Extended syllogistics Robert van Rooij ILLC 1
Reverse the standard picture Standard 1. Start with Propositional logic 2. Extend to Predicate logic, 3. Show that Syllogistics is insignificant part Should be (and used to be): 1. Start with Syllogistics 2. Extend to Propositional Logic 3. Show that Predicate logic is (almost) natural result 2
Aristotelian syllogistics • Terms: S , T , P , M (1-place-predicates) • Sentences: ∀ x [ Sx → Px ] – SaP ∃ x [ Sx ∧ Px ] – SiP – SeP negation of SaP – SoP negation of SiP • Syllogism: two premisses, one conclusion: Major premise ( P ), Minor premise ( S ), Conclusion S − P 3
Proof theory 1. The first 4 valid Syllogisms of the first figure: (a) Barbara 1 : MaP, SaM ⊢ SaP (b) Celarent 1 : MeP, SaM ⊢ SeP (c) Darii 1 : MaP, SiM ⊢ SiP (d) Ferio 1 : MeP, SiM ⊢ SoP 2. Law of Identity: ⊢ TaT . 3. Existential Import: ⊢ TiT . 4. Reductio per impossible : if the conclusion of a syllogism is false, at least one of the premisses is false as well. Γ , ¬ φ ⊢ ψ, ¬ ψ ⇒ Γ ⊢ φ . 4
Dictum de Omni = substitution principle Dictum de Omni : Whatever is truly said of all, is truly said of which that subject is affirmatively predicated (Barbara, Darii). Formally, DDO: MaP, Γ( M ) + ⊢ Γ( P ), with Γ( M ) + a sentence where M occurs not distributed S − aP + , S + iP + S + oP − S − eP − Dictum de Nullo implements Celarent and Ferio 5
Negative terms The proof system SYL of syllogistic reasoning consists of the following set of axioms and rules: (1) MaP, Γ( M ) + ⊢ Γ( P ) Dictum de Omni (2) ⊢ TaT Law of identity (3) ⊢ T ≡ T Double negation (4) SaP ⊢ PaS Contraposition (5) Γ , ¬ φ ⊢ ψ, ¬ ψ ⇒ Γ ⊢ φ Reductio per impossible (f) ⊢ ¬ ( TaT ) (or TiT ) Existential import 6
Empty and Transcedental terms Introduce ⊤ . Naturally valid: Sa ⊤ . By contraposition: ⊤ aS . But problem with TiT (existential import) (6) ⊢ ¬ ( TaT ), for all positive categorical terms T (7) ⊢ Sa ⊤ (8) SaS ⊢ SaP (only if S empty) Rule (8) doesn’t seem of much use, but will be important later. 7
Singular terms Traditional: Socrates is mortal, SaM . But still seems that SeM is contradictory with SaM . How is it that opposition is valid in the case of singular propositions [...] since elsewhere a univeral affirmative and a particular negative are opposed. Should we say that a singular proposition is equivalent to a particular and to a universal proposition? Yes, we should. (Leibniz) (9) for all singular terms I and terms P : IiP − | ⊢ IaP . ‘Everybody is smart, thus Plato is smart’ ⊤ aS, Pa ⊤ ⊢ DDO PaS . 8
Boolean algebra Leibniz: allow for Conjunctive terms If S and T are terms, ST is also a term. Interpretation: V M ( ST ) = V M ( S ) ∩ V M ( T ) SaPQ − | ⊢ SaP and SaQ Proof theory: (given by Leibniz!) This generates all of Boolean algebra! (invented by Leibniz!) and thus propositional logic 9
