Extended syllogistics Robert van Rooij ILLC 1 Reverse the - - PowerPoint PPT Presentation

Extended syllogistics

Robert van Rooij ILLC

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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

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Aristotelian syllogistics

• Terms: S, T, P, M (1-place-predicates)
• Sentences:

– SaP ∀x[Sx → Px] – SiP ∃x[Sx ∧ Px] – SeP negation of SaP – SoP negation of SiP

• Syllogism: two premisses, one conclusion:

Major premise (P), Minor premise (S), Conclusion S − P

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Proof theory

• 1. The first 4 valid Syllogisms of the first figure:

(a) Barbara1: MaP, SaM ⊢ SaP (b) Celarent1: MeP, SaM ⊢ SeP (c) Darii1: MaP, SiM ⊢ SiP (d) Ferio1: 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.

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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−eP − S+oP − Dictum de Nullo implements Celarent and Ferio

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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

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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.

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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.

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Boolean algebra

Leibniz: allow for Conjunctive terms If S and T are terms, ST is also a term. Interpretation: VM(ST) = VM(S) ∩ VM(T) Proof theory: SaPQ − | ⊢ SaP and SaQ (given by Leibniz!) This generates all of Boolean algebra! (invented by Leibniz!) and thus propositional logic

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Propositional Logic

Think of sentences as 0-ary terms. Interpretation: Denotation of any n-ary term is subset of Dn. Thus D0 = {}. D0 has two subsets: {} and ∅. If φ denotes {} it is true, and false otherwise (VM(⊤) = {}).

• VM(SaP) = { : VM(S) ⊆ VM(P)}
• VM(SiP) = { : VM(S) ∩ VM(P) = ∅}
• VM(SeP) = { : VM(S) ∩ VM(P) = ∅}
• VM(SoP) = { : VM(S) − VM(P) = ∅}

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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 VM(φ) = {} iff VM([⊤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 − | ⊢ ⊤0iP 0. (follows: ⊤0aP 0)

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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]

• r 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].

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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!!

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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 (S1aP 1)0.

• Combine monadic term S with n-ary term/relation R (and

connective ‘a’) ❀ n − 1-ary term (S1aRn)n−1. Semantics (S1aRn)n−1: {d1, ..., dn−1 : VM(S) ⊆ {dn ∈ D : d1, ..., dn ∈ VM(Rn)}.

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Relational sentences

• Every man loves a woman: Ma(WiL2)
• VM(Ma(WiL2)) = { : I(M) ⊆ {d ∈ D : d ∈ VM(WiL2)},

where VM(WiL2) = {d1 : I(W) ∩ {d2 ∈ D : d1, d2 ∈ I(L2)} = ∅}.

• There is woman who is loved by every man: Wi(MaL∪)
• where L∪ is the passive form of ‘love’: being loved by.
• In general, VM(R∪) = {d2, d1 : d1, d2 ∈ IM(R)}

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Reasoning with Relations 1

• Aristotle: ‘All wisdom is knowledge, Every good thing is object
• f some wisdom, thus, Every good thing is object of some

knowledge’.

• WaK, Ga(WiR) ⊢ Ga(KiR), with ‘R’ standing for ‘is object
• f’.
• 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−.

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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 ⊢ (PiL2)a(AiL2).
• 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. (PiL2)a(PiL2).

• Now (PiL2)a(AiL2) follows from PaA and (PiL2)a(PiL2) by

means of the Dictum the Omni

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Reasoning with Relations 3

• Frege: There is woman who is loved by every man:

Wi(MaL∪), thus Every man loves a woman: Ma(WiL2) (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’ . (14) Double passive: R∪∪ ≡ R (1′) Dictum de Omni: Γ(MaR)+, Θ(M)+ ⊢ Γ(Θ(R))

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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′)

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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 in, an

• 4. ‘All parents love their children’

❀ Quinean cropping

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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)

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Intensional semantics for syllogistics with complex terms

Robert van Rooij ILLC

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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)

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How to teach philosophical logic and semantics

Logic:

• 1. Start with Syllogistics (not with propositional logic)
• 2. Extend naturally to Propositional Logic and Boolean algebra

(add complex terms to SL with corresponding rules/axioms)

• 3. Extend further to natural decidable part of Predicate logic

(by adding relation terms to SL and some extra rules) Semantics

• 1. Start with minimal extensional models (partial orders)
• 2. Extend them to account for negative, and conjunctive terms
• 3. Follow with more natural intensional models

(as they were in vogue long before Frege)

• 4. Deal with relational terms.

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This talk

• Concentrate only on semantics
• Especially on the Leibnizian intensional models,
• and how to deal with the problem Leibniz could not solve:

How hey should be extended to deal with complex terms.

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Syllogistic structure

• Two terms, four types of sentences, syllogism

SaP (All S are P) SiP (some S are P)

• Modern: Use Set theory (Venn Diagrams), but not required
• U, ≤, M s.t.

U, ≤ is a partially ordered structure with minimal elements M (U, ∧ is a meet semi-lattice determined by U, ≤, M.)

• I is a function mapping each term T to an element of U − M
• I(SaP) = 1 iff I(S) ≤ I(P)

(SoP = 1 iff I(SaP) = 1) I(SiP) = 1 iff glb{I(S), I(P)} ∈ M (SeP = 1 iff I(SiP) = 1)

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Intensional semantics: Leibniz’ Predicate in Subject, trial I

• Leibniz used characteristic numbers, but isomorphic to this
• F a set of basic features, for each term T, I(T) ⊆ F.
• I(SaP) = 1

iff I(P) ⊆ I(S). I(SiP) = 1 iff I(S) ∩ I(P) = ∅.

