29 ideas
10757 | Henkin semantics has a second domain of predicates and relations (in upper case) [Rossberg] |
Full Idea: Henkin semantics (for second-order logic) specifies a second domain of predicates and relations for the upper case constants and variables. | |
From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3) | |
A reaction: This second domain is restricted to predicates and relations which are actually instantiated in the model. Second-order logic is complete with this semantics. Cf. Idea 10756. |
10759 | There are at least seven possible systems of semantics for second-order logic [Rossberg] |
Full Idea: In addition to standard and Henkin semantics for second-order logic, one might also employ substitutional or game-theoretical or topological semantics, or Boolos's plural interpretation, or even a semantics inspired by Lesniewski. | |
From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3) | |
A reaction: This is helpful in seeing the full picture of what is going on in these logical systems. |
10751 | Second-order logic needs the sets, and its consequence has epistemological problems [Rossberg] |
Full Idea: Second-order logic raises doubts because of its ontological commitment to the set-theoretic hierarchy, and the allegedly problematic epistemic status of the second-order consequence relation. | |
From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §1) | |
A reaction: The 'epistemic' problem is whether you can know the truths, given that the logic is incomplete, and so they cannot all be proved. Rossberg defends second-order logic against the second problem. A third problem is that it may be mathematics. |
10753 | Logical consequence is intuitively semantic, and captured by model theory [Rossberg] |
Full Idea: Logical consequence is intuitively taken to be a semantic notion, ...and it is therefore the formal semantics, i.e. the model theory, that captures logical consequence. | |
From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §2) | |
A reaction: If you come at the issue from normal speech, this seems right, but if you start thinking about the necessity of logical consequence, that formal rules and proof-theory seem to be the foundation. |
10752 | Γ |- S says S can be deduced from Γ; Γ |= S says a good model for Γ makes S true [Rossberg] |
Full Idea: Deductive consequence, written Γ|-S, is loosely read as 'the sentence S can be deduced from the sentences Γ', and semantic consequence Γ|=S says 'all models that make Γ true make S true as well'. | |
From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §2) | |
A reaction: We might read |= as 'true in the same model as'. What is the relation, though, between the LHS and the RHS? They seem to be mutually related to some model, but not directly to one another. |
10754 | In proof-theory, logical form is shown by the logical constants [Rossberg] |
Full Idea: A proof-theorist could insist that the logical form of a sentence is exhibited by the logical constants that it contains. | |
From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §2) | |
A reaction: You have to first get to the formal logical constants, rather than the natural language ones. E.g. what is the truth table for 'but'? There is also the matter of the quantifiers and the domain, and distinguishing real objects and predicates from bogus. |
10756 | A model is a domain, and an interpretation assigning objects, predicates, relations etc. [Rossberg] |
Full Idea: A standard model is a set of objects called the 'domain', and an interpretation function, assigning objects in the domain to names, subsets to predicate letters, subsets of the Cartesian product of the domain with itself to binary relation symbols etc. | |
From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3) | |
A reaction: The model actually specifies which objects have which predicates, and which objects are in which relations. Tarski's account of truth in terms of 'satisfaction' seems to be just a description of those pre-decided facts. |
10758 | If models of a mathematical theory are all isomorphic, it is 'categorical', with essentially one model [Rossberg] |
Full Idea: A mathematical theory is 'categorical' if, and only if, all of its models are isomorphic. Such a theory then essentially has just one model, the standard one. | |
From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3) | |
A reaction: So the term 'categorical' is gradually replacing the much-used phrase 'up to isomorphism'. |
10761 | Completeness can always be achieved by cunning model-design [Rossberg] |
Full Idea: All that should be required to get a semantics relative to which a given deductive system is complete is a sufficiently cunning model-theorist. | |
From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §5) |
10755 | A deductive system is only incomplete with respect to a formal semantics [Rossberg] |
Full Idea: No deductive system is semantically incomplete in and of itself; rather a deductive system is incomplete with respect to a specified formal semantics. | |
From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3) | |
