Ideas from 'Logical Consequence' by Vann McGee [2014], by Theme Structure

[found in 'Bloomsbury Companion to Philosophical Logic' (ed/tr Horsten,L/Pettigrew,R) [Bloomsbury 2014,978-1-4725-2303-0]].

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5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
Validity is explained as truth in all models, because that relies on the logical terms
                        Full Idea: A model of a language assigns values to non-logical terms. If a sentence is true in every model, its truth doesn't depend on those non-logical terms. Hence the validity of an argument comes from its logical form. Thus models explain logical validity.
                        From: Vann McGee (Logical Consequence [2014], 4)
                        A reaction: [compressed] Thus you get a rigorous account of logical validity by only allowing the rigorous input of model theory. This is the modern strategy of analytic philosophy. But is 'it's red so it's coloured' logically valid?
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
Natural language includes connectives like 'because' which are not truth-functional
                        Full Idea: Natural language includes connectives that are not truth-functional. In order for 'p because q' to be true, both p and q have to be true, but knowing the simpler sentences are true doesn't determine whether the larger sentence is true.
                        From: Vann McGee (Logical Consequence [2014], 2)
5. Theory of Logic / G. Quantification / 5. Second-Order Quantification
Second-order variables need to range over more than collections of first-order objects
                        Full Idea: To get any advantage from moving to second-order logic, we need to assign to second-order variables a role different from merely ranging over collections made up of things the first-order variables range over.
                        From: Vann McGee (Logical Consequence [2014], 7)
                        A reaction: Thus it is exciting if they range over genuine properties, but not so exciting if you merely characterise those properties as sets of first-order objects. This idea leads into a discussion of plural quantification.
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
An ontologically secure semantics for predicate calculus relies on sets
                        Full Idea: We can get a less ontologically perilous presentation of the semantics of the predicate calculus by using sets instead of concepts.
                        From: Vann McGee (Logical Consequence [2014], 4)
                        A reaction: The perilous versions rely on Fregean concepts, and notably Russell's 'concept that does not fall under itself'. The sets, of course, have to be ontologically secure, and so will involve the iterative conception, rather than naive set theory.
5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
Logically valid sentences are analytic truths which are just true because of their logical words
                        Full Idea: Logically valid sentences are a species of analytic sentence, being true not just in virtue of the meanings of their words, but true in virtue of the meanings of their logical words.
                        From: Vann McGee (Logical Consequence [2014], 4)
                        A reaction: A helpful link between logical truths and analytic truths, which had not struck me before.
5. Theory of Logic / K. Features of Logics / 3. Soundness
Soundness theorems are uninformative, because they rely on soundness in their proofs
                        Full Idea: Soundness theorems are seldom very informative, since typically we use informally, in proving the theorem, the very same rules whose soundness we are attempting to establish.
                        From: Vann McGee (Logical Consequence [2014], 5)
                        A reaction: [He cites Quine 1935]
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
The culmination of Euclidean geometry was axioms that made all models isomorphic
                        Full Idea: One of the culminating achievements of Euclidean geometry was categorical axiomatisations, that describe the geometric structure so completely that any two models of the axioms are isomorphic. The axioms are second-order.
                        From: Vann McGee (Logical Consequence [2014], 7)
                        A reaction: [He cites Veblen 1904 and Hilbert 1903] For most mathematicians, categorical axiomatisation is the best you can ever dream of (rather than a single true axiomatisation).
19. Language / F. Communication / 2. Assertion
A maxim claims that if we are allowed to assert a sentence, that means it must be true
                        Full Idea: If our linguistic conventions entitle us to assert a sentence, they thereby make it true, because of the maxim that 'truth is the norm of assertion'.
                        From: Vann McGee (Logical Consequence [2014], 8)
                        A reaction: You could only really deny that maxim if you had no belief at all in truth, but then you can assert anything you like (with full entitlement). Maybe you can assert anything you like as long as it doesn't upset anyone? Etc.