16 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'. |
| 8568 | A property is merely a constituent of laws of nature; temperature is just part of thermodynamics [Mellor] |
| Full Idea: Being a constituent of probabilistic laws of nature is all there is to being a property. There is no more to temperature than the thermodynamics and other laws they occur in. | |
| From: D.H. Mellor (Properties and Predicates [1991], 'Props') | |
| A reaction: How could thermodynamics be worked out without a prior concept of temperature? I think it is at least plausible to deny that there are any 'laws' of nature. But even Quine can't deny that some things are too hot to touch. |
| 8564 | There is obviously a possible predicate for every property [Mellor] |
| Full Idea: To every property there obviously corresponds a possible predicate applying to all and only those particulars with that property. | |
| From: D.H. Mellor (Properties and Predicates [1991], 'Intro') | |
| A reaction: This doesn't strike me as at all obvious. If nature dictates the properties, there may be vastly more than any human language could cope with. It is daft to say that a property can only exist if humanity can come up with a predicate for it. |
| 8566 | We need universals for causation and laws of nature; the latter give them their identity [Mellor] |
| Full Idea: I take the main reason for believing in contingent universals to be the roles they play in causation and in laws of nature, and those laws are what I take to give those universals their identity. | |
| From: D.H. Mellor (Properties and Predicates [1991], 'Props') | |
| A reaction: He agrees with Armstrong. Sounds a bit circular - laws are built on universals, and universals are identified by laws. It resembles a functionalist account of mental events. I think it is wrong. A different account of laws will be needed... |
| 8565 | If properties were just the meanings of predicates, they couldn't give predicates their meaning [Mellor] |
| Full Idea: One reason for denying that properties just are the meanings of our predicates is that, if they were, they could not give our predicates their meanings. | |
| From: D.H. Mellor (Properties and Predicates [1991], 'Props') | |
| A reaction: Neither way round sounds quite right to me. Predicate nominalism is wrong, but what is meant by a property 'giving' a predicate its meaning? It doesn't seem to allow room for error in our attempts to name the properties. |
| 15282 | Facts should be deducible from the theory and initial conditions, and prefer the simpler theory [Osiander, by Harré/Madden] |
| Full Idea: The two positivist criteria for a scientific theory are that the facts must be deducible from the theory together with initial conditions, and if there is more than one theory the simplest must be chosen. | |
| From: report of Andreas Osiander (Preface to 'De Revolutionibus' [1543]) by Harré,R./Madden,E.H. - Causal Powers 7.I | |
| A reaction: Harré and Madden cite this as a famous early statement of positivism. It seems to combine Hempel and Lewis very concisely. Wrong, of course. It does not, though, appear to mention 'laws'. |
| 8567 | Singular causation requires causes to raise the physical probability of their effects [Mellor] |
| Full Idea: Singular causation entails physical probabilities or chances. ...Causal laws require causes to raise their effects' chances, as when fires have a greater chance of occurring when explosions do. | |
| From: D.H. Mellor (Properties and Predicates [1991], 'Props') | |
| A reaction: It seems fairly obvious that a probability can be increased without actually causing something. Just after a harmless explosion is a good moment for arsonists, especially if Mellor will be the investigating officer. |