31 ideas
| 10888 | Sets can be defined by 'enumeration', or by 'abstraction' (based on a property) [Zalabardo] |
| Full Idea: We can define a set by 'enumeration' (by listing the items, within curly brackets), or by 'abstraction' (by specifying the elements as instances of a property), pretending that they form a determinate totality. The latter is written {x | x is P}. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.3) |
| 10889 | The 'Cartesian Product' of two sets relates them by pairing every element with every element [Zalabardo] |
|
Full Idea:
The 'Cartesian Product' of two sets, written A x B, is the relation which pairs every element of A with every element of B. So A x B = { |
|
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.6) |
| 10890 | A 'partial ordering' is reflexive, antisymmetric and transitive [Zalabardo] |
| Full Idea: A binary relation in a set is a 'partial ordering' just in case it is reflexive, antisymmetric and transitive. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.6) |
| 10886 | Determinacy: an object is either in a set, or it isn't [Zalabardo] |
| Full Idea: Principle of Determinacy: For every object a and every set S, either a is an element of S or a is not an element of S. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.2) |
| 10887 | Specification: Determinate totals of objects always make a set [Zalabardo] |
| Full Idea: Principle of Specification: Whenever we can specify a determinate totality of objects, we shall say that there is a set whose elements are precisely the objects that we have specified. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.3) | |
| A reaction: Compare the Axiom of Specification. Zalabardo says we may wish to consider sets of which we cannot specify the members. |
| 10897 | A first-order 'sentence' is a formula with no free variables [Zalabardo] |
| Full Idea: A formula of a first-order language is a 'sentence' just in case it has no free variables. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.2) |
| 10893 | Γ |= φ for sentences if φ is true when all of Γ is true [Zalabardo] |
| Full Idea: A propositional logic sentence is a 'logical consequence' of a set of sentences (written Γ |= φ) if for every admissible truth-assignment all the sentences in the set Γ are true, then φ is true. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.4) | |
| A reaction: The definition is similar for predicate logic. |
| 10899 | Γ |= φ if φ is true when all of Γ is true, for all structures and interpretations [Zalabardo] |
| Full Idea: A formula is the 'logical consequence' of a set of formulas (Γ |= φ) if for every structure in the language and every variable interpretation of the structure, if all the formulas within the set are true and the formula itself is true. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.5) |
| 10896 | Propositional logic just needs ¬, and one of ∧, ∨ and → [Zalabardo] |
| Full Idea: In propositional logic, any set containing ¬ and at least one of ∧, ∨ and → is expressively complete. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.8) |
| 10898 | The semantics shows how truth values depend on instantiations of properties and relations [Zalabardo] |
| Full Idea: The semantic pattern of a first-order language is the ways in which truth values depend on which individuals instantiate the properties and relations which figure in them. ..So we pair a truth value with each combination of individuals, sets etc. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.3) | |
| A reaction: So truth reduces to a combination of 'instantiations', which is rather like 'satisfaction'. |
| 10902 | We can do semantics by looking at given propositions, or by building new ones [Zalabardo] |
| Full Idea: We can look at semantics from the point of view of how truth values are determined by instantiations of properties and relations, or by asking how we can build, using the resources of the language, a proposition corresponding to a given semantic pattern. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.6) | |
| A reaction: The second version of semantics is model theory. |
| 10892 | We make a truth assignment to T and F, which may be true and false, but merely differ from one another [Zalabardo] |
| Full Idea: A truth assignment is a function from propositions to the set {T,F}. We will think of T and F as the truth values true and false, but for our purposes all we need to assume about the identity of these objects is that they are different from each other. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.4) | |
| A reaction: Note that T and F are 'objects'. This remark is important in understanding modern logical semantics. T and F can be equated to 1 and 0 in the language of a computer. They just mean as much as you want them to mean. |
| 10895 | 'Logically true' (|= φ) is true for every truth-assignment [Zalabardo] |
