way of dispensing with the second requirement, in the cases of complex entities like animals. If all samples are the same, then of course we can define a sample's essence through its kind, but not if samples differ in any way. |
| 13804 | A property is essential iff the object would not exist if it lacked that property [Forbes,G] |
| Full Idea: A property P is an essential property of an object x iff x could not exist and lack P, that is, as they say, iff x has P at every world at which x exists. | |
| From: Graeme Forbes (In Defense of Absolute Essentialism [1986], 1) | |
| A reaction: This immediately places the existence of x outside the normal range of its properties, so presumably 'existence is not a predicate', but that dictum may be doubted. As it stands this definition will include trivial and vacuous properties. |
| 13805 | Properties are trivially essential if they are not grounded in a thing's specific nature [Forbes,G] |
| Full Idea: Essential properties may be trivial or nontrivial. It is characteristic of P's being trivially essential to x that x's possession of P is not grounded in the specific nature of x. | |
| From: Graeme Forbes (In Defense of Absolute Essentialism [1986], 2) | |
| A reaction: This is where my objection to the modal view of essence arises. How is he going to explain 'grounded' and 'specific nature' without supplying an entirely different account of essence? |
| 12012 | Essential properties are those without which an object could not exist [Forbes,G] |
| Full Idea: An essential property of an object x is a property without possessing which x could not exist. | |
| From: Graeme Forbes (The Metaphysics of Modality [1985], 5.1) | |
| A reaction: This is certainly open to question. See Joan Kung's account of Aristotle on essence. I am necessarily more than eight years old (now), and couldn't exist without that property, but is the property part of my essence? |
| 13808 | A relation is essential to two items if it holds in every world where they exist [Forbes,G] |
| Full Idea: A relation R is essential to x and y (in that order) iff Rxy holds at every world where x and y both exist. | |
| From: Graeme Forbes (In Defense of Absolute Essentialism [1986], 2) | |
| A reaction: I find this bizarre. Not only does this seem to me to have nothing whatever to do with essence, but also the relation might hold even though it is a purely contingent matter. All rabbits are a reasonable distance from the local star. Essence of rabbit? |
| 13806 | Trivially essential properties are existence, self-identity, and de dicto necessities [Forbes,G] |
| Full Idea: The main groups of trivially essential properties are (a) existence, self-identity, or their consequences in S5; and (b) properties possessed in virtue of some de dicto necessary truth. | |
| From: Graeme Forbes (In Defense of Absolute Essentialism [1986], 2) | |
| A reaction: He adds 'extraneously essential' properties, which also strike me as being trivial, involving relations. 'Is such that 2+2=4' or 'is such that something exists' might be necessary, but they don't, I would say, have anything to do with essence. |
| 13807 | A property is 'extraneously essential' if it is had only because of the properties of other objects [Forbes,G] |
| Full Idea: P is 'extraneously essential' to x iff it is possessed by x at any world w only in virtue of the possession at w of certain properties by other objects. | |
| From: Graeme Forbes (In Defense of Absolute Essentialism [1986], 2) | |
| A reaction: I would say that these are the sorts of properties which have nothing to do with being essential, even if they are deemed to be necessary. |
| 12022 | Same parts does not ensure same artefact, if those parts could constitute a different artefact [Forbes,G] |
| Full Idea: Sameness of parts is not sufficient for identity of artefacts at a world, since the very same parts may turn up at different times as the parts of artefacts with different designs and functions. | |
| From: Graeme Forbes (The Metaphysics of Modality [1985], 7.2) | |
| A reaction: Thus the Ship of Theseus could be dismantled and turned into a barn (as happened with the 'Mayflower'). They could then be reconstituted as the ship, which would then have two beginnings (as Chris Hughes has pointed out). |
| 12025 | Artefacts have fuzzy essences [Forbes,G] |
| Full Idea: Artefacts can be ascribed fuzzy essences. ...We might say that it is essential to an artefact to have 'most' of its parts. | |
| From: Graeme Forbes (The Metaphysics of Modality [1985], 7.6) | |
