the relational syllogistic (and beyond)it can be shown (p-h and moss 09): theorem there is a nite...

The classical syllogistic The relational syllogistic The numerical syllogistic Two curiosities Conclusion

The Relational Syllogistic

(and beyond)

Ian Pratt-Hartmann

(Joint work with Lawrence S. Moss)

School of Computer Science

Manchester University,

Manchester, M13 9PL

email: [email protected]

NLCS, New Orleans: June, 2013


Outline

The classical syllogistic

The relational syllogistic

The numerical syllogistic

Two curiosities

Conclusion


• The classical syllogistic, denoted S, is the logic correspondingto the following fragment of English:

Every p is a q ∀x(p(x)→ q(x)) ∀(p, q)Some p is a q ∃x(p(x) ∧ q(x)) ∃(p, q)No p is a q ∀x(p(x)→ ¬q(x)) ∀(p, q̄)Some p is not a q ∃x(p(x) ∧ ¬q(x)) ∃(p, q̄)

• Syntax: non-logical signature of unary atoms p ∈ P;

` =: p | p̄ (literals)ϕ =: ∃(p, `) | ∀(p, `) (formulas of S)

• Semantics: an interpretation is a set A 6= ∅ with a functionp 7→ pA ⊆ A;

(p̄)A = A \ pA

A|=∀(p, `) i� pA ⊆ `A

A|=∃(p, `) i� pA ∩ `A 6= ∅.


• The classical syllogisms are proof-rules for S:

∀(p, q) ∃(o, p)∃(o, q)

Darii∀(p, q̄) ∃(o, p)∃(o, q̄)

Ferio

• They may be combined to yield derivations in the expectedway:

∃(artst, bkpr) ∀(artst, crpntr)∃(crpntr, bkpr)

Darii

∀(bkpr, dntst)∃(crpntr, dntst)

Ferio

• Any set X of such rules then induces a derivability relation, forinstance:

{∃(artst, bkpr),∀(artst, crpntr),∀(bkpr, dntst)} `X ∃(crpntr, dntst)


• A sequent with premises Φ and conclusion ψ is valid just incase, for any interpretation A, A |= Φ implies A |= ψ(more brie�y: just in case Φ |= ψ.)

• An absurdity, ⊥ is a formula of the form ∃(p, p̄).• We say that a derivation relation ` is

• sound if Φ ` ψ entails Φ |= ψ• complete if Φ |= ψ entails Φ ` ψ• refutation-complete if Φ |= ⊥ entails Φ ` ⊥

• For the classical syllogistic, the notions of validity andderivability can be made to coincide:

TheoremThere is a �nite set of rules X in S such that `X is sound andcomplete.


• These rules su�ce:

∀(q, `) ∃(p, q)∃(p, `)

(D1)∀(p, q) ∀(q, `)

∀(p, `)(B)

∃(p, `) ∀(p, q)∃(q, `)

(D2) ψ ψ̄ϕ (X)

∀(p, p̄)∀(p, `)

(A)

∀(q, ¯̀) ∃(p, `)∃(p, q̄)

(D3)∀(p, p)

(T)∃(p, `)∃(p, p)

(I)

• There are some di�erences from the syllogistic as understoodin classical times:

• ∀(p, q) does not entail ∃(p, q)• ∀(p, p), ∃(p, p) are allowed


• Completeness was �rst shown by (Corcoran 72) and (Smiley73).

• Important quali�cation: these authors allowedreductio-ad-absurdum (as a �nal step).

[

∃(q, o)

]

∀(o, p)∃(q, p)

Darii

∀(p, q̄)∃(q, q̄)

Ferio

∀(o, q̄)RAA

• We write{∀(o, p),∀(p, q̄)} X ∀(o, q̄)

where X is any rule-set containing Ferio and Darii.

• However, reductio-ad-absurdum is not actually needed for S(P-H and Moss 09).

• A completeness theorem was provided by ukasiewicz 39/50 for theclassical syllogistic together with Boolean sentence-connectives.


• Completeness was �rst shown by (Corcoran 72) and (Smiley73).

• Important quali�cation: these authors allowedreductio-ad-absurdum (as a �nal step).

[∃(q, o)] ∀(o, p)∃(q, p)

Darii

∀(p, q̄)∃(q, q̄)

Ferio

∀(o, q̄)RAA

• We write{∀(o, p),∀(p, q̄)} X ∀(o, q̄)

where X is any rule-set containing Ferio and Darii.

• However, reductio-ad-absurdum is not actually needed for S(P-H and Moss 09).

• A completeness theorem was provided by ukasiewicz 39/50 for theclassical syllogistic together with Boolean sentence-connectives.


• It is natural to ask what happens if we extend the language ofthe classical syllogistic

• Transitive verbs: Every artist admires every beekeeper• Numerical determiners: At least three artists are beekeepers• Adjectives: Every clever beekeeper is a carpenter

• Such sentences are considered throughout the history of logic,but determined e�orts only really began in the 19th Century:





Augustus DeMorgan





Augustus DeMorgan George Boole





Augustus DeMorgan George Boole William Stanley Jevons


Outline




Two curiosities

Conclusion


• The relational syllogistic, denoted R, extends the classicalsyllogistic with transitive verbs.

