25 ideas
| 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. |
| 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. |
| 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'. |
| 14064 | If a statue is identical with the clay of which it is made, that identity is contingent [Gibbard] |
| Full Idea: Under certain conditions a clay statue is identical with the piece of clay of which it is made, and if this is so then the identity is contingent. | |
| From: Allan Gibbard (Contingent Identity [1975], Intro) | |
| A reaction: This initiated the modern debate about statues, and it is an attack on Kripke's claim that if two things are identical, then they are necessarily identical. Kripke seems right about Hesperus and Phosphorus, but not about the statue. |
| 14066 | A 'piece' of clay begins when its parts stick together, separately from other clay [Gibbard] |
| Full Idea: A 'piece' of clay is a portion of clay which comes into existence when all of its parts come to be stuck to each other, and cease to be stuck to any clay which is not a part of the portion. | |
| From: Allan Gibbard (Contingent Identity [1975], I) | |
| A reaction: The sort of gormlessly elementary things that philosophers find themselves having to say, but this is a good basic assertion for a discussion of statue and clay, and I can't think of an objection to it. |
| 14067 | Clay and statue are two objects, which can be named and reasoned about [Gibbard] |
| Full Idea: The piece of clay and the statue are 'objects' - that is to say, they can be designated with proper names, and the logic we ordinarily use will still apply. | |
| From: Allan Gibbard (Contingent Identity [1975], I) | |
| A reaction: An interesting indication of the way that 'object' is used in modern analytic philosophy, which may not be the way that it is used in ordinary English. The number 'seven', for example, seems to be an object by this criterion. |
| 14069 | We can only investigate the identity once we have designated it as 'statue' or as 'clay' [Gibbard] |
| Full Idea: To ask meaningfully what that thing would be, we must designate it either as a statue or as a piece of clay. What that thing would be, apart from the way it is designated, is a question without meaning. | |
| From: Allan Gibbard (Contingent Identity [1975], III) | |
| A reaction: He obviously has a powerful point, but to suggest that we can only investigate a mysterious object once we have designated it as something sounds daft. It would ruin the fun of archaeology. |
| 14076 | Essentialism is the existence of a definite answer as to whether an entity fulfils a condition [Gibbard] |
| Full Idea: Essentialism for a class of entities is that for one entity and a condition which it fulfills, the question of whether it necessarily fulfills the condition has a definite answer apart from the way the entity is specified. | |
| From: Allan Gibbard (Contingent Identity [1975], VII) | |
| A reaction: Yet another definition of essentialism, but resting, as usual in modern discussions, entirely on the notion of necessity. Kit Fine's challenge is that if you investigate the source of the necessity, it turns out to be an essence. |
| 14077 | Essentialism for concreta is false, since they can come apart under two concepts [Gibbard] |
| Full Idea: Essentialism for the class of concrete things is false, since a statue necessarily fulfils a condition as 'Goliath', but only contingently fulfils it as 'lumpl'. On the other hand, essentialism for the class of individual concepts can be true. | |
| From: Allan Gibbard (Contingent Identity [1975], VII) | |
| A reaction: This rests on his definition of essentialism in Idea 14076. He rests his essentialism about concepts on an account given by Carnap ('Meaning and Necessity' §41). The essence of a statue and the essence of a lump of clay do seem distinct. |
| 14070 | A particular statue has sortal persistence conditions, so its origin defines it [Gibbard] |
| Full Idea: A proper name like 'Goliath' denotes a thing in the actual world, and invokes a sortal with certain persistence criteria. Hence its origin makes a statue the statue that it is, and if statues in different worlds have the same beginning, they are the same. | |
| From: Allan Gibbard (Contingent Identity [1975], III) | |
| A reaction: Too neat. There are vague, ambiguous and duplicated origins. Persistence criteria can shift during the existence of a thing (like a club which changes its own constitution). In replicated statues, what is the status of the mould? |
