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Single Idea 13357
[filed under theme 5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
]
Full Idea
The usual view of the meaning of truth-functors is that each is defined by its own truth-table, independently of any other truth-functor.
Gist of Idea
Truth-functors are usually held to be defined by their truth-tables
Source
David Bostock (Intermediate Logic [1997], 2.7)
Book Ref
Bostock,David: 'Intermediate Logic' [OUP 1997], p.46
The
72 ideas
from 'Intermediate Logic'
13821
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Extensionality is built into ordinary logic semantics; names have objects, predicates have sets of objects
[Bostock]
|
13346
|
Truth is the basic notion in classical logic
[Bostock]
|
13347
|
Validity is a conclusion following for premises, even if there is no proof
[Bostock]
|
13348
|
It seems more natural to express |= as 'therefore', rather than 'entails'
[Bostock]
|
13349
|
Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid'
[Bostock]
|
13350
|
'Assumptions' says that a formula entails itself (φ|=φ)
[Bostock]
|
13351
|
'Thinning' allows that if premisses entail a conclusion, then adding further premisses makes no difference
[Bostock]
|
13352
|
'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z
[Bostock]
|
13353
|
'Negation' says that Γ,¬φ|= iff Γ|=φ
[Bostock]
|
13354
|
'Conjunction' says that Γ|=φ∧ψ iff Γ|=φ and Γ|=ψ
[Bostock]
|
13355
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'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,ψ|=
[Bostock]
|
13356
|
The 'conditional' is that Γ|=φ→ψ iff Γ,φ|=ψ
[Bostock]
|
13421
|
'Disjunctive Normal Form' is ensuring that no conjunction has a disjunction within its scope
[Bostock]
|
13422
|
'Conjunctive Normal Form' is ensuring that no disjunction has a conjunction within its scope
[Bostock]
|
13357
|
Truth-functors are usually held to be defined by their truth-tables
[Bostock]
|
13359
|
Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers
[Bostock]
|
13358
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Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all
[Bostock]
|
13361
|
An expression is only a name if it succeeds in referring to a real object
[Bostock]
|
13360
|
In logic, a name is just any expression which refers to a particular single object
[Bostock]
|
13362
|
If an object has two names, truth is undisturbed if the names are swapped; this is Extensionality
[Bostock]
|
13363
|
A (modern) predicate is the result of leaving a gap for the name in a sentence
[Bostock]
|
13364
|
Interpretation by assigning objects to names, or assigning them to variables first
[Bostock, by PG]
|
13438
|
'Prenex normal form' is all quantifiers at the beginning, out of the scope of truth-functors
[Bostock]
|
13439
|
Venn Diagrams map three predicates into eight compartments, then look for the conclusion
[Bostock]
|
13612
|
Non-branching rules add lines, and branching rules need a split; a branch with a contradiction is 'closed'
[Bostock]
|
13611
|
Tableau proofs use reduction - seeking an impossible consequence from an assumption
[Bostock]
|
13541
|
For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ
[Bostock]
|
13540
|
A set of formulae is 'inconsistent' when there is no interpretation which can make them all true
[Bostock]
|
13542
|
A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula
[Bostock]
|
13613
|
A completed open branch gives an interpretation which verifies those formulae
[Bostock]
|
13543
|
A relation is not reflexive, just because it is transitive and symmetrical
[Bostock]
|
13544
|
Inconsistency or entailment just from functors and quantifiers is finitely based, if compact
[Bostock]
|
13545
|
Elementary logic cannot distinguish clearly between the finite and the infinite
[Bostock]
|
13623
|
The syntactic turnstile |- φ means 'there is a proof of φ' or 'φ is a theorem'
[Bostock]
|
13610
|
A logic with ¬ and → needs three axiom-schemas and one rule as foundation
[Bostock]
|
13616
|
The Deduction Theorem greatly simplifies the search for proof
[Bostock]
|
13614
|
MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment)
[Bostock]
|
13617
|
MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ
[Bostock]
|
13615
|
'Conditonalised' inferences point to the Deduction Theorem: If Γ,φ|-ψ then Γ|-φ→ψ
[Bostock]
|
13618
|
Compactness means an infinity of sequents on the left will add nothing new
[Bostock]
|
13620
|
Proof by Assumptions can always be reduced to Proof by Axioms, using the Deduction Theorem
[Bostock]
|
13619
|
Quantification adds two axiom-schemas and a new rule
[Bostock]
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13621
|
The Deduction Theorem and Reductio can 'discharge' assumptions - they aren't needed for the new truth
[Bostock]
|
13622
|
Axiom systems from Frege, Russell, Church, Lukasiewicz, Tarski, Nicod, Kleene, Quine...
[Bostock]
|
13753
|
Natural deduction takes proof from assumptions (with its rules) as basic, and axioms play no part
[Bostock]
|
13755
|
Excluded middle is an introduction rule for negation, and ex falso quodlibet will eliminate it
[Bostock]
|
13754
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Natural deduction rules for → are the Deduction Theorem (→I) and Modus Ponens (→E)
[Bostock]
|
13756
|
A tree proof becomes too broad if its only rule is Modus Ponens
[Bostock]
|
13757
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Unlike natural deduction, semantic tableaux have recipes for proving things
[Bostock]
|
13758
|
In natural deduction we work from the premisses and the conclusion, hoping to meet in the middle
[Bostock]
|
13759
|
Each line of a sequent calculus is a conclusion of previous lines, each one explicitly recorded
[Bostock]
|
13760
|
A sequent calculus is good for comparing proof systems
[Bostock]
|
13761
|
In a tableau proof no sequence is established until the final branch is closed; hypotheses are explored
[Bostock]
|
13762
|
Tableau rules are all elimination rules, gradually shortening formulae
[Bostock]
|
13801
|
An 'informal proof' is in no particular system, and uses obvious steps and some ordinary English
[Bostock]
|
13800
|
|= α=α and α=β |= φ(α/ξ ↔ φ(β/ξ) fix identity
[Bostock]
|
13803
|
If we are to express that there at least two things, we need identity
[Bostock]
|
13799
|
The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b)
[Bostock]
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13802
|
Relations can be one-many (at most one on the left) or many-one (at most one on the right)
[Bostock]
|
13812
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A 'zero-place' function just has a single value, so it is a name
[Bostock]
|
13811
|
A 'total' function ranges over the whole domain, a 'partial' function over appropriate inputs
[Bostock]
|
13813
|
Definite descriptions don't always pick out one thing, as in denials of existence, or errors
[Bostock]
|
13814
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Definite desciptions resemble names, but can't actually be names, if they don't always refer
[Bostock]
|
13816
|
Because of scope problems, definite descriptions are best treated as quantifiers
[Bostock]
|
13817
|
Definite descriptions are usually treated like names, and are just like them if they uniquely refer
[Bostock]
|
13815
|
Names do not have scope problems (e.g. in placing negation), but Russell's account does have that problem
[Bostock]
|
13820
|
The idea that anything which can be proved is necessary has a problem with empty names
[Bostock]
|
13818
|
If we allow empty domains, we must allow empty names
[Bostock]
|
13822
|
Fictional characters wreck elementary logic, as they have contradictions and no excluded middle
[Bostock]
|
13846
|
A 'free' logic can have empty names, and a 'universally free' logic can have empty domains
[Bostock]
|
13847
|
If non-existent things are self-identical, they are just one thing - so call it the 'null object'
[Bostock]
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13848
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We are only obliged to treat definite descriptions as non-names if only the former have scope
[Bostock]
|