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All the ideas for 'works', 'First-order Logic, 2nd-order, Completeness' and 'Thought and Reality'

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32 ideas

3. Truth / F. Semantic Truth / 2. Semantic Truth
Truth is part of semantics, since valid inference preserves truth [Dummett]
     Full Idea: The concept of truth belongs to semantics, since after all truth is what must be preserved by a valid deductive inference.
     From: Michael Dummett (Thought and Reality [1997], 2)
     A reaction: Does this conclusion follow? Compare 'nice taste belongs to cooking, since that is what cooking must preserve'. I don't like this. I take 'truth' to be a relevant concept to a discussion of a dog's belief that it is going to be taken for a walk.
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
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.
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.
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.
5. Theory of Logic / B. Logical Consequence / 2. Types of Consequence
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.
5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-
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.
5. Theory of Logic / D. Assumptions for Logic / 1. Bivalence
Language can violate bivalence because of non-referring terms or ill-defined predicates [Dummett]
     Full Idea: Two features of natural languages cause them to violate bivalence: singular terms (or proper names) which have a sense but fail to denote an object ('the centre of the universe'); and predicates which are not well defined for every object.
     From: Michael Dummett (Thought and Reality [1997], 4)
     A reaction: If we switch from sentences to propositions these problems might be avoided. If there is no reference, or a vague predicate, then there is (maybe) just no proposition being expressed which could be evaluated for truth.
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
The law of excluded middle is the logical reflection of the principle of bivalence [Dummett]
     Full Idea: The law of excluded middle is the reflection, within logic, of the principle of bivalence. It states that 'For any statement A, the statement 'A or not-A' is true'.
     From: Michael Dummett (Thought and Reality [1997], 5)
     A reaction: True-or-not-true is an easier condition to fulfil than true-or-false. The second says that 'false' is the only alternative, but the first allows other alternatives to 'true' (such as 'undecidable'). It is hard to challenge excluded middle. Somewhat true?
5. Theory of Logic / E. Structures of Logic / 1. Logical Form
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.
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
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.
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
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'.
5. Theory of Logic / K. Features of Logics / 4. Completeness
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)
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
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'.
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
Archimedes defined a straight line as the shortest distance between two points [Archimedes, by Leibniz]
     Full Idea: Archimedes gave a sort of definition of 'straight line' when he said it is the shortest line between two points.
     From: report of Archimedes (fragments/reports [c.240 BCE]) by Gottfried Leibniz - New Essays on Human Understanding 4.13
     A reaction: Commentators observe that this reduces the purity of the original Euclidean axioms, because it involves distance and measurement, which are absent from the purest geometry.
7. Existence / D. Theories of Reality / 2. Realism
Philosophers should not presume reality, but only invoke it when language requires it [Dummett]
     Full Idea: The philosopher's task is not to make a prior commitment for or against realism, but to discover how far realist considerations must be invoked in order to describe our understanding of our language: they may be invoked only if they must be invoked.
     From: Michael Dummett (Thought and Reality [1997], 6)
     A reaction: I don't see why the default position should be solipsism, or a commitment to Ockham's Razor. This is the Cartesian 'Enlightenment Project' approach to philosophy - that everything has to be proved. There is more to ontology than language.
7. Existence / D. Theories of Reality / 4. Anti-realism
We can't make sense of a world not apprehended by a mind [Dummett]
     Full Idea: We can make no clear sense of there being a world that is not apprehended by any mind.
     From: Michael Dummett (Thought and Reality [1997], 8)
     A reaction: I find Dummett's view quite baffling. It is no coincidence that Dummett is a theist, along (it seems) Berkeleian lines. I see no more problem with imagining such worlds than with imagining ships sunken long ago which will never be found.
7. Existence / D. Theories of Reality / 8. Facts / b. Types of fact
Since 'no bird here' and 'no squirrel here' seem the same, we must talk of 'atomic' facts [Dummett]
     Full Idea: What complex of objects constitutes the fact that there is no bird on the bough, and how is that distinct from no squirrel on the bough? This drives us to see the world as composed of 'atomic' facts, making complexes into compounds, not reality itself.