| 14073 | Claims on contingent identity seem to violate Leibniz's Law [Gibbard] |
| Full Idea: The most prominent objection to contingent identity (as in the case of the statue and its clay) is that it violates Leibniz's Law. | |
| From: Allan Gibbard (Contingent Identity [1975], V) | |
| A reaction: Depends what you mean by a property. The trickiest one would be that the statue has (right now) a disposition to be worth a lot, but the clay doesn't. But I don't think that is really a property of the statue. Properties are a muddle. |
| 14065 | Two identical things must share properties - including creation and destruction times [Gibbard] |
| Full Idea: For two things to be strictly identical, they must have all properties in common. That means, among other things, that they must start to exist at the same time and cease to exist at the same time. | |
| From: Allan Gibbard (Contingent Identity [1975], I) | |
| A reaction: I don't accept that coming into existence at time t is a 'property' of a thing. Coincident objects give you the notion of 'existing as' something, which complicates the whole story. |
| 14074 | Leibniz's Law isn't just about substitutivity, because it must involve properties and relations [Gibbard] |
| Full Idea: As a general law of substitutivity of identicals, Leibniz's Law is false. It is a law about properties and relations, that if two things are identical, they have the same properties and relations. It only works in contexts which attribute those. | |
| From: Allan Gibbard (Contingent Identity [1975], V) | |
| A reaction: I'm not convinced about relations, which are not intrinsic properties. Under different descriptions, the relations to human minds might differ. |
| 14072 | Possible worlds identity needs a sortal [Gibbard] |
| Full Idea: Identity across possible worlds makes sense only with respect to a sortal | |
| From: Allan Gibbard (Contingent Identity [1975], IV) | |
| A reaction: See Gibbard's other ideas from this paper. I fear that the sortal invoked is too uncertain and slippery to do any useful job, and I can't see any principled difficulty with naming something before you can think of a sortal for it. |
| 14078 | Only concepts, not individuals, can be the same across possible worlds [Gibbard] |
| Full Idea: It is meaningless to talk of the same concrete thing in different possible worlds, ...but it makes sense to speak of the same individual concept, which is just a function which assigns to each possible world in a set an individual in that world. | |
| From: Allan Gibbard (Contingent Identity [1975], VII) | |
| A reaction: A lovely bold response to the problem of transworld identity, but one which needs investigation. It sounds very promising to me. 'Aristotle' is a cocept, not a name. There is no separate category of 'names'. Wow. (Attach dispositions to concepts?). |
| 14079 | Kripke's semantics needs lots of intuitions about which properties are essential [Gibbard] |
| Full Idea: To use Kripke's semantics, one needs extensive intuitions that certain properties are essential and others accidental. | |
| From: Allan Gibbard (Contingent Identity [1975], X) | |
| A reaction: As usual, we could substitute the word 'necessary' for 'essential' without changing his meaning. If we are always referring to 'our' Hubert Humphrey is speculations about him, then nearly all of his properties will be necessary ones. |
| 14071 | Naming a thing in the actual world also invokes some persistence criteria [Gibbard] |
| Full Idea: The reference of a name in the actual world is fixed partly by invoking a set of persistence criteria which determine what thing it names. | |
| From: Allan Gibbard (Contingent Identity [1975], III) | |
| A reaction: This is offered as a modification to Kripke, to deal with the statue and clay. I fear that the 'persistence criteria' may be too vague, and too subject to possible change after the origin, to do the job required. |
| 6005 | Animals are dangerous and nourishing, and can't form contracts of justice [Hermarchus, by Sedley] |
| Full Idea: Hermarchus said that animal killing is justified by considerations of human safety and nourishment and by animals' inability to form contractual relations of justice with us. | |
| From: report of Hermarchus (fragments/reports [c.270 BCE]) by David A. Sedley - Hermarchus | |
| A reaction: Could the last argument be used to justify torturing animals? Or could we eat a human who was too brain-damaged to form contracts? |