| Full Idea: A propositional logic sentence is 'logically true', written |= φ, if it is true for every admissible truth-assignment. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.4) |
| 10900 | Logically true sentences are true in all structures [Zalabardo] |
| Full Idea: In first-order languages, logically true sentences are true in all structures. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.5) |
| 10894 | A sentence-set is 'satisfiable' if at least one truth-assignment makes them all true [Zalabardo] |
| Full Idea: A propositional logic set of sentences Γ is 'satisfiable' if there is at least one admissible truth-assignment that makes all of its sentences true. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.4) |
| 10901 | Some formulas are 'satisfiable' if there is a structure and interpretation that makes them true [Zalabardo] |
| Full Idea: A set of formulas of a first-order language is 'satisfiable' if there is a structure and a variable interpretation in that structure such that all the formulas of the set are true. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.5) |
| 10903 | A structure models a sentence if it is true in the model, and a set of sentences if they are all true in the model [Zalabardo] |
| Full Idea: A structure is a model of a sentence if the sentence is true in the model; a structure is a model of a set of sentences if they are all true in the structure. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.6) |
| 10891 | If a set is defined by induction, then proof by induction can be applied to it [Zalabardo] |
| Full Idea: Defining a set by induction enables us to use the method of proof by induction to establish that all the elements of the set have a certain property. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.3) |
| 23551 | It is necessary for a belief that it be held for a length of time [Fricker,M] |
| Full Idea: A mental state cannot count as a belief unless it has a reasonable life expectancy. It must be the sort of thing that one is disposed to assert not only now but in the future too. | |
| From: Miranda Fricker (Epistemic Injustice [2007], 2.3) | |
| A reaction: There are obvious counterexamples, where a firm belief is strongly formed, only to be dashed by a counterexample (such as a new witness in court) soon afterwards. That said, this idea is obviously correct. |
| 6871 | We can't only believe things if we are currently conscious of their justification - there are too many [Goldman] |
| Full Idea: Strong internalism says only current conscious states can justify beliefs, but this has the problem of Stored Beliefs, that most of our beliefs are stored in memory, and one's conscious state includes nothing that justifies them. | |
| From: Alvin I. Goldman (Internalism Exposed [1999], §2) | |
| A reaction: This point seems obviously correct, but one could still have a 'fairly strong' version, which required that you could always call into consciousness the justificiation for any belief that you happened to remember. |
| 6872 | Internalism must cover Forgotten Evidence, which is no longer retrievable from memory [Goldman] |
| Full Idea: Even weak internalism has the problem of Forgotten Evidence; the agent once had adequate evidence that she subsequently forgot; at the time of epistemic appraisal, she no longer has adequate evidence that is retrievable from memory. | |
| From: Alvin I. Goldman (Internalism Exposed [1999], §3) | |
| A reaction: This is certainly a basic problem for any account of justification. It will rule out any strict requirement that there be actual mental states available to support a belief. Internalism may be pushed to include non-conscious parts of the mind. |
| 6874 | Internal justification needs both mental stability and time to compute coherence [Goldman] |
| Full Idea: The problem for internalists of Doxastic Decision Interval says internal justification must avoid mental change to preserve the justification status, but must also allow enough time to compute the formal relations between beliefs. | |
| From: Alvin I. Goldman (Internalism Exposed [1999], §4) | |
| A reaction: The word 'compute' implies a rather odd model of assessing coherence, which seems instantaneous for most of us where everyday beliefs are concerned. In real mental life this does not strike me as a problem. |
| 23558 | Offering knowledge needs accuracy and sincerity; receiving it needs testimonial justice [Fricker,M] |
| Full Idea: Accuracy and Sincerity sustain trust as regards contributing knowledge to the pool; Testimonial Justice helps sustain trust as regards acquiring knowledge from the pool. | |
| From: Miranda Fricker (Epistemic Injustice [2007], 5.1) | |
| A reaction: Fricker's contribution is to show that acquiring knowledge has its own virtues, alongside discovering and communicating it. I take the underlying virtue to be absolute respect for all possible contributors. |