Propositional Logic Think of sentences as 0-ary terms. Interpretation: Denotation of any n -ary term is subset of D n . Thus D 0 = {��} . D 0 has two subsets: {��} and ∅ . If φ denotes {��} it is true, and false otherwise ( V M ( ⊤ ) = {��} ). • V M ( SaP ) = {�� : V M ( S ) ⊆ V M ( P ) } • V M ( SiP ) = {�� : V M ( S ) ∩ V M ( P ) � = ∅} • V M ( SeP ) = {�� : V M ( S ) ∩ V M ( P ) = ∅} • V M ( SoP ) = {�� : V M ( S ) − V M ( P ) � = ∅} 10
Propositional Logic: 2 Now we can form sentences like [ φ ] a [ ψ ] , [ φ ] i [ ψ ] , [ φ ] e [ ψ ] and [ φ ] o [ ψ ]. ‘ φ → ψ ’ ≡ [ φ ] a [ ψ ] ‘ φ ∧ ψ ≡ [ φ ] i [ ψ ], ‘ ¬ φ ’ ≡ [ φ ] e [ φ ], ‘ φ ∨ ψ ’ ≡ [ ¬ φ ] a [ ψ ] . Notice that because for 0-ary relation φ it holds that V M ( φ ) = {��} iff V M ([ ⊤ 0 ] i [ φ ]) = {��} , we can write ‘[ φ ]’ also as ‘[ ⊤ 0 ] i [ φ ]’. (10) 0-ary terms are not categorical and ⊤ 0 is a singular term. (11) P 0 − | ⊢ ⊤ 0 iP 0 . (follows: ⊤ 0 aP 0 ) 11
Propositional Logic: 3 • Modus Ponens: φ, φ → ψ ⊢ ψ [ ⊤ ] a [ φ ] , [ φ ] a [ ψ ] ⊢ DDO [ ⊤ ] a [ ψ ] • Modus Tollens: ¬ ψ, φ → ψ ⊢ ¬ φ [ ⊤ ] e [ ψ ] , [ φ ] a [ ψ ] ⊢ DDN [ ⊤ ] e [ φ ] or with DDO: [ ⊤ ] e [ ψ ] ⊢ Contrap [ ψ ] a [ ⊥ ] , [ φ ] a [ ψ ] ⊢ DDO [ φ ] a [ ⊥ ] • Hypothetical Syllogism: φ → ψ, ψ → χ ⊢ φ → χ : [ φ ] a [ ψ ] , [ ψ ] a [ χ ] ⊢ DDO [ φ ] a [ χ ] • Disjunctive Syllogism ( φ ∨ ψ, ¬ φ ⊢ ψ ): ([ ⊤ ] e [ φ ]) a [ ψ ] , [ ⊤ ] a ([ ⊤ ] e [ φ ]) ⊢ DDO [ ⊤ ] a [ ψ ]. 12
Propositional Logic: 4 • ‘ p ⊢ p ∨ p ’ ( p ≡ ⊤ ap pa ⊤ , ⊤ ap ⊢ DDO pap ) • ‘ p ∨ p ⊢ p ’, pap by (8): pa ⊥ . Via contraposition and double negation: ⊤ ap . Because ⊤ is a singular term it follows that p • ‘ p ∨ q ⊢ q ∨ p ’ (by contraposition and double negation) • ‘ p → q ⊢ ( r → p ) → ( r → q )’. follows, because one can proof the deduction theorem : Γ , p ⊢ q ⇒ Γ ⊢ paq But this is enough to show that our extended syllogistics incorporates propositional logic!! 13
Relations Syntax : Add new terms, i.e. n ary relations. • Combine monadic term S with 1-ary term P (and connective ‘ a ’, for instance) ❀ 0-ary term ( S 1 aP 1 ) 0 . • Combine monadic term S with n -ary term/relation R (and connective ‘ a ’) ❀ n − 1-ary term ( S 1 aR n ) n − 1 . Semantics ( S 1 aR n ) n − 1 : {� d 1 , ..., d n − 1 � : V M ( S ) ⊆ { d n ∈ D : � d 1 , ..., d n � ∈ V M ( R n ) } . 14
Relational sentences • Every man loves a woman: Ma ( WiL 2 ) • V M ( Ma ( WiL 2 )) = {�� : I ( M ) ⊆ { d ∈ D : � d � ∈ V M ( WiL 2 ) } , where V M ( WiL 2 ) = {� d 1 � : I ( W ) ∩ { d 2 ∈ D : � d 1 , d 2 � ∈ I ( L 2 ) } � = ∅} . • There is woman who is loved by every man: Wi ( MaL ∪ ) • where L ∪ is the passive form of ‘love’: being loved by . • In general, V M ( R ∪ ) = {� d 2 , d 1 � : � d 1 , d 2 � ∈ I M ( R ) } 15
