24 ideas
| 10301 | The axiom of choice is controversial, but it could be replaced [Shapiro] |
| Full Idea: The axiom of choice has a troubled history, but is now standard in mathematics. It could be replaced with a principle of comprehension for functions), or one could omit the variables ranging over functions. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], n 3) |
| 13331 | Part and whole contribute asymmetrically to one another, so must differ [Fine,K] |
| Full Idea: The whole identity of a part is relevant to whether it is a part, but the identity of the whole makes a part a part. The whole part belongs to the whole as a part. The standard account in terms of time-slices fails to respect this part/whole asymmetry. | |
| From: Kit Fine (Things and Their Parts [1999], §2) | |
| A reaction: Hard to follow, but I think the asymmetry is that the wholeness of the part contributes to the wholeness of the whole, while the wholeness of the whole contributes to the parthood of the part. Wholeness does different jobs in different directions. OK? |
| 10588 | First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [Shapiro] |
| Full Idea: Early study of first-order logic revealed a number of important features. Gödel showed that there is a complete, sound and effective deductive system. It follows that it is Compact, and there are also the downward and upward Löwenheim-Skolem Theorems. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) |
| 10298 | Some say that second-order logic is mathematics, not logic [Shapiro] |
| Full Idea: Some authors argue that second-order logic (with standard semantics) is not logic at all, but is a rather obscure form of mathematics. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.4) |
| 10299 | If the aim of logic is to codify inferences, second-order logic is useless [Shapiro] |
| Full Idea: If the goal of logical study is to present a canon of inference, a calculus which codifies correct inference patterns, then second-order logic is a non-starter. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.4) | |
| A reaction: This seems to be because it is not 'complete'. However, moves like plural quantification seem aimed at capturing ordinary language inferences, so the difficulty is only that there isn't a precise 'calculus'. |
| 10300 | Logical consequence can be defined in terms of the logical terminology [Shapiro] |
| Full Idea: Informally, logical consequence is sometimes defined in terms of the meanings of a certain collection of terms, the so-called 'logical terminology'. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.4) | |
| A reaction: This seems to be a compositional account, where we build a full account from an account of the atomic bits, perhaps presented as truth-tables. |
| 10290 | Second-order variables also range over properties, sets, relations or functions [Shapiro] |
| Full Idea: Second-order variables can range over properties, sets, or relations on the items in the domain-of-discourse, or over functions from the domain itself. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) |
| 10590 | Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro] |
| Full Idea: Upward Löwenheim-Skolem: if a set of first-order formulas is satisfied by a domain of at least the natural numbers, then it is satisfied by a model of at least some infinite cardinal. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) |
| 10296 | The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro] |
| Full Idea: Both of the Löwenheim-Skolem Theorems fail for second-order languages with a standard semantics | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.3.2) |
| 10297 | The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro] |
| Full Idea: The Löwenheim-Skolem theorem is usually taken as a sort of defect (often thought to be inevitable) of the first-order logic. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.4) | |
| A reaction: [He is quoting Wang 1974 p.154] |
| 10292 | Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro] |
| Full Idea: Downward Löwenheim-Skolem: a finite or denumerable set of first-order formulas that is satisfied by a model whose domain is infinite is satisfied in a model whose domain is the natural numbers | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) |
| 10294 | Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro] |
| Full Idea: Full second-order logic has all the expressive power needed to do mathematics, but has an unworkable model theory. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) | |
| A reaction: [he credits Cowles for this remark] Having an unworkable model theory sounds pretty serious to me, as I'm not inclined to be interested in languages which don't produce models of some sort. Surely models are the whole point? |
| 10591 | Logicians use 'property' and 'set' interchangeably, with little hanging on it [Shapiro] |
| Full Idea: In studying second-order logic one can think of relations and functions as extensional or intensional, or one can leave it open. Little turns on this here, and so words like 'property', 'class', and 'set' are used interchangeably. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.2.1) | |
| A reaction: Important. Students of the metaphysics of properties, who arrive with limited experience of logic, are bewildered by this attitude. Note that the metaphysics is left wide open, so never let logicians hijack the metaphysical problem of properties. |
| 13332 | Hierarchical set membership models objects better than the subset or aggregate relations do [Fine,K] |
| Full Idea: It is the hierarchical conception of sets and their members, rather than the linear conception of set and subset or of aggregate and component, that provides us with the better model for the structure of part-whole in its application to material things. | |
| From: Kit Fine (Things and Their Parts [1999], §5) | |
| A reaction: His idea is to give some sort of internal structure. He says of {a,b,c,d} that we can create subsets {a,b} and {c,d} from that. But {{a,b},{c,d}} has given member sets, and he is looking for 'natural' divisions between the members. |
| 13333 | The matter is a relatively unstructured version of the object, like a set without membership structure [Fine,K] |
