14 ideas
| 17884 | Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner] |
| Full Idea: There are many coherent stopping points in the hierarchy of increasingly strong mathematical systems, starting with strict finitism, and moving up through predicativism to the higher reaches of set theory. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], Intro) |
| 17893 | 'Reflection principles' say the whole truth about sets can't be captured [Koellner] |
| Full Idea: Roughly speaking, 'reflection principles' assert that anything true in V [the set hierarchy] falls short of characterising V in that it is true within some earlier level. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 2.1) |
| 14181 | Validity is where either the situation or the interpretation blocks true premises and false conclusion [Etchemendy, by Read] |
| Full Idea: The Representational account of validity says an argument is valid if there is no situation where the premises are true and the conclusion false. The Interpretation account says the premises are true and conclusion false under no interpretations. | |
| From: report of John Etchemendy (The Concept of Logical Consequence [1999]) by Stephen Read - Formal and Material Consequence 'Inval' | |
| A reaction: My immediate instinct is to want logic to be about situations, rather than interpretations. Situations are more about thought, where interpretations are more about language. I think our account of logic should have some applicability to animals. |
| 14180 | Etchemendy says fix the situation and vary the interpretation, or fix interpretations with varying situations [Etchemendy, by Read] |
| Full Idea: In Etchemendy's Interpretational Semantics (perhaps better called 'Substitutional') we keep the situation fixed and vary the interpretation; in Representational Semantics ('Modal'?) we keep interpretations fixed but consider varying situations. | |
| From: report of John Etchemendy (The Concept of Logical Consequence [1999]) by Stephen Read - Formal and Material Consequence 'Inval' | |
| A reaction: [compressed] These are semantic strategies for interpreting logic, so they are two ways you might go about assessing an argument. |
| 17894 | We have no argument to show a statement is absolutely undecidable [Koellner] |
| Full Idea: There is at present no solid argument to the effect that a given statement is absolutely undecidable. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 5.3) |
| 17890 | There are at least eleven types of large cardinal, of increasing logical strength [Koellner] |
| Full Idea: Some of the standard large cardinals (in order of increasing (logical) strength) are: inaccessible, Mahlo, weakly compact, indescribable, Erdös, measurable, strong, Wodin, supercompact, huge etc. (...and ineffable). | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) | |
| A reaction: [I don't understand how cardinals can have 'logical strength', but I pass it on anyway] |
| 17887 | PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner] |
| Full Idea: To the extent that we are justified in accepting Peano Arithmetic we are justified in accepting its consistency, and so we know how to expand the axiom system so as to overcome the limitation [of Gödel's Second Theorem]. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.1) | |
| A reaction: Each expansion brings a limitation, but then you can expand again. |
| 17891 | Arithmetical undecidability is always settled at the next stage up [Koellner] |
| Full Idea: The arithmetical instances of undecidability that arise at one stage of the hierarchy are settled at the next. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) |
| 23681 | The first motion or effect cannot be produced necessarily, so the First Cause must be a free agent [Reid] |
| Full Idea: That the first motion, or the first effect, whatever it be, cannot be produced necessarily, and, consequently, that the First Cause must be a free agent, has been demonstrated clearly and unanswerably. | |
| From: Thomas Reid (Essays on Active Powers 4: Liberty of Agents [1788], 8) | |
| A reaction: He has said that the First Cause can only be conceived by us as an 'agent'. If there is an agential First Cause, then he must be right. It is this need for God to be free which makes scepticism about free will unacceptable to many. |
| 23676 | A willed action needs reasonable understanding of what is to be done [Reid] |
| Full Idea: There can be no will without such a degree of understanding, at least, as gives the conception of that which we will. | |
| From: Thomas Reid (Essays on Active Powers 4: Liberty of Agents [1788], 1) | |
| A reaction: Presumably this 'conception' includes an understanding of the probable consequences, but they are of infinite complexity. I see this as an objection to 'ultimate' free will and responsibility, because there are only ever degrees of understanding. |
| 23680 | We are morally free, because we experience it, we are accountable, and we pursue projects [Reid] |
| Full Idea: I believe in moral liberty first because we have a natural conviction of belief that in many cases we act freely, second because we are accountable, and third because we can prosecute an end by a long series of means adapted. | |
| From: Thomas Reid (Essays on Active Powers 4: Liberty of Agents [1788], 5) | |
| A reaction: This is his final summary of why he believes in free will. Why didn't Plato and Aristotle have this natural belief? He could only believe we are 'accountable' because he believes in free will. Ants and bees pursue lengthy projects. Hm. |
| 23678 | A motive is merely an idea, like advice, and not a force for action [Reid] |
| Full Idea: A motive is equally incapable of action and of passion; because it is not a thing that exists, but a thing that is conceived. …Motives may be compared to advice or exhortation. | |
| From: Thomas Reid (Essays on Active Powers 4: Liberty of Agents [1788], 4) | |
| A reaction: We say people are motivated by greed or anger or love, which seems a bit stronger than mere advice. |
| 23677 | We all know that mere priority or constant conjunction do not have to imply causation [Reid] |
| Full Idea: Every man who understands the language knows that neither priority, nor constant conjunction, nor both taken together, imply efficiency. | |
| From: Thomas Reid (Essays on Active Powers 4: Liberty of Agents [1788], 2) | |
| A reaction: This invites the question of how we do know causal events, if none of our experiences are enough to prove it. Reid says we have an innate knowledge that all events are caused, but that isn't much help. The presence of power? |
| 23679 | The principle of the law of nature is that matter is passive, and is acted upon [Reid] |
| Full Idea: The law of nature respecting matter is grounded upon this principle: That matter is an inert, inactive substance, which does not act, but is acted upon. | |
| From: Thomas Reid (Essays on Active Powers 4: Liberty of Agents [1788], 5) | |
| A reaction: A clear statement (alongside Euler's) of the 18th century view, still with us, but strikes me as entirely wrong. Their view needs the active power of God to drive the laws. Matter has intrinsic primitive powers, and laws describe patterns of behaviour. |