12 ideas
| 10282 | Logic is the study of sound argument, or of certain artificial languages (or applying the latter to the former) [Hodges,W] |
| Full Idea: A logic is a collection of closely related artificial languages, and its older meaning is the study of the rules of sound argument. The languages can be used as a framework for studying rules of argument. | |
| From: Wilfrid Hodges (First-Order Logic [2001], 1.1) | |
| A reaction: [Hodges then says he will stick to the languages] The suspicion is that one might confine the subject to the artificial languages simply because it is easier, and avoids the tricky philosophical questions. That approximates to computer programming. |
| 10283 | A formula needs an 'interpretation' of its constants, and a 'valuation' of its variables [Hodges,W] |
| Full Idea: To have a truth-value, a first-order formula needs an 'interpretation' (I) of its constants, and a 'valuation' (ν) of its variables. Something in the world is attached to the constants; objects are attached to variables. | |
| From: Wilfrid Hodges (First-Order Logic [2001], 1.3) |
| 10284 | There are three different standard presentations of semantics [Hodges,W] |
| Full Idea: Semantic rules can be presented in 'Tarski style', where the interpretation-plus-valuation is reduced to the same question for simpler formulas, or the 'Henkin-Hintikka style' in terms of games, or the 'Barwise-Etchemendy style' for computers. | |
| From: Wilfrid Hodges (First-Order Logic [2001], 1.3) | |
| A reaction: I haven't yet got the hang of the latter two, but I note them to map the territory. |
| 10285 | I |= φ means that the formula φ is true in the interpretation I [Hodges,W] |
| Full Idea: I |= φ means that the formula φ is true in the interpretation I. | |
| From: Wilfrid Hodges (First-Order Logic [2001], 1.5) | |
| A reaction: [There should be no space between the vertical and the two horizontals!] This contrasts with |-, which means 'is proved in'. That is a syntactic or proof-theoretic symbol, whereas |= is a semantic symbol (involving truth). |
| 10288 | Down Löwenheim-Skolem: if a countable language has a consistent theory, that has a countable model [Hodges,W] |
| Full Idea: Downward Löwenheim-Skolem (the weakest form): If L is a first-order language with at most countably many formulas, and T is a consistent theory in L. Then T has a model with at most countably many elements. | |
| From: Wilfrid Hodges (First-Order Logic [2001], 1.10) |
| 10289 | Up Löwenheim-Skolem: if infinite models, then arbitrarily large models [Hodges,W] |
| Full Idea: Upward Löwenheim-Skolem: every first-order theory with infinite models has arbitrarily large models. | |
| From: Wilfrid Hodges (First-Order Logic [2001], 1.10) |
| 10287 | If a first-order theory entails a sentence, there is a finite subset of the theory which entails it [Hodges,W] |
| Full Idea: Compactness Theorem: suppose T is a first-order theory, ψ is a first-order sentence, and T entails ψ. Then there is a finite subset U of T such that U entails ψ. | |
| From: Wilfrid Hodges (First-Order Logic [2001], 1.10) | |
| A reaction: If entailment is possible, it can be done finitely. |
| 10286 | A 'set' is a mathematically well-behaved class [Hodges,W] |
| Full Idea: A 'set' is a mathematically well-behaved class. | |
| From: Wilfrid Hodges (First-Order Logic [2001], 1.6) |
| 8412 | A causal interaction is when two processes intersect, and correlated modifications persist afterwards [Salmon] |
| Full Idea: When two processes intersect, and they undergo correlated modifications which persist after the intersection, I shall say that the intersection is a causal interaction. I take this as a fundamental causal concept. | |
| From: Wesley Salmon (Causality: Production and Propagation [1980], §4) | |
| A reaction: There may be a problem individuating processes, just as there is for events. I like this approach to causation, which is ontologically sparse, and fits in with the scientific worldview. Change of properties sounds precise, but isn't. Stick to processes. |
| 8413 | Cause must come first in propagations of causal interactions, but interactions are simultaneous [Salmon] |
| Full Idea: In a typical cause-effect situation (a 'propagation') cause must precede effect, for propagation over a finite time interval is an essential feature. In an 'interaction', an intersection of processes resulting in change, we have simultaneity. | |
| From: Wesley Salmon (Causality: Production and Propagation [1980], §8) | |
| A reaction: This takes the direction of time as axiomatic, and quite right too. Salmon isn't addressing the real difficulty, though, which is that the resultant laws are usually held to be time-reversible, which is a bit of a puzzle. |
| 8411 | Instead of localised events, I take enduring and extended processes as basic to causation [Salmon] |
| Full Idea: I propose to approach causality by taking processes rather than events as basic entities. Events are relatively localised in space and time, while processes have much greater temporal duration, and, in many cases, much greater spatial extent. | |
| From: Wesley Salmon (Causality: Production and Propagation [1980], §2) | |
| A reaction: This strikes me as an incredibly promising proposal, not just in our understanding of causation, but for our general metaphysics and understanding of nature. See Idea 4931, for example. Vague events and processes blend into one another. |
| 7482 | Resurrection developed in Judaism as a response to martyrdoms, in about 160 BCE [Anon (Dan), by Watson] |
| Full Idea: The idea of resurrection in Judaism seems to have first developed around 160 BCE, during the time of religious martyrdom, and as a response to it (the martyrs were surely not dying forever?). It is first mentioned in the book of Daniel. | |
| From: report of Anon (Dan) (27: Book of Daniel [c.165 BCE], Ch.7) by Peter Watson - Ideas | |
| A reaction: Idea 7473 suggests that Zoroaster beat them to it by 800 years. |