| Full Idea: The wood is, as it were, a relatively unstructured version of the tree, just as the set {a,b,c,d} is an unstructured counterpart of the set {{a,b},{c,d}}. | |
| From: Kit Fine (Things and Their Parts [1999], §5) | |
| A reaction: He is trying to give a modern logicians' account of the Aristotelian concept of 'form' (as applied to matter). It is part of the modern project that objects must be connected to the formalism of mereology or set theory. If it works, are we thereby wiser? |
| 13326 | A 'temporary' part is a part at one time, but may not be at another, like a carburetor [Fine,K] |
| Full Idea: First, a thing can be a part in a way that is relative to a time, for example, that a newly installed carburettor is now part of my car, whereas earlier it was not. (This will be called a 'temporary' part). | |
| From: Kit Fine (Things and Their Parts [1999], Intro) | |
| A reaction: [Cf Idea 13327 for the 'second' concept of part] I'm immediately uneasy. Being a part seems to be a univocal concept. He seems to be distinguishing parts which are necessary for identity from those which aren't. Fine likes to define by example. |
| 13327 | A 'timeless' part just is a part, not a part at some time; some atoms are timeless parts of a water molecule [Fine,K] |
| Full Idea: Second, an object can be a part of another in a way that is not relative to time ('timeless'). It is not appropriate to ask when it is a part. Thus pants and jacket are parts of the suit, atoms of a water molecule, and two pints part of a quart of milk. | |
| From: Kit Fine (Things and Their Parts [1999], Intro) | |
| A reaction: [cf Idea 13326 for the other concept of 'part'] Again I am uneasy that 'part' could have two meanings. A Life Member is a member in the same way that a normal paid up member is a member. |
| 13329 | An 'aggregative' sum is spread in time, and exists whenever a component exists [Fine,K] |
| Full Idea: In the 'aggregative' understanding of a sum, it is spread out in time, so that exists whenever any of its components exists (just as it is located at any time wherever any of its components are located). | |
| From: Kit Fine (Things and Their Parts [1999], §1) | |
| A reaction: This works particularly well for something like an ancient forest, which steadily changes its trees. On that view, though, the ship which has had all of its planks replaced will be the identical single sum of planks all the way through. Fine agrees. |
| 13330 | An 'compound' sum is not spread in time, and only exists when all the components exists [Fine,K] |
| Full Idea: In the 'compound' notion of sum, the mereological sum is spread out only in space, not also in time. For it to exist at a time, all of its components must exist at the time. | |
| From: Kit Fine (Things and Their Parts [1999], §1) | |
| A reaction: It is hard to think of anything to which this applies, apart from for a classical mereologist. Named parts perhaps, like Tom, Dick and Harry. Most things preserve sum identity despite replacement of parts by identical components. |
| 13328 | Two sorts of whole have 'rigid embodiment' (timeless parts) or 'variable embodiment' (temporary parts) [Fine,K] |
| Full Idea: I develop a version of hylomorphism, in which the theory of 'rigid embodiment' provides an account of the timeless relation of part, and the theory of 'variable embodiment' is an account of the temporary relation. We must accept two new kinds of whole. | |
| From: Kit Fine (Things and Their Parts [1999], Intro) | |
| A reaction: [see Idea 13326 and Idea 13327 for the two concepts of 'part'] This is easier to take than the two meanings for 'part'. Since Aristotle, everyone has worried about true wholes (atoms, persons?) and looser wholes (houses). |
| 19566 | Epistemology does not just concern knowledge; all aspects of cognitive activity are involved [Kvanvig] |
| Full Idea: Epistemology is not just knowledge. There is enquiring, reasoning, changes of view, beliefs, assumptions, presuppositions, hypotheses, true beliefs, making sense, adequacy, understanding, wisdom, responsible enquiry, and so on. | |
| From: Jonathan Kvanvig (Truth is not the Primary Epistemic Goal [2005], 'What') | |
| A reaction: [abridged] Stop! I give in. His topic is whether truth is central to epistemology. Rivals seem to be knowledge-first, belief-first, and justification-first. I'm inclined to take justification as the central issue. Does it matter? |
| 19568 | Making sense of things, or finding a good theory, are non-truth-related cognitive successes [Kvanvig] |
| Full Idea: There are cognitive successes that are not obviously truth related, such as the concepts of making sense of the course of experience, and having found an empirically adequate theory. | |
| From: Jonathan Kvanvig (Truth is not the Primary Epistemic Goal [2005], 'Epistemic') | |
| A reaction: He is claiming that truth is not the main aim of epistemology. He quotes Marian David for the rival view. Personally I doubt whether the concepts of 'making sense' or 'empirical adequacy' can be explicated without mentioning truth. |
| 19567 | The 'defeasibility' approach says true justified belief is knowledge if no undermining facts could be known [Kvanvig] |
| Full Idea: The 'defeasibility' approach says that having knowledge requires, in addition to justified true belief, there being no true information which, if learned, would result in the person in question no longer being justified in believing the claim. | |
| From: Jonathan Kvanvig (Truth is not the Primary Epistemic Goal [2005], 'Epistemic') | |
| A reaction: I take this to be an externalist view, since it depends on information of which the cognizer may be unaware. A defeater may yet have an undiscovered counter-defeater. The only real defeater is the falsehood of the proposition. |
| 19570 | Reliabilism cannot assess the justification for propositions we don't believe [Kvanvig] |
| Full Idea: The most serious problem for reliabilism is that it cannot explain adequately the concept of propositional justification, the kind of justification one might have for a proposition one does not believe, or which one disbelieves. | |
| From: Jonathan Kvanvig (Truth is not the Primary Epistemic Goal [2005], Notes 2) | |
| A reaction: I don't understand this (though I pass it on anyway). Why can't the reliabilist just offer a critique of the reliability of the justification available for the dubious proposition? |