Combining Texts

All the ideas for 'Actualism and Possible Worlds', 'Human Personality' and 'Intermediate Logic'

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97 ideas

3. Truth / A. Truth Problems / 3. Value of Truth
Genius and love of truth are always accompanied by great humility [Weil]
4. Formal Logic / A. Syllogistic Logic / 2. Syllogistic Logic
Venn Diagrams map three predicates into eight compartments, then look for the conclusion [Bostock]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL
'Disjunctive Normal Form' is ensuring that no conjunction has a disjunction within its scope [Bostock]
'Conjunctive Normal Form' is ensuring that no disjunction has a conjunction within its scope [Bostock]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL
'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,ψ|= [Bostock]
'Assumptions' says that a formula entails itself (φ|=φ) [Bostock]
'Thinning' allows that if premisses entail a conclusion, then adding further premisses makes no difference [Bostock]
The 'conditional' is that Γ|=φ→ψ iff Γ,φ|=ψ [Bostock]
'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z [Bostock]
'Negation' says that Γ,¬φ|= iff Γ|=φ [Bostock]
'Conjunction' says that Γ|=φ∧ψ iff Γ|=φ and Γ|=ψ [Bostock]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL
A logic with ¬ and → needs three axiom-schemas and one rule as foundation [Bostock]
4. Formal Logic / E. Nonclassical Logics / 6. Free Logic
A 'free' logic can have empty names, and a 'universally free' logic can have empty domains [Bostock]
5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
Truth is the basic notion in classical logic [Bostock]
Elementary logic cannot distinguish clearly between the finite and the infinite [Bostock]
Fictional characters wreck elementary logic, as they have contradictions and no excluded middle [Bostock]
5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-
The syntactic turnstile |- φ means 'there is a proof of φ' or 'φ is a theorem' [Bostock]
5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
Validity is a conclusion following for premises, even if there is no proof [Bostock]
It seems more natural to express |= as 'therefore', rather than 'entails' [Bostock]
Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid' [Bostock]
5. Theory of Logic / B. Logical Consequence / 5. Modus Ponens
MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment) [Bostock]
MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ [Bostock]
5. Theory of Logic / D. Assumptions for Logic / 4. Identity in Logic
The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b) [Bostock]
|= α=α and α=β |= φ(α/ξ ↔ φ(β/ξ) fix identity [Bostock]
If we are to express that there at least two things, we need identity [Bostock]
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
Truth-functors are usually held to be defined by their truth-tables [Bostock]
5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
A 'zero-place' function just has a single value, so it is a name [Bostock]
A 'total' function ranges over the whole domain, a 'partial' function over appropriate inputs [Bostock]
5. Theory of Logic / F. Referring in Logic / 1. Naming / a. Names
In logic, a name is just any expression which refers to a particular single object [Bostock]
5. Theory of Logic / F. Referring in Logic / 1. Naming / e. Empty names
An expression is only a name if it succeeds in referring to a real object [Bostock]
5. Theory of Logic / F. Referring in Logic / 2. Descriptions / b. Definite descriptions
Definite desciptions resemble names, but can't actually be names, if they don't always refer [Bostock]
Because of scope problems, definite descriptions are best treated as quantifiers [Bostock]
Definite descriptions are usually treated like names, and are just like them if they uniquely refer [Bostock]
We are only obliged to treat definite descriptions as non-names if only the former have scope [Bostock]
Definite descriptions don't always pick out one thing, as in denials of existence, or errors [Bostock]
5. Theory of Logic / F. Referring in Logic / 2. Descriptions / c. Theory of definite descriptions
Names do not have scope problems (e.g. in placing negation), but Russell's account does have that problem [Bostock]
5. Theory of Logic / G. Quantification / 1. Quantification
'Prenex normal form' is all quantifiers at the beginning, out of the scope of truth-functors [Bostock]
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
If we allow empty domains, we must allow empty names [Bostock]
5. Theory of Logic / H. Proof Systems / 1. Proof Systems
An 'informal proof' is in no particular system, and uses obvious steps and some ordinary English [Bostock]
5. Theory of Logic / H. Proof Systems / 2. Axiomatic Proof
Quantification adds two axiom-schemas and a new rule [Bostock]
Axiom systems from Frege, Russell, Church, Lukasiewicz, Tarski, Nicod, Kleene, Quine... [Bostock]
5. Theory of Logic / H. Proof Systems / 3. Proof from Assumptions
'Conditonalised' inferences point to the Deduction Theorem: If Γ,φ|-ψ then Γ|-φ→ψ [Bostock]
Proof by Assumptions can always be reduced to Proof by Axioms, using the Deduction Theorem [Bostock]
The Deduction Theorem and Reductio can 'discharge' assumptions - they aren't needed for the new truth [Bostock]
The Deduction Theorem greatly simplifies the search for proof [Bostock]
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
Natural deduction takes proof from assumptions (with its rules) as basic, and axioms play no part [Bostock]
Excluded middle is an introduction rule for negation, and ex falso quodlibet will eliminate it [Bostock]
In natural deduction we work from the premisses and the conclusion, hoping to meet in the middle [Bostock]
Natural deduction rules for → are the Deduction Theorem (→I) and Modus Ponens (→E) [Bostock]
5. Theory of Logic / H. Proof Systems / 5. Tableau Proof
Tableau proofs use reduction - seeking an impossible consequence from an assumption [Bostock]
A completed open branch gives an interpretation which verifies those formulae [Bostock]
Non-branching rules add lines, and branching rules need a split; a branch with a contradiction is 'closed' [Bostock]
In a tableau proof no sequence is established until the final branch is closed; hypotheses are explored [Bostock]
Unlike natural deduction, semantic tableaux have recipes for proving things [Bostock]
A tree proof becomes too broad if its only rule is Modus Ponens [Bostock]
Tableau rules are all elimination rules, gradually shortening formulae [Bostock]
5. Theory of Logic / H. Proof Systems / 6. Sequent Calculi
Each line of a sequent calculus is a conclusion of previous lines, each one explicitly recorded [Bostock]
A sequent calculus is good for comparing proof systems [Bostock]
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
Interpretation by assigning objects to names, or assigning them to variables first [Bostock, by PG]
5. Theory of Logic / I. Semantics of Logic / 5. Extensionalism
Extensionality is built into ordinary logic semantics; names have objects, predicates have sets of objects [Bostock]
If an object has two names, truth is undisturbed if the names are swapped; this is Extensionality [Bostock]
5. Theory of Logic / K. Features of Logics / 2. Consistency
For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ [Bostock]
A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula [Bostock]
A set of formulae is 'inconsistent' when there is no interpretation which can make them all true [Bostock]
5. Theory of Logic / K. Features of Logics / 6. Compactness
Inconsistency or entailment just from functors and quantifiers is finitely based, if compact [Bostock]
Compactness means an infinity of sequents on the left will add nothing new [Bostock]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all [Bostock]
Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers [Bostock]
7. Existence / A. Nature of Existence / 3. Being / a. Nature of Being
Necessary beings (numbers, properties, sets, propositions, states of affairs, God) exist in all possible worlds [Plantinga]
8. Modes of Existence / A. Relations / 4. Formal Relations / a. Types of relation
Relations can be one-many (at most one on the left) or many-one (at most one on the right) [Bostock]
A relation is not reflexive, just because it is transitive and symmetrical [Bostock]
9. Objects / D. Essence of Objects / 1. Essences of Objects
Socrates is a contingent being, but his essence is not; without Socrates, his essence is unexemplified [Plantinga]
9. Objects / F. Identity among Objects / 5. Self-Identity
If non-existent things are self-identical, they are just one thing - so call it the 'null object' [Bostock]
10. Modality / A. Necessity / 6. Logical Necessity
The idea that anything which can be proved is necessary has a problem with empty names [Bostock]
10. Modality / E. Possible worlds / 1. Possible Worlds / a. Possible worlds
Possible worlds clarify possibility, propositions, properties, sets, counterfacts, time, determinism etc. [Plantinga]
10. Modality / E. Possible worlds / 1. Possible Worlds / d. Possible worlds actualism
Plantinga's actualism is nominal, because he fills actuality with possibilia [Stalnaker on Plantinga]
16. Persons / B. Nature of the Self / 7. Self and Body / a. Self needs body
What is sacred is not a person, but the whole physical human being [Weil]
18. Thought / A. Modes of Thought / 1. Thought
The mind is imprisoned and limited by language, restricting our awareness of wider thoughts [Weil]
19. Language / C. Assigning Meanings / 3. Predicates
A (modern) predicate is the result of leaving a gap for the name in a sentence [Bostock]
19. Language / C. Assigning Meanings / 8. Possible Worlds Semantics
Plantinga has domains of sets of essences, variables denoting essences, and predicates as functions [Plantinga, by Stalnaker]
Plantinga's essences have their own properties - so will have essences, giving a hierarchy [Stalnaker on Plantinga]
19. Language / D. Propositions / 1. Propositions
Are propositions and states of affairs two separate things, or only one? I incline to say one [Plantinga]
21. Aesthetics / A. Aesthetic Experience / 6. The Sublime
Beauty is an attractive mystery, leaving nothing to be desired [Weil]
22. Metaethics / B. Value / 1. Nature of Value / f. Ultimate value
All we need are the unity of justice, truth and beauty [Weil]
22. Metaethics / B. Value / 2. Values / c. Life
The sacred in every human is their expectation of good rather than evil [Weil]
22. Metaethics / B. Value / 2. Values / g. Love
Everything which originates in love is beautiful [Weil]
22. Metaethics / B. Value / 2. Values / j. Evil
Evil is transmitted by comforts and pleasures, but mostly by doing harm to people [Weil]
24. Political Theory / D. Ideologies / 8. Socialism
It is not more money which the wretched members of society need [Weil]
24. Political Theory / D. Ideologies / 9. Communism
The problem of the collective is not suppression of persons, but persons erasing themselves [Weil]
25. Social Practice / B. Equalities / 1. Grounds of equality
People absurdly claim an equal share of things which are essentially privileged [Weil]
25. Social Practice / C. Rights / 1. Basis of Rights
Rights are asserted contentiously, and need the backing of force [Weil]
Giving centrality to rights stifles all impulses of charity [Weil]
25. Social Practice / D. Justice / 1. Basis of justice
The spirit of justice needs the full attention of truth, and that attention is love [Weil]
Justice (concerning harm) is distinct from rights (concerning inequality) [Weil]
25. Social Practice / D. Justice / 3. Punishment / d. Reform of offenders
The only thing in society worse than crime is repressive justice [Weil]
Punishment aims at the good for men who don't desire it [Weil]
28. God / A. Divine Nature / 6. Divine Morality / c. God is the good
The only choice is between supernatural good, or evil [Weil]