Ideas from 'Intermediate Logic' by David Bostock [1997], 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
Venn Diagrams map three predicates into eight compartments, then look for the conclusion
Aristotle's said some Fs are G or some Fs are not G, forgetting that there might be no Fs
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
'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
'Assumptions' says that a formula entails itself (φ|=φ)
'Thinning' allows that if premisses entail a conclusion, then adding further premisses makes no difference
'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z
'Negation' says that Γ,φ|= iff Γ|=φ
'Conjunction' says that Γ|=φ∧ψ iff Γ|=φ and Γ|=ψ
'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,ψ|=
The 'conditional' is that Γ|=φ→ψ iff Γ,φ|=ψ
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
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
5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
Truth is the basic notion in classical logic
Elementary logic cannot distinguish clearly between the finite and the infinite
Fictional characters wreck elementary logic, as they have contradictions and no excluded middle
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'
5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
Validity is a conclusion following for premises, even if there is no proof
It seems more natural to express |= as 'therefore', rather than 'entails'
Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid'
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)
MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ
5. Theory of Logic / D. Assumptions for Logic / 4. Identity in Logic
|= α=α and α=β |= φ(α/ξ ↔ φ(β/ξ) fix identity
If we are to express that there at least two things, we need identity
The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b)
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
5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
A 'total' function ranges over the whole domain, a 'partial' function over appropriate inputs
A 'zero-place' function just has a single value, so it is a name
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
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
5. Theory of Logic / F. Referring in Logic / 2. Descriptions / b. Definite descriptions
Definite descriptions don't always pick out one thing, as in denials of existence, or errors
We are only obliged to treat definite descriptions as non-names if only the former have scope
Definite desciptions resemble names, but can't actually be names, if they don't always refer
Because of scope problems, definite descriptions are best treated as quantifiers
Definite descriptions are usually treated like names, and are just like them if they uniquely refer
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
5. Theory of Logic / G. Quantification / 1. Quantification
'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
If we allow empty domains, we must allow empty names
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
5. Theory of Logic / H. Proof Systems / 2. Axiomatic Proof
Quantification adds two axiom-schemas and a new rule
Axiom systems from Frege, Russell, Church, Lukasiewicz, Tarski, Nicod, Kleene, Quine...
5. Theory of Logic / H. Proof Systems / 3. Proof from Assumptions
The Deduction Theorem greatly simplifies the search for proof
'Conditonalised' inferences point to the Deduction Theorem: If Γ,φ|-ψ then Γ|-φ→ψ
Proof by Assumptions can always be reduced to Proof by Axioms, using the Deduction Theorem
The Deduction Theorem and Reductio can 'discharge' assumptions - they aren't needed for the new truth
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
Excluded middle is an introduction rule for negation, and ex falso quodlibet will eliminate it
In natural deduction we work from the premisses and the conclusion, hoping to meet in the middle
Natural deduction rules for → are the Deduction Theorem (→I) and Modus Ponens (→E)
5. Theory of Logic / H. Proof Systems / 5. Tableau Proof
Tableau proofs use reduction - seeking an impossible consequence from an assumption
Non-branching rules add lines, and branching rules need a split; a branch with a contradiction is 'closed'
A completed open branch gives an interpretation which verifies those formulae
In a tableau proof no sequence is established until the final branch is closed; hypotheses are explored
Tableau rules are all elimination rules, gradually shortening formulae
Unlike natural deduction, semantic tableaux have recipes for proving things
A tree proof becomes too broad if its only rule is Modus Ponens
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
A sequent calculus is good for comparing proof systems
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
Interpretation by assigning objects to names, or assigning them to variables first
5. Theory of Logic / I. Semantics of Logic / 6. Extensionalism
Extensionality is built into ordinary logic semantics; names have objects, predicates have sets of objects
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
A set of formulae is 'inconsistent' when there is no interpretation which can make them all true
A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula
For 'negation-consistent', there is never |-(S)φ and |-(S)φ
5. Theory of Logic / K. Features of Logics / 6. Compactness
Inconsistency or entailment just from functors and quantifiers is finitely based, if compact
Compactness means an infinity of sequents on the left will add nothing new
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / f. Mathematical induction
Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all
Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
Mathematics has no special axioms of its own, but follows from principles of logic (with definitions)
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)
A relation is not reflexive, just because it is transitive and symmetrical
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'
10. Modality / A. Necessity / 6. Logical Necessity
The idea that anything which can be proved is necessary has a problem with empty names
19. Language / A. Language / 6. Predicates
A (modern) predicate is the result of leaving a gap for the name in a sentence