Combining Texts

All the ideas for 'Animal Rights and Wrongs', 'Foundations without Foundationalism' and 'An American Indian model of the Universe'

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86 ideas

3. Truth / F. Semantic Truth / 1. Tarski's Truth / b. Satisfaction and truth
Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro]
4. Formal Logic / A. Syllogistic Logic / 1. Aristotelian Logic
Aristotelian logic is complete [Shapiro]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set
A set is 'transitive' if contains every member of each of its members [Shapiro]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
Choice is essential for proving downward Löwenheim-Skolem [Shapiro]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / a. Sets as existing
Are sets part of logic, or part of mathematics? [Shapiro]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
It is central to the iterative conception that membership is well-founded, with no infinite descending chains [Shapiro]
Russell's paradox shows that there are classes which are not iterative sets [Shapiro]
Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets [Shapiro]
4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element [Shapiro]
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
There is no 'correct' logic for natural languages [Shapiro]
Logic is the ideal for learning new propositions on the basis of others [Shapiro]
5. Theory of Logic / A. Overview of Logic / 2. History of Logic
Bernays (1918) formulated and proved the completeness of propositional logic [Shapiro]
Can one develop set theory first, then derive numbers, or are numbers more basic? [Shapiro]
Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order [Shapiro]
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
First-order logic was an afterthought in the development of modern logic [Shapiro]
The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed [Shapiro]
Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable [Shapiro]
The notion of finitude is actually built into first-order languages [Shapiro]
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Second-order logic is better than set theory, since it only adds relations and operations, and nothing else [Shapiro, by Lavine]
Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? [Shapiro]
Henkin semantics has separate variables ranging over the relations and over the functions [Shapiro]
In standard semantics for second-order logic, a single domain fixes the ranges for the variables [Shapiro]
Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics [Shapiro]
5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
Semantic consequence is ineffective in second-order logic [Shapiro]
If a logic is incomplete, its semantic consequence relation is not effective [Shapiro]
5. Theory of Logic / E. Structures of Logic / 1. Logical Form
Finding the logical form of a sentence is difficult, and there are no criteria of correctness [Shapiro]
5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro]
5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction
'Satisfaction' is a function from models, assignments, and formulas to {true,false} [Shapiro]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
Semantics for models uses set-theory [Shapiro]
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro]
Categoricity can't be reached in a first-order language [Shapiro]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [Shapiro]
Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro]
The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro]
Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails [Shapiro]
5. Theory of Logic / K. Features of Logics / 3. Soundness
'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence [Shapiro]
5. Theory of Logic / K. Features of Logics / 4. Completeness
We can live well without completeness in logic [Shapiro]
5. Theory of Logic / K. Features of Logics / 6. Compactness
Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro]
Compactness is derived from soundness and completeness [Shapiro]
5. Theory of Logic / K. Features of Logics / 9. Expressibility
A language is 'semantically effective' if its logical truths are recursively enumerable [Shapiro]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / d. Natural numbers
Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
First-order arithmetic can't even represent basic number theory [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
Some reject formal properties if they are not defined, or defined impredicatively [Shapiro]
8. Modes of Existence / B. Properties / 10. Properties as Predicates
Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro]
11. Knowledge Aims / A. Knowledge / 4. Belief / b. Elements of beliefs
Having beliefs involves recognition, expectation and surprise [Scruton]
11. Knowledge Aims / A. Knowledge / 4. Belief / f. Animal beliefs
If an animal has beliefs, that implies not only that it can make mistakes, but that it can learn from them [Scruton]
12. Knowledge Sources / B. Perception / 1. Perception
Perception (which involves an assessment) is a higher state than sensation [Scruton]
13. Knowledge Criteria / E. Relativism / 5. Language Relativism
Hopi consistently prefers verbs and events to nouns and things [Whorf]
14. Science / B. Scientific Theories / 4. Paradigm
Scientific thought is essentially a specialised part of Indo-European languages [Whorf]
15. Nature of Minds / B. Features of Minds / 1. Consciousness / d. Purpose of consciousness
There is consciousness whenever behaviour must be explained in terms of mental activity [Scruton]
16. Persons / A. Concept of a Person / 2. Persons as Responsible
Our concept of a person is derived from Roman law [Scruton]
17. Mind and Body / B. Behaviourism / 4. Behaviourism Critique
Conditioning may change behaviour without changing the mind [Scruton]
18. Thought / A. Modes of Thought / 3. Emotions / c. Role of emotions
An emotion is a motive which is also a feeling [Scruton]
18. Thought / A. Modes of Thought / 5. Rationality / c. Animal rationality
Do we use reason to distinguish people from animals, or use that difference to define reason? [Scruton]
22. Metaethics / A. Ethics Foundations / 1. Nature of Ethics / a. Preconditions for ethics
All moral life depends ultimately on piety, which is our recognition of our own dependence [Scruton]
23. Ethics / B. Contract Ethics / 1. Contractarianism
Kant's Moral Law is the rules rational beings would accept when trying to live by agreement [Scruton]
23. Ethics / C. Virtue Theory / 3. Virtues / a. Virtues
The modern virtues are courage, prudence, wisdom, temperance, justice, charity and loyalty [Scruton]
23. Ethics / C. Virtue Theory / 3. Virtues / c. Justice
Only just people will drop their own self-interests when faced with an impartial verdict [Scruton]
23. Ethics / C. Virtue Theory / 3. Virtues / f. Compassion
Sympathy can undermine the moral order just as much as crime does [Scruton]
23. Ethics / D. Deontological Ethics / 2. Duty
That which can only be done by a callous person, ought not to be done [Scruton]
23. Ethics / D. Deontological Ethics / 3. Universalisability
As soon as we drop self-interest and judge impartially, we find ourselves agreeing about conflicts [Scruton]
23. Ethics / E. Utilitarianism / 1. Utilitarianism
Utilitarianism merely guides us (by means of sympathy) when the moral law is silent [Scruton]
Morality is not a sort of calculation, it is what sets the limits to when calculation is appropriate [Scruton]
Utilitarianism says we can't blame Stalin yet, but such a theory is a sick joke [Scruton]
Utilitarianism is wrong precisely because it can't distinguish animals from people [Scruton]
25. Social Practice / F. Life Issues / 6. Animal Rights
Brutal animal sports are banned because they harm the personality of the watcher [Scruton]
Many breeds of animals have needs which our own ancestors planted in them [Scruton]
Introducing a natural means of controlling animal population may not be very compassionate [Scruton]
We favour our own animals over foreign ones because we see them as fellow citizens [Scruton]
Animals command our sympathy and moral concern initially because of their intentionality [Scruton]
Letting your dog kill wild rats, and keeping rats for your dog to kill, are very different [Scruton]
Many of the stranger forms of life (e.g. worms) interest us only as a species, not as individuals [Scruton]
An animal has individuality if it is nameable, and advanced animals can respond to their name [Scruton]
I may avoid stepping on a spider or flower, but fellow-feeling makes me protect a rabbit [Scruton]
Lucky animals are eaten by large predators, the less lucky starve, and worst is death by small predators [Scruton]
We can easily remove the risk of suffering from an animal's life, but we shouldn't do it [Scruton]
Sheep and cattle live comfortable lives, and die an enviably easy death [Scruton]
Concern for one animal may harm the species, if the individual is part of a bigger problem [Scruton]
Animals are outside the community of rights, but we still have duties towards them [Scruton]
27. Natural Reality / D. Time / 2. Passage of Time / e. Tensed (A) series
The Hopi have no concept of time as something flowing from past to future [Whorf]