Some artist hates every beekeeperNo beekeeper hates every artistSome artist is not a beekeeper

∃(art,∀(bkpr, hate)),∀(bkpr,∃(artst, hate))∃(art, bkpr).

• Syntax: add a non-logical signature of binary atoms r ∈ R;

t =: r | r̄ (binary literals)c =: ` | ∀(p, t) | ∃(p, t) (c-terms)ϕ =: ∃(p, c) | ∀(p, c) (formulas of R)

• Semantics: A also de�nes a function r 7→ rA ⊆ (A× A).

(r̄)A = (A× A) \ rA

(∀(p, t))A = {a ∈ A | 〈a, b〉 ∈ tA for all b ∈ pA}(∃(p, t))A = {a ∈ A | 〈a, b〉 ∈ tA for some b ∈ pA}


• We can write syllogism-like rules for R just as for S:

∀(o, ∃(p, r)) ∀(p, q)∀(o,∃(q, r))

(∃∀)

• Then we have the derivation

∃(artst, ∀(bkpr, hate))

∀(bkpr,∃(artst, hate)) [∀(art, bkpr)]1

∀(bkpr,∃(bkpr, hate))(∃∀)

[∀(art, bkpr)]1

∀(artst,∃(bkpr, hate))∃(artst, artst)∃(art, bkpr)

(RAA)1.

• It can be shown (P-H and Moss 09):

TheoremThere is a �nite set of rules X in R such that `X is sound andrefutation-complete.

• A completeness theorem was provided by (Nishihara, Morita andIwata 90) for R together with Boolean sentence-connectives.


• RAA is required for R (P-H and Moss 09):TheoremThere exists no �nite set X of syllogistic rules in R such that`X is sound and complete.

Dowód.Let Γn be the set of sentences

∀(pi ,∃(pi+1, r)) (1 ≤ i < n)∀(p1, ∀(pn, r))∀(pi , p̄j) (1 ≤ i < j ≤ n)

Then Γn |= ∀(p1, ∃(pn, r)), but Γn \ {∀(pi ,∃(pi+1, r))} has nonon-trivial R-consequences!


• The above notation suggests natural extensions of S and R:

∀(p̄, q) Every non-p is a q∃(p̄, q̄) Some non-p is not a q∀(p,∃(q̄, r)) Every p rs some non-q . . .

• Formally, we de�ne the languages S† and R† as follows:

e =: ` | ∀(`, t) | ∃(`, t) (e-terms)ϕ =: ∃(`,m) | ∀(`,m) (formulas of S†)ϕ =: ∃(`, e) | ∀(`, e) (formulas of R†)

• Here is a valid argument in R†:Every artist hates every beekeeperEvery artist hates every non-beekeeperEvery artist hates every carpenter

∀(art,∀(bkpr, hate))∀(art,∀(bkpr, hate))∀(art,∀(cptr, hate))


• The following can be shown (PH and Moss 09)

TheoremThere exists a �nite set X of syllogistic rules in S† such that`X is sound and complete.

TheoremThere exists no �nite set X of syllogistic rules in R† such that

X is sound and complete.

• Thus adding `noun-level' negation does not change theproof-theoretic situation for the classical syllogistic, but it does

for the relational syllogistic.


• It is worth noting the complexity of satis�ability for thelanguages considered so far:

• The existence of syllogistic proof-systems of various kindsimposes upper bounds on complexity:

• The satis�ability problem for any syllogistic language for whichthere exists a sound and (refutation-) complete syllogisticsystem `X is in PTime.

• The satis�ability problem for any syllogistic language for whichthere exists a sound and complete syllogistic system X is inNPTime.

• The following are straightforward to establish:• The satis�ability problems for S, S†, and R are allNLogSpace-complete

• The satis�ability problem for R† is ExpTime-complete.• Thus, the existence of a sound and complete X for R† wouldentail NPTime = ExpTime.


Outline




Two curiosities

Conclusion


• The numerical syllogistic, denoted N , extends the classicalsyllogistic with counting determiners

At most C ps are qs ∃≤C (p, q)More than C ps are qs ∃>C (p, q)At most C ps are not qs ∃≤C (p, q̄)More than C ps are not qs ∃>C (p, q̄)

• Syntax:

ϕ =: ∃>C (p, `) | ∃≤C (p, `) (formulas of N )

• Unlike the languages considered above, N has in�nitely manysentence-forms.

• We denote by Sk the fragment of N in which all numericalsubscripts are bounded by k .


• A valid argument in N (actually, in S12):More than twelve artists are beekeepers

At most three beekeepers are carpenters

At most four gardeners are not carpenters

≥ 13

≤ 3

≤ 4


• A valid argument in N (actually, in S12):More than twelve artists are beekeepers

At most three beekeepers are carpenters

At most four gardeners are not carpenters

More than �ve artists are not gardeners.