| 6873 | Coherent justification seems to require retrieving all our beliefs simultaneously [Goldman] |
| Full Idea: The problem of Concurrent Retrieval is a problem for internalism, notably coherentism, because an agent could ascertain coherence of her entire corpus only by concurrently retrieving all of her stored beliefs. | |
| From: Alvin I. Goldman (Internalism Exposed [1999], §3) | |
| A reaction: Sounds neat, but not very convincing. Goldman is relying on scepticism about short-term memory, but all belief and knowledge will collapse if we go down that road. We couldn't do simple arithmetic if Goldman's point were right. |
| 6875 | Reliability involves truth, and truth is external [Goldman] |
| Full Idea: Reliability involves truth, and truth (on the usual assumption) is external. | |
| From: Alvin I. Goldman (Internalism Exposed [1999], §6) | |
| A reaction: As an argument for externalism this seems bogus. I am not sure that truth is either 'internal' or 'external'. How could the truth of 3+2=5 be external? Facts are mostly external, but I take truth to be a relation between internal and external. |
| 23550 | Burge says we are normally a priori entitled to believe testimony [Fricker,M] |
| Full Idea: In Tyler Burge's view we have an a priori entitlement for believing what others tell us, other things being equal. | |
| From: Miranda Fricker (Epistemic Injustice [2007], 1.3 n11) | |
| A reaction: [Burge 'Content Preservation' 1992] Close to Davidson's Charity (that without a default assumption of truth-speaking language won't work at all). Davidson is right about casual conversation, but for important testimony Burge should be more cautious. |
| 23552 | We assess testimonial probabilities by the speaker, the listener, the facts, and the circumstances [Fricker,M] |
| Full Idea: A person should receive the word of his interlocutor in the light of the probability that someone like that would (be able and willing to) to tell someone like him the truth about something like this in circumstances like these. | |
| From: Miranda Fricker (Epistemic Injustice [2007], 3.2) | |
| A reaction: That's a pretty good summary of the rational response to testimony. I can't think of any other factors. |
| 23553 | Testimonial judgement is not logical, but produces reasons and motivations [Fricker,M] |
| Full Idea: Moral/testimonial judgement is non-inferential, uncodifiable, intrinsically motivating, intrinsically reason-giving, and typically has an emotional aspect. | |
| From: Miranda Fricker (Epistemic Injustice [2007], 3.3) | |
| A reaction: Fricker's compressed summary of her findings about testimony. The first words indicate her belief that assessment of testimony is a moral affair. |
| 23555 | Assessing credibility involves the impact of both the speaker's and the listener's social identity [Fricker,M] |
| Full Idea: For a hearer to identify the impact of identity power in their credibility judgements they must be alert to the impact not only of the speaker's social identity, but also the impact of the own social identity on their credibility judgements. | |
| From: Miranda Fricker (Epistemic Injustice [2007], 4.1) | |
| A reaction: [why are all sentences in academic writing twice as long as they need to be? - that question is deeper than it looks!] This is a salutary warning. Not just 'what are my prejudices?', but also 'what is this person willing to tell a person like me?'. |
| 23554 | Judgements can be unreflective and non-inferential, yet rational, by being sensitive to experience [Fricker,M] |
| Full Idea: Our idea of testimonial sensibility is a spontaneous critical sensitivity permanently in training and adapting to experience. …This gives us a picture of how judgements can be rational yet unreflective, critical yet non-inferential. | |
| From: Miranda Fricker (Epistemic Injustice [2007], 3.4) | |
| A reaction: Love this. I want to connect human reasoning to good judgement by animals, and I offer the word 'sensible' to bridge the gap. Dogs and scientists can be sensible. Fricker spells out more fully what I have in mind, with reference to testimony. |
| 23557 | To judge agents in remote times and cultures we need a moral resentment weaker than blame [Fricker,M] |
| Full Idea: I think that identifying forms of moral resentment that fall short of blame but which are agent-directed is the key to achieving appropriate moral response across historical and cultural distance. | |
| From: Miranda Fricker (Epistemic Injustice [2007], 4.2) | |
| A reaction: Very good. Simple blame for horrible actions performed in remote rather horrible societies is pointless. But switching off moral sensibilities when reading history and anthropology looks like a slippery slope, so 'moral resentment' is nice. |