15 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) |
| 4483 | If abstract terms are sets of tropes, 'being a unicorn' and 'being a griffin' turn out identical [Loux] |
| Full Idea: If trope theorists say abstract singular terms name sets of tropes, what is the referent of 'is a unicorn'? The only candidate is the null set (with no members), but there is just one null set, so 'being a unicorn' and 'being a griffin' will be identical. | |
| From: Michael J. Loux (Metaphysics: contemporary introduction [1998], p.86) | |
| A reaction: Not crucial, I would think, given that a unicorn is just a horse with a horn. Hume explains how we do that, combining ideas which arose from actual tropes. |
| 4481 | Austere nominalists insist that the realist's universals lack the requisite independent identifiability [Loux] |
| Full Idea: Austere nominalists insist that the realist's universals lack the requisite independent identifiability. | |
| From: Michael J. Loux (Metaphysics: contemporary introduction [1998], p.60) | |
| A reaction: Plato's view seems to be that we don't identify universals independently. We ascend The Line, or think about the shadows in The Cave, and infer the universals from an array of particulars (by dialectic). |
| 4477 | Universals come in hierarchies of generality [Loux] |
| Full Idea: Universals come in hierarchies of generality. | |
| From: Michael J. Loux (Metaphysics: contemporary introduction [1998], p.24) | |
| A reaction: If it is possible to state facts about universals, this obviously encourages a rather Platonic approach to them, as existent things with properties. But maybe the hierarchies are conventional, not natural. |
| 4482 | Austere nominalism has to take a host of things (like being red, or human) as primitive [Loux] |
| Full Idea: In return for a one-category ontology (with particulars but no universals), the austere nominalist is forced to take a whole host of things (like being red, or triangular, or human) as unanalysable or primitive. | |
| From: Michael J. Loux (Metaphysics: contemporary introduction [1998], p.68) | |
| A reaction: I see that 'red' might have to be primitive, but being human can just be a collection of particulars. It is no ontologically worse to call them 'primitive' than to say they exist. |
| 4478 | Nominalism needs to account for abstract singular terms like 'circularity'. [Loux] |
| Full Idea: Nominalists have been very concerned to provide an account of the role of abstract singular terms (such as 'circularity'). | |
| From: Michael J. Loux (Metaphysics: contemporary introduction [1998], p.34) | |
| A reaction: Whether this is a big problem depends on our view of abstraction. If it only consists of selecting one property of an object and reifying it, then we can give a nominalist account of properties, and the problem is solved. |
| 4480 | Times and places are identified by objects, so cannot be used in a theory of object-identity [Loux] |
| Full Idea: Any account of the identity of material objects which turns on the identity of places and times must face the objection that the identity of places and times depends, in turn, on the identities of the objects located at them. | |
| From: Michael J. Loux (Metaphysics: contemporary introduction [1998], p.56) | |
| A reaction: This may be a benign circle, in which we concede that there are two basic interdependent concepts of objects and space-time. If you want to define identity - in terms of what? |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
| From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
| A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |