### Ideas from 'Intermediate Logic' by David Bostock , by Theme Structure

#### [found in 'Intermediate Logic' by Bostock,David [OUP 1997,0-19-875142-7]].

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###### 4. Formal Logic / A. Syllogistic Logic / 2. Syllogistic Logic
 13439 Venn Diagrams map three predicates into eight compartments, then look for the conclusion
###### 4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL
 13421 'Disjunctive Normal Form' is ensuring that no conjunction has a disjunction within its scope
 13422 'Conjunctive Normal Form' is ensuring that no disjunction has a conjunction within its scope
###### 4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL
 13356 The 'conditional' is that Γ|=φ→ψ iff Γ,φ|=ψ
 13353 'Negation' says that Γ,¬φ|= iff Γ|=φ
 13354 'Conjunction' says that Γ|=φ∧ψ iff Γ|=φ and Γ|=ψ
 13355 'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,ψ|=
 13350 'Assumptions' says that a formula entails itself (φ|=φ)
 13351 'Thinning' allows that if premisses entail a conclusion, then adding further premisses makes no difference
 13352 'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z
###### 4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL
 13610 A logic with ¬ and → needs three axiom-schemas and one rule as foundation
###### 4. Formal Logic / E. Nonclassical Logics / 6. Free Logic
 13846 A 'free' logic can have empty names, and a 'universally free' logic can have empty domains
###### 5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
 13346 Truth is the basic notion in classical logic
 13545 Elementary logic cannot distinguish clearly between the finite and the infinite
 13822 Fictional characters wreck elementary logic, as they have contradictions and no excluded middle
###### 5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-
 13623 The syntactic turnstile |- φ means 'there is a proof of φ' or 'φ is a theorem'
###### 5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
 13349 Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid'
 13347 Validity is a conclusion following for premises, even if there is no proof
 13348 It seems more natural to express |= as 'therefore', rather than 'entails'
###### 5. Theory of Logic / B. Logical Consequence / 5. Modus Ponens
 13617 MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ
 13614 MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment)
###### 5. Theory of Logic / D. Assumptions for Logic / 4. Identity in Logic
 13799 The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b)
 13800 |= α=α and α=β |= φ(α/ξ ↔ φ(β/ξ) fix identity
 13803 If we are to express that there at least two things, we need identity
###### 5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
 13357 Truth-functors are usually held to be defined by their truth-tables
###### 5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
 13812 A 'zero-place' function just has a single value, so it is a name
 13811 A 'total' function ranges over the whole domain, a 'partial' function over appropriate inputs
###### 5. Theory of Logic / F. Referring in Logic / 1. Naming / a. Names
 13360 In logic, a name is just any expression which refers to a particular single object
###### 5. Theory of Logic / F. Referring in Logic / 1. Naming / e. Empty names
 13361 An expression is only a name if it succeeds in referring to a real object
###### 5. Theory of Logic / F. Referring in Logic / 2. Descriptions / b. Definite descriptions
 13813 Definite descriptions don't always pick out one thing, as in denials of existence, or errors
 13814 Definite desciptions resemble names, but can't actually be names, if they don't always refer
 13816 Because of scope problems, definite descriptions are best treated as quantifiers
 13817 Definite descriptions are usually treated like names, and are just like them if they uniquely refer
 13848 We are only obliged to treat definite descriptions as non-names if only the former have scope
###### 5. Theory of Logic / F. Referring in Logic / 2. Descriptions / c. Theory of definite descriptions
 13815 Names do not have scope problems (e.g. in placing negation), but Russell's account does have that problem
###### 5. Theory of Logic / G. Quantification / 1. Quantification
 13438 'Prenex normal form' is all quantifiers at the beginning, out of the scope of truth-functors
###### 5. Theory of Logic / G. Quantification / 2. Domain of Quantification
 13818 If we allow empty domains, we must allow empty names
###### 5. Theory of Logic / H. Proof Systems / 1. Proof Systems
 13801 An 'informal proof' is in no particular system, and uses obvious steps and some ordinary English
###### 5. Theory of Logic / H. Proof Systems / 2. Axiomatic Proof
 13619 Quantification adds two axiom-schemas and a new rule
 13622 Axiom systems from Frege, Russell, Church, Lukasiewicz, Tarski, Nicod, Kleene, Quine...
###### 5. Theory of Logic / H. Proof Systems / 3. Proof from Assumptions
 13615 'Conditonalised' inferences point to the Deduction Theorem: If Γ,φ|-ψ then Γ|-φ→ψ
 13620 Proof by Assumptions can always be reduced to Proof by Axioms, using the Deduction Theorem
 13621 The Deduction Theorem and Reductio can 'discharge' assumptions - they aren't needed for the new truth
 13616 The Deduction Theorem greatly simplifies the search for proof
###### 5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
 13753 Natural deduction takes proof from assumptions (with its rules) as basic, and axioms play no part
 13755 Excluded middle is an introduction rule for negation, and ex falso quodlibet will eliminate it
 13758 In natural deduction we work from the premisses and the conclusion, hoping to meet in the middle
 13754 Natural deduction rules for → are the Deduction Theorem (→I) and Modus Ponens (→E)
###### 5. Theory of Logic / H. Proof Systems / 5. Tableau Proof
 13611 Tableau proofs use reduction - seeking an impossible consequence from an assumption
 13612 Non-branching rules add lines, and branching rules need a split; a branch with a contradiction is 'closed'
 13613 A completed open branch gives an interpretation which verifies those formulae
 13756 A tree proof becomes too broad if its only rule is Modus Ponens
 13757 Unlike natural deduction, semantic tableaux have recipes for proving things
 13762 Tableau rules are all elimination rules, gradually shortening formulae
 13761 In a tableau proof no sequence is established until the final branch is closed; hypotheses are explored
###### 5. Theory of Logic / H. Proof Systems / 6. Sequent Calculi
 13759 Each line of a sequent calculus is a conclusion of previous lines, each one explicitly recorded
 13760 A sequent calculus is good for comparing proof systems
###### 5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
 13364 Interpretation by assigning objects to names, or assigning them to variables first [PG]
###### 5. Theory of Logic / I. Semantics of Logic / 5. Extensionalism
 13821 Extensionality is built into ordinary logic semantics; names have objects, predicates have sets of objects
 13362 If an object has two names, truth is undisturbed if the names are swapped; this is Extensionality
###### 5. Theory of Logic / K. Features of Logics / 2. Consistency
 13541 For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ
 13542 A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula
 13540 A set of formulae is 'inconsistent' when there is no interpretation which can make them all true
###### 5. Theory of Logic / K. Features of Logics / 6. Compactness
 13544 Inconsistency or entailment just from functors and quantifiers is finitely based, if compact
 13618 Compactness means an infinity of sequents on the left will add nothing new
###### 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
 13359 Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers
 13358 Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all
###### 8. Modes of Existence / A. Relations / 4. Formal Relations / a. Types of relation
 13802 Relations can be one-many (at most one on the left) or many-one (at most one on the right)
 13543 A relation is not reflexive, just because it is transitive and symmetrical
###### 9. Objects / F. Identity among Objects / 5. Self-Identity
 13847 If non-existent things are self-identical, they are just one thing - so call it the 'null object'
###### 10. Modality / A. Necessity / 6. Logical Necessity
 13820 The idea that anything which can be proved is necessary has a problem with empty names
###### 19. Language / C. Assigning Meanings / 3. Predicates
 13363 A (modern) predicate is the result of leaving a gap for the name in a sentence