The classical syllogistic
The relational syllogistic
The numerical syllogistic
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
Two curiosities
Conclusion
The classical syllogistic
The relational syllogistic
The numerical syllogistic
Outline
The classical syllogistic
The relational syllogistic
The numerical syllogistic
Two curiosities
Conclusion
Two curiosities
Conclusion
The classical syllogistic
•
The relational syllogistic
∀x (p (x ) → q (x ))
∃x (p (x ) ∧ q (x ))
∀x (p (x ) → ¬q (x ))
∃x (p (x ) ∧ ¬q (x ))
∀(p , q )
∃(p , q )
∀(p , q̄ )
∃(p , q̄ )
Syntax: non-logical signature of unary atoms p ∈ P;
` =: p | p̄
ϕ =: ∃(p , `) | ∀(p , `)
•
Two curiosities
Conclusion
The classical syllogistic, denoted S , is the logic corresponding
to the following fragment of English:
Every p is a q
Some p is a q
No p is a q
Some p is not a q
•
The numerical syllogistic
(literals)
(formulas of S )
Semantics: an interpretation is a set A 6= ∅ with a function
p 7→ p A ⊆ A;
(p̄ )A = A \ p A
A|=∀(p , `) i p A ⊆ `A
A|=∃(p , `) i p A ∩ `A 6= ∅.
The classical syllogistic
•
The relational syllogistic
Darii
Conclusion
∀(p , q̄ ) ∃(o , p )
∃(o , q̄ )
Ferio
They may be combined to yield derivations in the expected
way:
∃(artst, bkpr) ∀(artst, crpntr)
∃(crpntr, bkpr)
Darii
∃(crpntr, dntst)
•
Two curiosities
The classical syllogisms are proof-rules for S :
∀(p , q ) ∃(o , p )
∃(o , q )
•
The numerical syllogistic
∀(bkpr, dntst)
Ferio
Any set X of such rules then induces a derivability relation, for
instance:
{∃(artst, bkpr), ∀(artst, crpntr), ∀(bkpr, dntst)} `X ∃(crpntr, dntst)
The classical syllogistic
•
•
•
The numerical syllogistic
Two curiosities
Conclusion
A sequent with premises Φ and conclusion ψ is valid just in
case, for any interpretation A, A |= Φ implies A |= ψ
(more briey: just in case Φ |= ψ .)
An absurdity, ⊥ is a formula of the form ∃(p , p̄ ).
We say that a derivation relation ` is
•
•
•
•
The relational syllogistic
Φ ` ψ entails Φ |= ψ
Φ |= ψ entails Φ ` ψ
refutation-complete if Φ |= ⊥ entails Φ ` ⊥
sound if
complete if
For the classical syllogistic, the notions of validity and
derivability can be made to coincide:
Theorem
There is a nite set of rules X in S such that `X is sound and
complete.
The classical syllogistic
•
The relational syllogistic
Two curiosities
Conclusion
These rules suce:
∀(q , `)
∃(p , q )
∃(p , `)
∃(p , `)
∀(p , q )
∃(q , `)
¯
∀(q , `)
∃(p , `)
∃(p , q̄ )
•
The numerical syllogistic
∀(p , q )
(D1)
(D2)
∀(q , `)
∀(p , `)
ψ
(D3)
ψ̄
ϕ
∀(p , p̄ )
∀(p , `)
(X)
∀(p , p )
(B)
(T)
(A)
∃(p , `)
∃(p , p )
(I)
There are some dierences from the syllogistic as understood
in classical times:
• ∀(p , q ) does not entail ∃(p , q )
• ∀(p , p ), ∃(p , p ) are allowed
The classical syllogistic
•
•
The relational syllogistic
The numerical syllogistic
Conclusion
Completeness was rst shown by (Corcoran 72) and (Smiley
73).
Important qualication: these authors allowed
reductio-ad-absurdum (as a nal step).
∃(q , o ) ∀(o , p )
Darii
∃(q , p )
∀(p , q̄ )
∃(q , q̄ )
•
Two curiosities
Ferio
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 Šukasiewicz 39/50 for the
classical syllogistic together with Boolean sentence-connectives.
The classical syllogistic
•
•
The relational syllogistic
The numerical syllogistic
Conclusion
Completeness was rst shown by (Corcoran 72) and (Smiley
73).
Important qualication: these authors allowed
reductio-ad-absurdum (as a nal step).
[∃(q , o )] ∀(o , p )
Darii
∃(q , p )
∀(p , q̄ )
∃(q , q̄ )
RAA
∀(o , q̄ )
•
Two curiosities
Ferio
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 Šukasiewicz 39/50 for the
classical syllogistic together with Boolean sentence-connectives.
The classical syllogistic
•
The numerical syllogistic
Two curiosities
Conclusion
It is natural to ask what happens if we extend the language of
the classical syllogistic
•
•
•
•
The relational 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 eorts only really began in the 19th Century:
The classical syllogistic
•
The numerical syllogistic
Two curiosities
Conclusion
It is natural to ask what happens if we extend the language of
the classical syllogistic
•
•
•
•
The relational 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 eorts only really began in the 19th Century:
Augustus De
Morgan
The classical syllogistic
•
The numerical syllogistic
Two curiosities
Conclusion
It is natural to ask what happens if we extend the language of
the classical syllogistic
•
•
•
•
The relational 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 eorts only really began in the 19th Century:
Augustus De
Morgan
George Boole
The classical syllogistic
•
The numerical syllogistic
Two curiosities
Conclusion
It is natural to ask what happens if we extend the language of
the classical syllogistic
•
•
•
•
The relational 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 eorts only really began in the 19th Century:
Augustus De
Morgan
George Boole
William Stanley Jevons
The classical syllogistic
The relational syllogistic
The numerical syllogistic
Outline
The classical syllogistic
The relational syllogistic
The numerical syllogistic
Two curiosities
Conclusion
Two curiosities
Conclusion
The classical syllogistic
•
The relational syllogistic
The numerical syllogistic
The relational syllogistic, denoted R, extends the classical
syllogistic with transitive verbs.
Some artist hates every beekeeper
No beekeeper hates every artist
Some 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
c =: ` | ∀(p , t ) | ∃(p , t )
ϕ =: ∃(p , c ) | ∀(p , c )
•
Two curiosities
(binary literals)
(c-terms)
(formulas of R)
Semantics: A also denes a function r 7→ r A ⊆ (A × A).
(r̄ )A = (A × A) \ r A
(∀(p , t ))A = {a ∈ A | ha, bi ∈ t A for all b ∈ p A }
(∃(p , t ))A = {a ∈ A | ha, bi ∈ t A for some b ∈ p A }
Conclusion
The classical syllogistic
•
The relational syllogistic
The numerical syllogistic
(∃∀)
Then we have the derivation
∀(bkpr, ∃(artst, hate)) [∀(art, bkpr)]1
∀(bkpr, ∃(bkpr, hate))
∃(artst, ∀(bkpr, hate))
∃(artst, artst)
∃(art, bkpr)
•
Conclusion
We can write syllogism-like rules for R just as for S :
∀(o , ∃(p , r )) ∀(p , q )
∀(o , ∃(q , r ))
•
Two curiosities
(∃∀)
[∀(art, bkpr)]1
∀(artst, ∃(bkpr, hate))
(RAA)1
.
It can be shown (P-H and Moss 09):
Theorem
There is a nite set of rules X in R such that `X is sound and
refutation-complete.
•
A completeness theorem was provided by (Nishihara, Morita and
Iwata 90) for R together with Boolean sentence-connectives.
The classical syllogistic
•
The relational syllogistic
The numerical syllogistic
Two curiosities
Conclusion
RAA is required for R (P-H and Moss 09):
Theorem
There 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 ))
∀(p1 , ∀(pn , r ))
∀(pi , p̄j )
(1 ≤ i < n)
(1 ≤ i < j ≤ n)
Then Γn |= ∀(p , ∃(pn , r )), but Γn \ {∀(pi , ∃(pi + , r ))} has no
non-trivial R-consequences!
1
1
The classical syllogistic
•
The relational syllogistic
Every non-p is a q
Some non-p is not a q
Every p r s some non-q
Conclusion
...
Formally, we dene the languages S † and R† as follows:
e =: ` | ∀(`, t ) | ∃(`, t )
ϕ =: ∃(`, m) | ∀(`, m)
ϕ =: ∃(`, e ) | ∀(`, e )
•
Two curiosities
The above notation suggests natural extensions of S and R:
∀(p̄ , q )
∃(p̄ , q̄ )
∀(p , ∃(q̄ , r ))
•
The numerical syllogistic
(e-terms)
(formulas of S † )
(formulas of R† )
Here is a valid argument in R† :
Every artist hates every beekeeper
Every artist hates every non-beekeeper
Every artist hates every carpenter
∀(art, ∀(bkpr, hate))
∀(art, ∀(bkpr, hate))
∀(art, ∀(cptr, hate))
The classical syllogistic
•
The relational syllogistic
The numerical syllogistic
Two curiosities
Conclusion
The following can be shown (PH and Moss 09)
Theorem
There exists a nite set X of syllogistic rules in S † such that
`X is sound and complete.
Theorem
There 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 the
proof-theoretic situation for the classical syllogistic, but it does
for the relational syllogistic.
The classical syllogistic
•
•
The relational syllogistic
The numerical syllogistic
Two curiosities
Conclusion
It is worth noting the complexity of satisability for the
languages considered so far:
The existence of syllogistic proof-systems of various kinds
imposes upper bounds on complexity:
•
The satisability problem for any syllogistic language for which
there exists a sound and (refutation-) complete syllogistic
system
•
`X
is in PTime.
The satisability problem for any syllogistic language for which
there exists a sound and complete syllogistic system
X
is in
NPTime.
•
The following are straightforward to establish:
•
The satisability problems for
S , S †,
and
R
are all
NLogSpace-complete
•
•
The satisability problem for
R†
is ExpTime-complete.
Thus, the existence of a sound and complete for R† would
entail NPTime = ExpTime.
X
The classical syllogistic
The relational syllogistic
The numerical syllogistic
Outline
The classical syllogistic
The relational syllogistic
The numerical syllogistic
Two curiosities
Conclusion
Two curiosities
Conclusion
The classical syllogistic
•
•
The relational syllogistic
The numerical syllogistic
•
Conclusion
The numerical syllogistic, denoted N , extends the classical
syllogistic with counting determiners
Syntax:
At most C p s are q s
More than C p s are q s
At most C p s are not q s
More than C p s are not q s
ϕ =: ∃>C (p , `) | ∃≤C (p , `)
•
Two curiosities
∃≤C (p , q )
∃>C (p , q )
∃≤C (p , q̄ )
∃>C (p , q̄ )
(formulas of N )
Unlike the languages considered above, N has innitely many
sentence-forms.
We denote by Sk the fragment of N in which all numerical
subscripts are bounded by k .
The classical syllogistic
•
The relational syllogistic
The numerical syllogistic
Two curiosities
A valid argument in N (actually, in S ):
More than twelve artists are beekeepers
At most three beekeepers are carpenters
At most four gardeners are not carpenters
12
≤3
≥ 13
≤4
Conclusion
The classical syllogistic
•
The relational syllogistic
The numerical syllogistic
Two curiosities
A valid argument in N (actually, in S ):
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.
12
≤3
≥ 13
≤4
Conclusion
The classical syllogistic
•
The relational syllogistic
The numerical syllogistic
Two curiosities
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 eort to work out a set of
syllogism-like rules for N .
So have a number of twentieth-century authors . . .
Conclusion
The classical syllogistic
•
The relational syllogistic
The numerical syllogistic
Two curiosities
Conclusion
The following can be shown (PH09, PH 13)
Theorem
There exists no nite set X of syllogistic rules in N such that
X is sound and complete.
Theorem
There exists no nite set X of syllogistic rules in S1 such that
X is sound and complete.
•
•
The satisability problem for S is NPTime-hard, and for N is
in NPTime under binary coding (Eisenbrand and Shmonin 07).
Adding `noun-level' negation makes no dierence here.
1
The classical syllogistic
•
The relational syllogistic
The numerical syllogistic
Two curiosities
Of course, we could add numbers to R and R† , as well, to
formalize 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
Conclusion
The classical syllogistic
•
The relational syllogistic
The numerical syllogistic
Two curiosities
Of course, we could add numbers to R and R† , as well, to
formalize 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
Conclusion
The classical syllogistic
•
•
The relational syllogistic
The numerical syllogistic
Two curiosities
Conclusion
Of course, we could add numbers to R and R† , as well, to
formalize 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), and
again lacks any sound and complete (indirect) syllogistic
proof-system.
The classical syllogistic
The relational syllogistic
The numerical syllogistic
Outline
The classical syllogistic
The relational syllogistic
The numerical syllogistic
Two curiosities
Conclusion
Two curiosities
Conclusion
The classical syllogistic
The relational syllogistic
The numerical syllogistic
Two curiosities
•
We have seen that R admits no sound and complete direct
syllogistic rule-system.
But does it have any extensions which do?
The answer is yes. We consider statements of the form
•
If there are p s, then there are q s
Syntax of RE :
ϕ =: ∃(p , `) | ∀(p , `) |
(p , q )
Semantics:
A |=
(p , q ) i p A 6= ∅ ⇒ q A 6= ∅.
Æ
•
(p , q )
Æ
•
Æ
•
Conclusion
The classical syllogistic
The relational syllogistic
The numerical syllogistic
Theorem
• There is a nite set of rules
and complete.
Conclusion
X in RE such that ` is sound
X
The rules set is BIG. The best I could do has 70 rules including
∀(q , ∀(o , t̄ ))
Æ
∀(o 0 , ∀(q , t ))
(q , o 0 )
∀(o 0 , p )
∃(p , q̄ )
Æ
•
Two curiosities
(q , o )
∀(o , p )
∃(p , p )
The classical syllogistic
•
The relational syllogistic
The numerical syllogistic
Various philosophers since (including maybe Aristotle) have
wondered why you cannot say
Every p is every q
No p is every q
•
Some p is every q
Some p is not every q ,
or even
Every p is some q
No p is some q
•
Two curiosities
Some p is some q
Some p is no q ,
all of which sound extremely strange.
Conclusion
The classical syllogistic
•
The numerical syllogistic
Two curiosities
Conclusion
Things become a little clearer if we reformulate the classical
syllogistic, thus:
Every
No
•
The relational syllogistic
p
p
is identical to some
is identical to any
q
q
Some
Some
p
p
is identical to some
q
is not identical to any
q.
because the second quantier here can be meaningfully
dualized:
Every
No
p
p
is identical to every
is identical to every
q
q
Some
Some
p
p
is identical to every
q
is not identical to every
q.
The classical syllogistic
•
The relational syllogistic
The numerical syllogistic
This suggests an unnatural extension of S , say H, featuring
forms such as
∀(p , ∀q )
•
Similarly, we could have
∀(p , ∀q̄ )
•
Two curiosities
H† ,
∃(p , ∀q )
allowing noun-negation:
∃(p , ∀q̄ )
etc.
Quick test: what does
Every artist is every beekeeper
mean?
Conclusion
The classical syllogistic
•
The relational syllogistic
The numerical syllogistic
Similarly, we could have
∀(p , ∀q̄ )
•
•
Conclusion
This suggests an unnatural extension of S , say H, featuring
forms such as
∀(p , ∀q )
•
Two curiosities
H† ,
∃(p , ∀q )
allowing noun-negation:
∃(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.
The classical syllogistic
The relational syllogistic
The numerical syllogistic
Two curiosities
•
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 Sir
William Hamilton (Bart.) who got into a big argument with
De Morgan about quantication of the predicate.
Conclusion
The classical syllogistic
The relational syllogistic
The numerical syllogistic
Two curiosities
Conclusion
Theorem
There is no nite set of rules that is sound and complete for H
without reductio-ad-absurdum.
Theorem
There is a nite set of rules that is sound and complete for H, as
long as reductio-ad-absurdum is allowed as a last step.
Theorem
Unless PTime= NPTime, there is no nite set of rules that is
sound and complete for H† , with reductio-ad-absurdum allowed (as
a last step).
Theorem
There is a nite set of rules that is sound and complete for H† with
reductio-ad-absurdum allowed unconditionally.
The classical syllogistic
The relational syllogistic
The numerical syllogistic
Outline
The classical syllogistic
The relational syllogistic
The numerical syllogistic
Two curiosities
Conclusion
Two curiosities
Conclusion
The classical syllogistic
•
The relational syllogistic
The numerical syllogistic
Two curiosities
We have investigated the existence of sound and complete
syllogistic proof-systems
S
S†
R
RE
R†
Sz (z ≥ 1)
N
H
H†
`X
`X
X `X
No X
No X
No X
X X
NLogSpace
NLogSpace
NLogSpace
≤
PTime
ExpTime
NPTime
NPTime
≤
PTime
NPTime
Conclusion
The classical syllogistic
•
•
The relational syllogistic
The numerical syllogistic
Two curiosities
The classical syllogistic is a logic whose salience derives from
the syntax of certain natural languages.
By considering larger fragments of natural languages, we
obtain more expressive logics whose properties we can
investigate.
•
ditransitive verbs, anaphora, relative clauses, conjunctions,
ellipsis
•
•
•
•
Conclusion
comparatives and expressions of quantity
generalized quantication
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.