• This didn’t work, because
• 1. SaP intensional (predicate in subject),
• 2. but SiP extensional.
• Can improve if also SiP given intensional semantics.

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Leibniz’ successfull trial (simple frgmnt)

• I(P) = P +, P −, where P +/P − is set of basic features any
• bject instantiating P must/cannot have
• P + ∩ P − = ∅

P ⊑ Q iff P + ⊆ Q+ and P − ⊑ Q− P|Q iff P + ∩ Q− = ∅ or P − ∩ Q+ = ∅

• Individual I∗: maximal set of positive & negative features.
• SaP

iff P ⊑ S SeP iff S|P ⇒ SiP iff not S|P iff there is an I∗: S ⊑ I∗ and P ⊑ I∗.

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Goals of this talk

Make Leibniz’ dream come true: A: Give intensional semantics analogue of Leibniz’ earlier trials ‘challenge’: how to give SiP an intensional semantics? B: Extend this to account for complex terms C: Give pair-like intensional analysis for complex terms

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A: Improve on Leibniz’ 1st trials

• Idea: Add an incompatibility relation ⊥ between features.
• ∆ = set of subsets of F which contain such mutually

incompatible elements: ∆ = {S ⊆ F : ∃x, y ∈ S : x⊥y}.

• I(SaP) = 1

iff I(P) ⊆ I(S). I(SiP) = 1 iff I(S) ∪ I(P) ∈ ∆. Thus, SiP is true iff S and P do not contain mutually incompatible features.

• Implicit semantics of conjunctive terms: I(SP) = I(S) ∪ I(P)

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For B and C: Add Negation

• If we add negative terms to syllogistics, they need to satisfy
• 1. Double negation: P = P
• 2. Contraposition: SaP |

= PaS

• Notice that once we have ∧ and P, we also have ∨
• Natural proposal: Use a DeMorgan lattice:
• 1. U, ≤ a distributive lattice (∧ and ∨ as inf and sup)
• 2. x = x
• 3. x ≤ y → y ≤ x

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B: Lentzen’s ‘intensional’ analysis

• Idea: term denotes set of sets of individuals, a filter
• Vp(TT ′) = {X ⊆ D : VL(T) ∩ VL(T ′) ⊆ X}
• Vp(T) = {X ⊆ D : VL(T) ⊆ X}
• Works, but not really intensional:
• Makes crucial use of individuals to acocunt for complex terms
• That is exactly what one doesn’t want

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B1: Boolean with features Lift quickly

• F/I is set of all consistent/Maximal consistent subsets of F.
• If T primitive, then IB(T) ∈ F

and VB(T) = {X ∈ F : I(T) ⊆ X}

• VB(TT ′)

= VB(T) ∩ VB(T ′)

• VB(T)

= F − VB(T)

• SaP

iff VB(S) ⊆ VB(P) SiP iff VB(S) ∩ VB(P) = ∅ thus SiP iff ∃I ∈ I : min(VB(S)) ⊑ I & min(VB(P)) ⊑ I

• where X ⊑ Y

iff ∃X ∈ X : X ⊆ Y

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• One undesirable feature: System too rich:
• Syllogistics with complex terms need not yet be Boolean

(even if double negation elimination/introduction)

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B2: DeMorgan Idea: lift late

• IDM(T) is consistent set of sets of features.
• IDM(TT ′)

= {X ∪ Y : X ∈ IDM(T) & Y ∈ IDM(T ′)}

• IDM(T)

= {X : X ∈ IDM(T)}, where X = {{x} : x ∈ X}, and X ∧ Y = {X ∪ Y : X ∈ X, Y ∈ Y}

• VDM(T) = {Y ⊆ F : ∃X ∈ IDM(T) : X ⊆ Y }

DM Algebra

• I(SaP) = 1

iff VDM(S) ⊆ VDM(P) I(SiP) = 1 iff VDM(S) ∩ VDM(P) = ∅

• Boolean: only look at maximal (consistent) sets.

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C: Leibniz’ paired semantics

• Recall: T = T +, T −
• where T + is set of features any T-individual must have,
• and T − is set of features no T-individual can have.
• Leibniz’ problem:

How to account for negative and conjunctive terms? ⇒ Leibniz dropped project, but not necessary!

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C: Lifted Leibniz’ Paired semantics

• for primitive P: Ip(T) = T +, T − as in Leibniz, but

Vp(T) = {X ⊆ F : I+

p (T) ⊆ X}, {X ⊆ F : I− p (T) ⊆ X}

• Vp(TT ′)

= V +

p (T) ∩ V + p (T ′), V − p (T) ∪ V − p (T ′)

• Vp(T)

= V −

p (T), V + p (T)

• Vp(SaP) = 1

iff V +

p (S) ⊆ V + p (P),

= 0 iff Vp(SoP) = 1 Vp(SiP) = 1 iff V +

p (S) ∩ V + p (P) = ∅, = 0 iff Vp(SeP) = 1

Vp(SoP) = 1 iff V +

p (S) ∩ V − p (P) = ∅, = 0 iff Vp(SaP) = 1

Vp(SeP) = 1 iff V −

p (S) ∪ V − p (P) = F,

= 0 iff Vp(SiP) = 1

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Conclusions

• Recall: it is a small step from Syllogistics to Propositional logic
• And propositional logic need not be classical
• Think of features as situations, and there you go
• The featured analysis is not far from logic atomistic semantic

analysis of propositional logic

• Indeed, one can easily see Leibniz as a forerunner of

Wittgenstein (in many respects)

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