Nominalistic Logic (Extended Abstract)

Nominalistic Logic (Extended Abstract) Jørgen Villadsen Department of Informatics and Mathematical Modeling T ec hnical U niversi ty of Denmark Nominalistic Logic (NL) is a ne w pres ent ation o f Paul Gilmore’s Int ensional Type Theory (ITT) as a sequent calculus tog ether with a succinct nominal- ization a xiom (N) that permits names of predic a tes as individuals in cer tain cases. The logic has a flexible comprehension a x iom, but no extensionality axiom and no infinity axiom, although ax iom N is the key to the deriv ation of Peano’s p ostulates for the natur a l n um bers. References: ITT Paul Gilmore: An In tensional T yp e The ory: Motivation and Cu t-Elimination. Journal of Symbolic Logic 383-4 0 0 200 1 . NL Jørgen Villa dsen: Nominalistic L o gic: F r om Naive S et The ory to Int ensional T yp e T he ory. P ages 57 –85 in Klaus Rob ering (editor ): New Appr oaches to Classes and Concepts. V olume 14, Studies in Lo g ic, College Publications 200 8 . c and x, y , z , . . . , x n , y n , z n , . . . range ov er cons tant s and coun tably infinitely many v ariables, respec tively , and p, q , r, s, t, . . . , p n , q n , r n , s n , t n , . . . range ov er terms pro duced by the g r ammar: t : : = pt | λx.p | x | c pt 1 · · · t n stands for ( pt 1 ) · · · t n λx 1 · · · x n .p stands for λx 1 . · · · λx n .p (used for all v ar iable-binding op erators ). A v ar iable x occurs b ound (free) in a term if x is (not) in the scope of a λx . The term p [ t/x ] is p with ev ery free occur rence of x replaced wit h t . α -renaming of the v ar iable x to the v ariable y in λx.p is λy .p [ y / x ] assuming y is not free in p and does not b ecome bo und in p . β -r eduction of ( λx.p ) t is p [ t/x ] assuming no free v ariable in t bec o mes bo und. η -reduction of λx.px is p assuming x is not free in p . s ∼ t if t is s with a series of α -renamings. s ≻ t if t is s with either a β -reduction or an η -re ductio n. 1 τ , . . . , τ n , . . . and σ , . . . , σ n , . . . range o v er t ypes and predicate types, resp ectively , pr o duced b y the grammar: τ : : = σ | ı σ : : = τ σ | o τ 1 · · · τ n σ stands for τ 1 · · · ( τ n σ ) It is assumed that ev ery constant and v ariable ha s a unique t yp e such that there a re infinitely man y v ariables for each t ype The type system t : τ deter mines that a term t has t ype τ by the conditions: x : τ if x has t ype τ (v ar ) c : τ if c has t ype τ (con) p : τ σ and t : τ gives pt : σ (app) x : τ and p : σ gives λx.p : τ σ (abs) p : ı if p is nominalizable (nom) p is nominalizable if p : σ and x : ı for every f ree v ar iable x in p p is a form ula if p : o (a sentence is a closed form ula) There is a constant of type ooo and for every type τ a constan t of type ( τ o ) o The constants and v ariables m ust ha ve appro priate t ypes in the abbreviations : ¬ p : = cpp Here c means “neither . . . nor . . . ” . p ∨ q : = ¬ cpq Ditto . p ∧ q : = ¬ ( ¬ p ∨ ¬ q ) ∃ x.p : = ¬ cλx.p Here c means “no . . . exists”. ∀ x.p : = ¬∃ x. ¬ p s 6 = t : = ( λxy . ∃ z .z x ∧ ¬ z y ) st s = t : = ¬ ( s 6 = t ) p → q : = ¬ p ∨ q p ↔ q : = ( p → q ) ∧ ( q → p ) The op erator priority is as follows from high to lo w: ¬ = ∧ ∨ → ↔ λ . V ariable-binding oper ators, lik e ∃ and ∀ , ha v e the sa me priority a s λ . Other op erators , like 6 =, ha v e the same priority as = (or ¬ if unary). ∨ and ∧ are left asso c iative and → and ↔ are right ass o ciative. . = and 6 . = are similar to = and 6 =, respectively , but alw ays has t ype ııo . 2 Θ , Γ a nd ∆ rang e o ver p o s sible empty formula sequences The sequen t calculus Γ ⊢ ∆ ha s sequences on both sides in the rules: p ⊢ q if p ∼ q or p ≻ q (S) Γ ⊢ ∆ = = = = = = = = = = = = = = = = = = p, Γ ⊢ ∆ Γ ⊢ ∆, p (T) Θ, p, q , Γ ⊢ ∆ Θ, q , p, Γ ⊢ ∆ Γ ⊢ ∆, p, q , Θ Γ ⊢ ∆, q , p, Θ (E) p, p, Γ ⊢ ∆ p, Γ ⊢ ∆ Γ ⊢ ∆, p, p Γ ⊢ ∆, p (C) Γ ⊢ ∆, p, q cpq , Γ ⊢ ∆ ⊢ p , q , cpq (P) cp, pt ⊢ px, Γ ⊢ ∆ x not free in p , Γ or ∆ Γ ⊢ ∆, cp (Q) ⊢ p = q ↔ p . = q (N) The left and right rules are placed next to each o ther. Double lines indicate that the rule w orks in both directio ns. Multiple conclusio ns indicate multiple rules eac h with a sing le conclusion. ⊤ : = ∃ x.x T ruth ⊥ : = ¬⊤ F alsit y s ≡ t : = ( λxy . ∀ z 1 · · · z n .xz 1 · · · z n ↔ y z 1 · · · z n ) st Equiv alence s 6≡ t : = ¬ ( s ≡ t ) t ′ : = ( λxy .x . = y ) t Successor 0 : = λx.x 6 . = x Zero 1 : = 0 ′ 2 : = 1 ′ 3 : = 2 ′ . . . 3