| A reaction: I think I prefer to accept the idea that essences are unstable things, in all cases. For all we know, electrons might subtly change their general character, or cease to be uniform, tomorrow. Essences explain, and what needs explaining changes. |
| 13809 | One might be essentialist about the original bronze from which a statue was made [Forbes,G] |
| Full Idea: In the case of artefacts, there is an essentialism about original matter; for instance, it would be said of any particular bronze statue that it could not have been cast from a totally different quantity of bronze. | |
| From: Graeme Forbes (In Defense of Absolute Essentialism [1986], 3) | |
| A reaction: Forbes isn't endorsing this, and it doesn't sound convincing. He quotes the thought 'I wish I had made this pot from a different piece of clay'. We might corrupt a statue by switching bronze, but I don't think the sculptor could do so. |
| 12020 | An individual might change their sex in a world, but couldn't have differed in sex at origin [Forbes,G] |
| Full Idea: In the time of a single world, the same individual can undergo a change of sex, but it is less clear that an individual of one sex could have been, from the outset, an individual of another. | |
| From: Graeme Forbes (The Metaphysics of Modality [1985], 6.5) | |
| A reaction: I don't find this support for essentiality of origin very persuasive. I struggle with these ideas. Given my sex yesterday, then presumably I couldn't have had a different sex yesterday. Given that pigs can fly, pigs can fly. What am I missing? |
| 11888 | Identities must hold because of other facts, which must be instrinsic [Forbes,G, by Mackie,P] |
| Full Idea: Forbes has two principles of identity, which we can call the No Bare Identities Principle (identities hold in virtue of other facts), and the No Extrinsic Determination Principle (that only intrinsic facts of a thing establish identity). | |
| From: report of Graeme Forbes (The Metaphysics of Modality [1985], 127-8) by Penelope Mackie - How Things Might Have Been 2.7 | |
| A reaction: The job of the philosopher is to prise apart the real identities of things from the way in which we conceive of identities. I take these principles to apply to real identities, not conceptual identities. |
| 17742 | Scotus based modality on semantic consistency, instead of on what the future could allow [Walicki] |
| Full Idea: The link between time and modality was severed by Duns Scotus, who proposed a notion of possibility based purely on the notion of semantic consistency. 'Possible' means for him logically possible, that is, not involving contradiction. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History B.4) |
| 12003 | De re modal formulae, unlike de dicto, are sensitive to transworld identities [Forbes,G] |
| Full Idea: The difference between de re and de dicto formulae is a difference between formulae which are, and formulae which are not, sensitive to the identities of objects at various worlds. | |
| From: Graeme Forbes (The Metaphysics of Modality [1985], 3.1) |
| 12028 | De re necessity is a form of conceptual necessity, just as de dicto necessity is [Forbes,G] |
| Full Idea: De re necessity does not differ from de dicto necessity in respect of how it arises: it is still a form of conceptual necessity. | |
| From: Graeme Forbes (The Metaphysics of Modality [1985], 9.4) | |
| A reaction: [Forbes proceeds to argue for this claim] Forbes defends a form of essentialism, but takes the necessity to arise from a posteriori truths because of the a priori involvement of other concepts (rather as Kripke argues). |
| 13810 | The source of de dicto necessity is not concepts, but the actual properties of the thing [Forbes,G] |
| Full Idea: It is widely held that the source of de dicto necessity is in concepts, ..but I deny this... even with simple de dicto necessities, the source of the necessity is to be found in the properties to which the predicates of the de dicto truth refer. | |
| From: Graeme Forbes (In Defense of Absolute Essentialism [1986], 3) | |
| A reaction: It is normal nowadays to say this about de re necessities, but this is more unusual. |
| 12008 | Unlike places and times, we cannot separate possible worlds from what is true at them [Forbes,G] |
| Full Idea: There is no means by which we might distinguish a possible world from what is true at it. ...Whereas our ability to separate a place, or a time, from its occupier is crucial to realism about places and times, as is a distance relation. | |
| From: Graeme Forbes (The Metaphysics of Modality [1985], 4.2) | |
| A reaction: He is objecting to Lewis's modal realism. I'm not fully convinced. It depends whether we are discussing real ontology or conceptual space. In the latter I see no difference between times and possible worlds. In ontology, a 'time' is weird. |
