17 ideas
| 17750 | The first clear proof of the consistency of the first order predicate logic was in 1928 [Hilbert/Ackermann, by Walicki] |
| Full Idea: The first clear proof of the consistency of the first order predicate logic is found in the 1928 book of Hilbert and Ackermann. | |
| From: report of Hilbert,D/Ackermann,W (Principles of Theoretical Logic [1928]) by Michal Walicki - Introduction to Mathematical Logic History E.2.1 |
| 15375 | If terms change their designations in different states, they are functions from states to objects [Fitting] |
| Full Idea: The common feature of every designating term is that designation may change from state to state - thus it can be formalized by a function from states to objects. | |
| From: Melvin Fitting (Intensional Logic [2007], 3) | |
| A reaction: Specifying the objects sounds OK, but specifying states sounds rather tough. |
| 15376 | Intensional logic adds a second type of quantification, over intensional objects, or individual concepts [Fitting] |
| Full Idea: To first order modal logic (with quantification over objects) we can add a second kind of quantification, over intensions. An intensional object, or individual concept, will be modelled by a function from states to objects. | |
| From: Melvin Fitting (Intensional Logic [2007], 3.3) |
| 15378 | Awareness logic adds the restriction of an awareness function to epistemic logic [Fitting] |
| Full Idea: Awareness logic enriched Hintikka's epistemic models with an awareness function, mapping each state to the set of formulas we are aware of at that state. This reflects some bound on the resources we can bring to bear. | |
| From: Melvin Fitting (Intensional Logic [2007], 3.6.1) | |
| A reaction: [He cites Fagin and Halpern 1988 for this] |
| 15379 | Justication logics make explicit the reasons for mathematical truth in proofs [Fitting] |
| Full Idea: In justification logics, the logics of knowledge are extended by making reasons explicit. A logic of proof terms was created, with a semantics. In this, mathematical truths are known for explicit reasons, and these provide a measure of complexity. | |
| From: Melvin Fitting (Intensional Logic [2007], 3.6.1) |
| 11026 | Classical logic is deliberately extensional, in order to model mathematics [Fitting] |
| Full Idea: Mathematics is typically extensional throughout (we write 3+2=2+3 despite the two terms having different meanings). ..Classical first-order logic is extensional by design since it primarily evolved to model the reasoning of mathematics. | |
| From: Melvin Fitting (Intensional Logic [2007], §1) |
| 11028 | λ-abstraction disambiguates the scope of modal operators [Fitting] |
| Full Idea: λ-abstraction can be used to abstract and disambiguate a predicate. De re is [λx◊P(x)](f) - f has the possible-P property - and de dicto is ◊[λxP(x)](f) - possibly f has the P-property. Also applies to □. | |
| From: Melvin Fitting (Intensional Logic [2007], §3.3) | |
| A reaction: Compare the Barcan formula. Originated with Church in the 1930s, and Carnap 1947, but revived by Stalnaker and Thomason 1968. Because it refers to the predicate, it has a role in intensional versions of logic, especially modal logic. |
| 15377 | Definite descriptions pick out different objects in different possible worlds [Fitting] |
| Full Idea: Definite descriptions pick out different objects in different possible worlds quite naturally. | |
| From: Melvin Fitting (Intensional Logic [2007], 3.4) | |
| A reaction: A definite description can pick out the same object in another possible world, or a very similar one, or an object which has almost nothing in common with the others. |
| 23438 | Full rationality must include morality [Foot] |
| Full Idea: You haven't got a full idea of rationality until you've got morality within it. | |
| From: Philippa Foot (Interview with Philippa Foot [2003], p.35) | |
| A reaction: Does this mean that mathematical proofs are not rational, or that they are moral? |
| 23437 | Practical reason is goodness in choosing actions [Foot] |
| Full Idea: Practical rationality is goodness in respect of reason for actions, just as rationality of thinking is goodness in respect of beliefs. | |
| From: Philippa Foot (Interview with Philippa Foot [2003], p.35) | |
| A reaction: It is very Greek to think that rationality involves goodness. There seems to be a purely instrumental form of practical reason that just gets from A to B, as when giving accurate street directions to someone. |
| 23436 | It is an odd Humean view to think a reason to act must always involve caring [Foot] |
| Full Idea: One would need a very special, very Humean, view about reasons for actions to think a man doesn't have a reason unless he cares. | |
| From: Philippa Foot (Interview with Philippa Foot [2003], p.34-5) | |
| A reaction: She says she used to believe this, but was wrong. It is hard to imagine acting for reasons if they don't care about anything at all (even that it's their job). But then people just do care about things. |
| 23431 | Human defects are just like plant or animal defects [Foot] |
| Full Idea: We describe defects in human beings in the same way as we do defects in plants and animals. …You cannot talk about a river as being defective. | |
| From: Philippa Foot (Interview with Philippa Foot [2003], p.33) | |
| A reaction: This is a much clearer commitment to naturalistic ethics than I have found in her more academic writings. My opinion of Foot (which was already high) went up when I read this interview. …She says vice is a defect of the will. |
| 23432 | Concepts such as function, welfare, flourishing and interests only apply to living things [Foot] |
| Full Idea: There are concepts which apply only to living things, considered in their own right, which would include function, welfare, flourishing, interests and the good of something. | |
| From: Philippa Foot (Interview with Philippa Foot [2003], p.33) | |
| A reaction: This is a very Aristotelian view, with which I entirely agree. The central concept is function. |
| 23433 | Humans need courage like a plant needs roots [Foot] |
| Full Idea: A plant needs strong roots in the same way human beings need courage. | |
| From: Philippa Foot (Interview with Philippa Foot [2003], p.33) | |
| A reaction: I'm not quite convince by the analogy, but I strongly agree with her basic approach. |
| 23434 | There is no fact-value gap in 'owls should see in the dark' [Foot] |
| Full Idea: If you say 'an owl should be able to see in the dark' …you're not going to think that there's a gap between facts and evaluation. | |
| From: Philippa Foot (Interview with Philippa Foot [2003], p.33) | |
| A reaction: I take this to be a major and fundamental idea, which pinpoints the failure of Humeans to understand the world correctly. There is always total nihilism, of course, but that is a sort of blindness to how things are. Demanding 'proof' of values is crazy. |
| 23439 | Principles are not ultimate, but arise from the necessities of human life [Foot] |
| Full Idea: I don't believe in ultimate principles that must be simply affirmed or denied, but rather in an appeal to the necessities of human life. | |
| From: Philippa Foot (Interview with Philippa Foot [2003], p.37) | |
| A reaction: I agree. Humans have a strong tendency to elevate anything which they consider important into an absolute (such as the value of life, or freedom). |
| 23435 | If you demonstrate the reason to act, there is no further question of 'why should I?' [Foot] |
| Full Idea: You lose the sense of 'should' if you go on saying 'why should I?' when you've finished the argument about what is rational to do, what you've got reason to do. | |
| From: Philippa Foot (Interview with Philippa Foot [2003], P.34) | |
| A reaction: Some people reify the concept of duty, so that they do what is required without caring about the reason. I suppose that would wither if they were shown that no reason exists. |