Reasoning with Relations 1 • Aristotle: ‘All wisdom is knowledge, Every good thing is object of some wisdom, thus, Every good thing is object of some knowledge’. • WaK, Ga ( WiR ) ⊢ Ga ( KiR ), with ‘ R ’ standing for ‘is object of’. • Follows by Dictum de Omni, if ‘ W ’ occurs positively in ‘ Ga ( WiR )’! • If P positive in Γ, then P negative in Γ, otherwise positively. If ( SaR ) positive in Γ, then S − aR + , otherwise S + aR − . If ( SiR ) positive in Γ, then S + iR + , otherwise S − iR − . 16
Reasoning with Relations 2 • Leibniz: ‘Every thing which is a painting is an art (or shorter, painting is an art), thus everyone who learns a thing which is a painting learns a thing which is an art’ (or shorter: everyone who learns painting learns an art). • PaA ⊢ ( PiL 2 ) a ( AiL 2 ). • Leibniz: can account for if we add the extra (and tautological) premiss ‘Everybody who learns a thing which is a painting learns a thing which is a painting’, i.e. ( PiL 2 ) a ( PiL 2 ). • Now ( PiL 2 ) a ( AiL 2 ) follows from PaA and ( PiL 2 ) a ( PiL 2 ) by means of the Dictum the Omni 17
Reasoning with Relations 3 • Frege: There is woman who is loved by every man: Wi ( MaL ∪ ), thus Every man loves a woman: Ma ( WiL 2 ) (13) Oblique Conversion: Sa ( S ′ aR ) ≡ S ′ a ( SaR ∪ ) from ‘every man loves every woman’ we infer that ‘every woman is loved by every man’ . R ∪∪ ≡ R (14) Double passive: (1 ′ ) Dictum de Omni : Γ( MaR ) + , Θ( M ) + ⊢ Γ(Θ( R )) 18
Reasoning with Relations 4 1. Wi ( MaL ∪ ) premiss 2. ( MaL ∪ ) a ( MaL ∪ ) a tautology (everyone loved by every man is loved by every man) 3. Ma (( MaL ∪ ) aL ∪∪ ) from 2 and (13) (with S = ( MaL ∪ ) and S ′ = M ) 4. Ma (( MaL ∪ ) aL ) by 3 and (14), substitution of L for L ∪∪ 5. Ma ( WiL ) by 1 and 4, by Dictum de Omni (1 ′ ) 19
Relation with predicate logic 1. This is not yet FOL, but an interesting (variable-free) fragment 2. Quine, Purdy: a (the maximal?) decidable fragment of it. 3. (At least) three men are sick’ Numerical quantors i n , a n ❀ 4. ‘All parents love their children ’ ❀ Quinean cropping 20
Why interesting? • Historical interest (or counterfactual history) • Decidability and Complexity • More natural for natural language (no misleading form) • Suggest different ontology/model theory 1. Mereology instead of set theory 2. Intensional interpretation (predicate in subject) 21
Intensional semantics for syllogistics with complex terms Robert van Rooij ILLC 22
It is an old dispute whether formal logic should concern itself mainly with intensions or extensions. In general, logicians whose training was mainly philosophical have decided for intensions, while those whose training was mainly mathematical have decided for extensions. (intorduction to second edition of Principia Mathematica, Russell & Whitehead) 23
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