| 12009 | The problem with possible worlds realism is epistemological; we can't know properties of possible objects [Forbes,G] |
| Full Idea: The main objection to realism about worlds is from epistemology. Knowledge of properties of objects requires experience of these objects, which must be within the range of our sensory faculties, but only concrete actual objects achieve that. | |
| From: Graeme Forbes (The Metaphysics of Modality [1985], 4.2) | |
| A reaction: This pinpoints my dislike of the whole possible worlds framework, ontologically speaking. I seem to be an actualist. I take possibilities to be inferences to the best explanation from the powers we know of in the actual world. We experience potentiality. |
| 12007 | Possible worlds are points of logical space, rather like other times than our own [Forbes,G] |
| Full Idea: Someone impressed by the parallel between tense and modal operators ...might suggest that just as we can speak of places and times forming their own manifolds or spaces, so we can say that worlds are the points of logical space. | |
| From: Graeme Forbes (The Metaphysics of Modality [1985], 4.2) | |
| A reaction: I particularly like the notion of worlds being "points of logical space", and am inclined to remove it from this context and embrace it as the correct way to understand possible worlds. We must understand logical or conceptual space. |
| 12011 | Transworld identity concerns the limits of possibility for ordinary things [Forbes,G] |
| Full Idea: An elucidation of transworld identity can be regarded as an elucidation of the boundaries of possibility for ordinary things. | |
| From: Graeme Forbes (The Metaphysics of Modality [1985], 5.1) | |
| A reaction: I presume that if we don't search for some such criterion, we just have to face the possibility that Aristotle could have been a poached egg in some possible world. To know the bounds of possibility, study the powers of actual objects. |
| 12016 | The problem of transworld identity can be solved by individual essences [Forbes,G] |
| Full Idea: The motivation for investigating individual essences should be obvious, since if every object has such an essence, the problem of elucidating transworld identity can be solved. | |
| From: Graeme Forbes (The Metaphysics of Modality [1985], 5.1) | |
| A reaction: It is important that, if necessary, the identities be 'individual', and not just generic, by sortal, or natural kind. We want to reason about (and explain) truths at the fine-grained level of the individual, not just at the broad level of generalisation. |
| 12004 | Counterpart theory is not good at handling the logic of identity [Forbes,G] |
| Full Idea: The outstanding technical objection to counterpart-theoretic semantics concerns its handling of the logic of identity. In quantified S5 (the orthodox semantics) a = b → □(a = b) is valid, but 'a' must not attach to two objects. | |
| From: Graeme Forbes (The Metaphysics of Modality [1985], 3.5) |
| 12021 | Haecceitism attributes to each individual a primitive identity or thisness [Forbes,G] |
| Full Idea: Haecceitism attributes to each individual a primitive identity or thisness, as opposed to the sort of essentialism that gives non-trivial conditions sufficient for transworld identity. | |
| From: Graeme Forbes (The Metaphysics of Modality [1985], 6.6) | |
| A reaction: 'Haecceitism' is the doctrine that things have primitive identity. A 'haecceity' is a postulated property which actually does the job. The key point of the view is that whatever it is is 'primitive', and not complex, or analysable. I don't believe it. |
| 12029 | We believe in thisnesses, because we reject bizarre possibilities as not being about that individual [Forbes,G] |
| Full Idea: The natural response to an unreasonable hypothesis of possibility for an object x, that in such a state of affairs it would not be x which satisfies the conditions, is evidence that we do possess concepts of thisness for individuals. | |
| From: Graeme Forbes (The Metaphysics of Modality [1985], 9.4) | |
| A reaction: We may have a 'concept' of thisness, but we needn't be committed to the 'existence' of a thisness. There is a fairly universal intuition that cessation of existence of an entity when it starts to change can be a very vague matter. |
| 19941 | Thou shalt love thy neighbour as thyself [Anon (Leviticus)] |
| Full Idea: Thou shalt love thy neighbour as thyself. | |
| From: Anon (Lev) (03: Book of Leviticus [c.700 BCE], 19.18) | |
| A reaction: Most Christians think Jesus originated this thought. Interestingly, this precedes Socrates, who taught a similar idea. |