≥ 13

≤ 3

≤ 4


• Evidently,

∀(p, `) ≡ ∃≤0(p, ¯̀)∃(p, `) ≡ ∃>0(p, `).

• Thus:• S notational variant of S0;• N is the union of all the Sk .

• De Morgan made a concerted e�ort to work out a set ofsyllogism-like rules for N .

• So have a number of twentieth-century authors . . .


• The following can be shown (PH09, PH 13)

TheoremThere exists no �nite set X of syllogistic rules in N such that


TheoremThere exists no �nite set X of syllogistic rules in S1 such that


• The satis�ability problem for S1 is NPTime-hard, and for N isin NPTime under binary coding (Eisenbrand and Shmonin 07).

• Adding `noun-level' negation makes no di�erence here.


• Of course, we could add numbers to R and R†, as well, toformalize such arguments as

At most one artist admires at most seven beekeepers

At most two carpenters admire at most eight dentists

At most three artists admire at least seven electricians

At most four beekeepers are not electricians

At most �ve dentists are not electricians

At most one beekeeper is a dentist

At most six artists are carpenters

• This logic is NExpTime-complete (under binary coding), andagain lacks any sound and complete (indirect) syllogistic

proof-system.


Outline




Two curiosities

Conclusion


• We have seen that R admits no sound and complete directsyllogistic rule-system.

• But does it have any extensions which do?• The answer is yes. We consider statements of the form

If there are ps, then there are qs

Æ

(p, q)

• Syntax of RE :

ϕ =: ∃(p, `) | ∀(p, `) |

Æ

(p, q)

• Semantics:

A |=

Æ

(p, q) i� pA 6= ∅ ⇒ qA 6= ∅.


•TheoremThere is a �nite set of rules X in RE such that `X is soundand complete.

• The rules set is BIG. The best I could do has 70 rules including

∀(o′, ∀(q, t)) ∀(q, ∀(o, t̄))

Æ

(q, o′) ∀(o′, p)

Æ

(q, o) ∀(o, p) ∃(p, p)

∃(p, q̄)


• Various philosophers since (including maybe Aristotle) havewondered why you cannot say

Every p is every q Some p is every q

No p is every q Some p is not every q,

• or even

Every p is some q Some p is some q

No p is some q Some p is no q,

• all of which sound extremely strange.


• Things become a little clearer if we reformulate the classicalsyllogistic, thus:

Every p is identical to some q Some p is identical to some qNo p is identical to any q Some p is not identical to any q.

• because the second quanti�er here can be meaningfullydualized:

Every p is identical to every q Some p is identical to every qNo p is identical to every q Some p is not identical to every q.


• This suggests an unnatural extension of S, say H, featuringforms such as

∀(p,∀q) ∃(p,∀q)• Similarly, we could have H†, allowing noun-negation:

∀(p, ∀q̄) ∃(p, ∀q̄) etc.• Quick test: what does

Every artist is every beekeeper

mean?

• Quick answer:Either there are no artists or no beekeepers or there is a unique

artist and a unique beekeeper and they are identical.


• Example validity in H†:Every artist is every artist

Every non-artist is every non-artist

Some beekeeper is not a carpenter

Some carpenter is not a dentist

Every dentist is a beekeeper

• By the way, H stands for Hamiltonian syllogistic, after SirWilliam Hamilton (Bart.) who got into a big argument with

De Morgan about quanti�cation of the predicate.


TheoremThere is no �nite set of rules that is sound and complete for Hwithout reductio-ad-absurdum.

TheoremThere is a �nite set of rules that is sound and complete for H, aslong as reductio-ad-absurdum is allowed as a last step.

TheoremUnless PTime= NPTime, there is no �nite set of rules that is

sound and complete for H†, with reductio-ad-absurdum allowed (asa last step).

TheoremThere is a �nite set of rules that is sound and complete for H† withreductio-ad-absurdum allowed unconditionally.


Outline




Two curiosities

Conclusion


• We have investigated the existence of sound and completesyllogistic proof-systems

S `X NLogSpaceS† `X NLogSpaceR �X� NLogSpaceRE `X ≤ PTimeR† No X ExpTimeSz (z ≥ 1) No X NPTimeN No X NPTimeH �X� ≤ PTimeH† X NPTime


• The classical syllogistic is a logic whose salience derives fromthe syntax of certain natural languages.

• By considering larger fragments of natural languages, weobtain more expressive logics whose properties we caninvestigate.

• ditransitive verbs, anaphora, relative clauses, conjunctions,ellipsis

• comparatives and expressions of quantity• generalized quanti�cation• tense and temporal expressions

• Some pointers to people who have worked on similar logics:• F.B. Fitch, P. Suppes, D. McAllister, R. Givan, W. Purdy• N. Francez, Y. Fyodorov, S. Winter,• L. Moss, B. MacCartney, T. Icard• J. Szymanik, M. Sevenster.

The classical syllogisticThe relational syllogisticThe numerical syllogisticTwo curiositiesConclusion

the relational syllogistic (and beyond)it can be shown (p-h and moss 09): theorem there is a nite...

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