A reaction: This important point indicates that a system might be complete with one semantics and incomplete with another. E.g. second-order logic can be made complete by employing a 'Henkin semantics'. |
17518 | Counting 'coin in this box' may have coin as the unit, with 'in this box' merely as the scope [Ayers] |
Full Idea: If we count the concept 'coin in this box', we could regard coin as the 'unit', while taking 'in this box' to limit the scope. Counting coins in two boxes would be not a difference in unit (kind of object), but in scope. | |
From: M.R. Ayers (Individuals without Sortals [1974], 'Counting') | |
A reaction: This is a very nice alternative to the Fregean view of counting, depending totally on the concept, and rests more on a natural concept of object. I prefer Ayers. Compare 'count coins till I tell you to stop'. |
17516 | If counting needs a sortal, what of things which fall under two sortals? [Ayers] |
Full Idea: If we accepted that counting objects always presupposes some sortal, it is surely clear that the class of objects to be counted could be designated by two sortals rather than one. | |
From: M.R. Ayers (Individuals without Sortals [1974], 'Realist' vii) | |
A reaction: His nice example is an object which is both 'a single piece of wool' and a 'sweater', which had better not be counted twice. Wiggins struggles to argue that there is always one 'substance sortal' which predominates. |
17520 | Events do not have natural boundaries, and we have to set them [Ayers] |
Full Idea: In order to know which event has been ostensively identified by a speaker, the auditor must know the limits intended by the speaker. ...Events do not have natural boundaries. | |
From: M.R. Ayers (Individuals without Sortals [1974], 'Concl') | |
A reaction: He distinguishes events thus from natural objects, where the world, to a large extent, offers us the boundaries. Nice point. |
16730 | If matter is entirely atoms, anything else we notice in it can only be modes [Gassendi] |
Full Idea: Since these atoms are the whole of the corporeal matter or substance that exists in bodies, if we conceive or notice anything else to exist in these bodies, that is not a substance but only some kind of mode of the substance. | |
From: Pierre Gassendi (Syntagma [1658], II.1.6.1), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 22.4 | |
A reaction: If the atoms have a few qualities of their own, are they just modes? If they are genuine powers, then there can be emergent powers, which are rather more than mere 'modes'. |
17519 | To express borderline cases of objects, you need the concept of an 'object' [Ayers] |
Full Idea: The only explanation of the power to produce borderline examples like 'Is this hazelnut one object or two?' is the possession of the concept of an object. | |
From: M.R. Ayers (Individuals without Sortals [1974], 'Counting') |
17511 | Recognising continuity is separate from sortals, and must precede their use [Ayers] |
Full Idea: The recognition of the fact of continuity is logically independent of the possession of sortal concepts, whereas the formation of sortal concepts is at least psychologically dependent upon the recognition of continuity. | |
From: M.R. Ayers (Individuals without Sortals [1974], Intro) | |
A reaction: I take this to be entirely correct. I might add that unity must also be recognised. |
17510 | Speakers need the very general category of a thing, if they are to think about it [Ayers] |
Full Idea: If a speaker indicates something, then in order for others to catch his reference they must know, at some level of generality, what kind of thing is indicated. They must categorise it as event, object, or quality. Thinking about something needs that much. | |
From: M.R. Ayers (Individuals without Sortals [1974], Intro) | |
A reaction: Ayers defends the view that such general categories are required, but not the much narrower sortal terms defended by Geach and Wiggins. I'm with Ayers all the way. 'What the hell is that?' |
17522 | We use sortals to classify physical objects by the nature and origin of their unity [Ayers] |
Full Idea: Sortals are the terms by which we intend to classify physical objects according to the nature and origin of their unity. | |
From: M.R. Ayers (Individuals without Sortals [1974], 'Concl') | |
A reaction: This is as opposed to using sortals for the initial individuation. I take the perception of the unity to come first, so resemblance must be mentioned, though it can be an underlying (essentialist) resemblance. |
17515 | Seeing caterpillar and moth as the same needs continuity, not identity of sortal concepts [Ayers] |
Full Idea: It is unnecessary to call moths 'caterpillars' or caterpillars 'moths' to see that they can be the same individual. It may be that our sortal concepts reflect our beliefs about continuity, but our beliefs about continuity need not reflect our sortals. | |
From: M.R. Ayers (Individuals without Sortals [1974], 'Realist' vi) | |