     From: Michael Dummett (Thought and Reality [1997], 1)
     A reaction: [He cites early Wittgenstein as an example] But 'no patch of red here' (or sense-datum) seems identical to 'no patch of green here'. I suppose you could catalogue all the atomic facts, and note that red wasn't among them. But you could do that for birds.
7. Existence / D. Theories of Reality / 8. Facts / c. Facts and truths
We know we can state facts, with true statements [Dummett]
     Full Idea: One thing we know about facts, namely that we can state them. Whenever we make some true statement, we state some fact.
     From: Michael Dummett (Thought and Reality [1997], 1)
     A reaction: Then facts become boring, and are subsumed within the problem of what 'true' means. Personally I have a concept of facts which includes unstatable facts. The physical basis of melancholy I take to be a complex fact which is beyond our powers.
7. Existence / D. Theories of Reality / 10. Vagueness / d. Vagueness as linguistic
'That is red or orange' might be considered true, even though 'that is red' and 'that is orange' were not [Dummett]
     Full Idea: A statement of the form 'that is red or orange', said of something on the borderline between the two colours, might rank as true, although neither 'that is red' nor 'that is orange' was true.
     From: Michael Dummett (Thought and Reality [1997], 5)
     A reaction: It seems to me that the problem here would be epistemological rather than ontological. One of the two is clearly true, but sometimes we can't decide which. How can anyone say 'It isn't red and it isn't orange, but it is either red or orange'?
12. Knowledge Sources / D. Empiricism / 5. Empiricism Critique
Empirical and a priori knowledge are not distinct, but are extremes of a sliding scale [Dummett]
     Full Idea: Our sentences cannot be divided into two classes, empirical and a priori, the truth of one to be decided by observation, the other by ratiocination. They lie on a scale, with observational sentences at one end, and mathematical ones at the other.
     From: Michael Dummett (Thought and Reality [1997], 5)
     A reaction: The modern post-Kantian dissolution of the rationalist-empiricist debate. I would say that mathematical sentences require no empirical evidence (for their operation, rather than foundation), but a bit of reasoning is involved in observation.
18. Thought / A. Modes of Thought / 1. Thought
A theory of thought will include propositional attitudes as well as propositions [Dummett]
     Full Idea: A comprehensive theory of thought will include such things as judgement and belief, as well as the mere grasp of propositions.
     From: Michael Dummett (Thought and Reality [1997], 4)
     A reaction: This seems to make any theory of thought a neat two-stage operation. Beware of neatness. While propositions might be explained using concepts, syntax and truth, the second stage looks faintly daunting. See Idea 2209, for example.
The theories of meaning and understanding are the only routes to an account of thought [Dummett]
     Full Idea: For the linguistic philosopher, the theory of meaning, and the theory of understanding that is built upon it, form the only route to a philosophical account of thought.
     From: Michael Dummett (Thought and Reality [1997], 4)
     A reaction: I am of the party that thinks thought is prior to language (esp. because of animals), but Dummett's idea does not deny this. He may well be right that this is the 'only route'. We can only hope to give an account of human thought.
18. Thought / E. Abstraction / 8. Abstractionism Critique
To 'abstract from' is a logical process, as opposed to the old mental view [Dummett]
     Full Idea: The phrase 'abstracted from' does not refer to the mental process of abstraction by disregarding features of concrete objects, in which many nineteenth century thinkers believed; it is a logical (not mental) process of concept-formation.
     From: Michael Dummett (Thought and Reality [1997], 1)
     A reaction: I take Frege's attack on 'psychologism' to be what dismissed the old view (Idea 5816). Could one not achieve the same story by negating properties in quantified logical expressions, instead of in the mind?
19. Language / A. Nature of Meaning / 4. Meaning as Truth-Conditions
To know the truth-conditions of a sentence, you must already know the meaning [Dummett]
     Full Idea: You can know the condition for a sentence to be true only when you know what the sentence means.
     From: Michael Dummett (Thought and Reality [1997], 3)
     A reaction: This makes the truth-conditions theory of meaning circular, and is Dummett's big objection to Davidson's view. The composition of a sentence creates a model of a world. Truth-conditions may only presuppose knowledge of concepts.
