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Single Idea 13363

[filed under theme 19. Language / C. Assigning Meanings / 3. Predicates ]

Full Idea

A simple way of approaching the modern notion of a predicate is this: given any sentence which contains a name, the result of dropping that name and leaving a gap in its place is a predicate. Very different from predicates in Aristotle and Kant.

Gist of Idea

A (modern) predicate is the result of leaving a gap for the name in a sentence

Source

David Bostock (Intermediate Logic [1997], 3.2)

Book Ref

Bostock,David: 'Intermediate Logic' [OUP 1997], p.74

A Reaction

This concept derives from Frege. To get to grips with contemporary philosophy you have to relearn all sorts of basic words like 'predicate' and 'object'.

The 72 ideas from 'Intermediate Logic'

13821 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 '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]
13358 Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all [Bostock]
13359 Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers [Bostock]
13362 If an object has two names, truth is undisturbed if the names are swapped; this is Extensionality [Bostock]
13360 In logic, a name is just any expression which refers to a particular single object [Bostock]
13361 An expression is only a name if it succeeds in referring to a real object [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]
13540 A set of formulae is 'inconsistent' when there is no interpretation which can make them all true [Bostock]
13541 For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ [Bostock]
13542 A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula [Bostock]
13543 A relation is not reflexive, just because it is transitive and symmetrical [Bostock]
13613 A completed open branch gives an interpretation which verifies those formulae [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]
13617 MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ [Bostock]
13614 MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment) [Bostock]
13616 The Deduction Theorem greatly simplifies the search for proof [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]
13619 Quantification adds two axiom-schemas and a new rule [Bostock]
13620 Proof by Assumptions can always be reduced to Proof by Axioms, using the Deduction Theorem [Bostock]
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 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 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]
13762 Tableau rules are all elimination rules, gradually shortening formulae [Bostock]
13761 In a tableau proof no sequence is established until the final branch is closed; hypotheses are explored [Bostock]
13801 An 'informal proof' is in no particular system, and uses obvious steps and some ordinary English [Bostock]
13802 Relations can be one-many (at most one on the left) or many-one (at most one on the right) [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]
13812 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 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]
13848 We are only obliged to treat definite descriptions as non-names if only the former have scope [Bostock]