A reaction: Something that metamorphosed through 15 different stages could hardly required 15 different sortals before we recognised the fact. Ayers is right. |
17517 | Could the same matter have more than one form or principle of unity? [Ayers] |
Full Idea: The abstract question arises of whether the same matter could be subject to more than one principle of unity simultaneously, or unified by more than one 'form'. | |
From: M.R. Ayers (Individuals without Sortals [1974], 'Realist' vii) | |
A reaction: He suggests that the unity of the sweater is destroyed by unravelling, and the unity of the thread by cutting. |
17513 | If there are two objects, then 'that marble, man-shaped object' is ambiguous [Ayers] |
Full Idea: The statue is marble and man-shaped, but so is the piece of marble. So not only are the two objects in the same place, but two marble and man-shaped objects in the same place, so 'that marble, man-shaped object' must be ambiguous or indefinite. | |
From: M.R. Ayers (Individuals without Sortals [1974], 'Prob') | |
A reaction: It strikes me as basic that it can't be a piece of marble if you subtract its shape, and it can't be a statue if you subtract its matter. To treat a statue as an object, separately from its matter, is absurd. |
17523 | Sortals basically apply to individuals [Ayers] |
Full Idea: Sortals, in their primitive use, apply to the individual. | |
From: M.R. Ayers (Individuals without Sortals [1974], 'Concl') | |
A reaction: If the sortal applies to the individual, any essence must pertain to that individual, and not to the class it has been placed in. |
17521 | You can't have the concept of a 'stage' if you lack the concept of an object [Ayers] |
Full Idea: It would be impossible for anyone to have the concept of a stage who did not already possess the concept of a physical object. | |
From: M.R. Ayers (Individuals without Sortals [1974], 'Concl') |
17514 | Temporal 'parts' cannot be separated or rearranged [Ayers] |
Full Idea: Temporally extended 'parts' are still mysteriously inseparable and not subject to rearrangement: a thing cannot be cut temporally in half. | |
From: M.R. Ayers (Individuals without Sortals [1974], 'Prob') | |
A reaction: A nice warning to anyone accepting a glib analogy between spatial parts and temporal parts. |
17509 | Some say a 'covering concept' completes identity; others place the concept in the reference [Ayers] |
Full Idea: Some hold that the 'covering concept' completes the incomplete concept of identity, determining the kind of sameness involved. Others strongly deny the identity itself is incomplete, and locate the covering concept within the necessary act of reference. | |
From: M.R. Ayers (Individuals without Sortals [1974], Intro) | |
A reaction: [a bit compressed; Geach is the first view, and Quine the second; Wiggins is somewhere between the two] |
17512 | If diachronic identities need covering concepts, why not synchronic identities too? [Ayers] |
Full Idea: Why are covering concepts required for diachronic identities, when they must be supposed unnecessary for synchronic identities? | |
From: M.R. Ayers (Individuals without Sortals [1974], 'Prob') |
16619 | We observe qualities, and use 'induction' to refer to the substances lying under them [Gassendi] |
Full Idea: Nothing beyond qualities is perceived by the senses. …When we refer to the substance in which the qualities inhere, we do this through induction, by which we reason that some subject lies under the quality. | |
From: Pierre Gassendi (Syntagma [1658], II.1.6.1), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 07.1 | |
A reaction: He talks of 'induction' (in an older usage), but he seems to mean abduction, since he never makes any observations of the substances being proposed. |
16593 | Atoms are not points, but hard indivisible things, which no force in nature can divide [Gassendi] |
Full Idea: The vulgar think atoms lack parts and are free of all magnitude, and hence nothing other than a mathematical point, but it is something solid and hard and compact, as to leave no room for division, separation and cutting. No force in nature can divide it. | |
From: Pierre Gassendi (Syntagma [1658], II.1.3.5), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 03.2 | |
A reaction: If you gloatingly think the atom has now been split, ask whether electrons and quarks now fit his description. Pasnau notes that though atoms are indivisible, they are not incorruptible, and could go out of existence, or be squashed. |
16729 | How do mere atoms produce qualities like colour, flavour and odour? [Gassendi] |
Full Idea: If the only material principles of things are atoms, having only size, shape, and weight, or motion, then why are so many additional qualities created and existing within the things: color, heat, flavor, odor, and innumerable others? | |
From: Pierre Gassendi (Syntagma [1658], II.1.5.7), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 22.4 | |
A reaction: This is pretty much the 'hard question' about the mind-body relation. Bacon said that heat was just motion of matter. I would say that this problem is gradually being solved in my lifetime. |