19. Language / A. Nature of Meaning / 5. Meaning as Verification
A justificationist theory of meaning leads to the rejection of classical logic [Dummett]
     Full Idea: If we adopt a justificationist theory of meaning, we must reject the universal law of excluded middle, and with it classical logic (which rests on the two-valued semantics of bivalence). We admit only intuitionist logic, which preserves justifiability.
     From: Michael Dummett (Thought and Reality [1997], 5)
     A reaction: This is Dummett's philosophy in a very neat nutshell. He seems to have started by accepting Brouwer's intuitionism, and then working back to language. It all implies anti-realism. I don't buy it.
Verificationism could be realist, if we imagined the verification by a superhuman power [Dummett]
     Full Idea: There is a possible route to realism, which has been called 'ideal verificationism', if we base our grasping the understanding and truth of a range of sentences on the procedure that would be available to an imagined being with superhuman powers.
     From: Michael Dummett (Thought and Reality [1997], 5)
     A reaction: This is actually a slippery slope for verificationists, as soon as they allow that verification could be done by other people. A verifier might turn up who had telepathy, or x-ray vision, or could see quarks...
If truths about the past depend on memories and current evidence, the past will change [Dummett]
     Full Idea: If justificationists succumb to the temptation for statements in the past, we shall view their senses as given by present memories and present traces of past events; but this will force us into a view of the past as itself changing.
     From: Michael Dummett (Thought and Reality [1997], 6)
     A reaction: Obviously Dummett attempts to sidestep this problem, but it strikes me as powerful support for the realist view about the past. How can we not be committed to the view that there are facts about the past quite unconnected to our verifying abilities?
19. Language / A. Nature of Meaning / 6. Meaning as Use
We could only guess the meanings of 'true' and 'false' when sentences were used [Dummett]
     Full Idea: Even if we guessed that the two words denoted the two truth-values, we should not know which stood for the value 'true' and which for the value 'false' until we knew how the sentences were in practice used.
     From: Michael Dummett (Thought and Reality [1997], 4)
     A reaction: These types of problem are always based on the idea that some one item must have logical priority in the process, but there is a lot of room for benign circularity in the development of mental and linguistic functions.
19. Language / A. Nature of Meaning / 7. Meaning Holism / a. Sentence meaning
Sentences are the primary semantic units, because they can say something [Dummett]
     Full Idea: While words are semantic atoms, sentences remain the primary semantic units, in the sense of the smallest bits of language by means of which it is possible to say anything.
     From: Michael Dummett (Thought and Reality [1997], 3)
     A reaction: Syncategorematic terms (look it up!) may need sentences, but most nouns and verbs can communicate quite a lot on their own. Whether words or sentences come first may not be a true/false issue.
19. Language / D. Propositions / 1. Propositions
We can't distinguish a proposition from its content [Dummett]
     Full Idea: No distinction can be drawn between a proposition and its content; no two distinct propositions can have the same content.
     From: Michael Dummett (Thought and Reality [1997], 3)
     A reaction: And one proposition cannot have two possible contents (ambiguity). Are we to say that a proposition supervenes on its content, or that proposition and content are identical? Ockham favours the latter.
27. Natural Reality / D. Time / 1. Nature of Time / d. Time as measure
Time is the measure of change, so we can't speak of time before all change [Dummett]
     Full Idea: Time is the measure of change, and it makes no sense to speak of how things were before there was anything that changed.
     From: Michael Dummett (Thought and Reality [1997], 8)
     A reaction: Something creating its own measure sounds like me marking my own exam papers. If an object appears, then inverts five seconds later, how can the inversion create the five seconds? How does that differ from inverting ten seconds later?
27. Natural Reality / D. Time / 1. Nature of Time / h. Presentism
If Presentism is correct, we cannot even say that the present changes [Dummett]
     Full Idea: If Presentism is correct - the doctrine that there is nothing at all, save what holds good at the present moment - then we cannot even say that the present changes, because that requires that things are not now as they were some time ago.
     From: Michael Dummett (Thought and Reality [1997], 2)
     A reaction: Presumably we can compare our present memory with our present experience. See Idea 6668. The logic (very ancient!) is that the present has not duration at all, and so no experiences can occur during it.