A presentation from my official defense of my Master Thesis, Sofia University “St. Kliment Ohridski”, Supervisor: Prof. Dimitar Vakarelov A publication based on my thesis: https://link.springer.com/chapter/10.1007/978-3-319-97879-6_11

1. Modal Syllogistic
Master Thesis Defence
Tsvetan Vasilev
Supervisor: Prof. Dimitar Vakarelov
Sofia University “St. Kliment Ohridski”
Faculty of Mathematics and Computer Science
Department of Mathematical Logic
26 May 2016
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2. Abstract
In this work we introduce the Aristotle’s syllogistic and its modal extension
from the standpoint of modern formal logic. We represent the Aristotle’s
non-modal syllogistic as a quantifier-free extension of propositional logic,
following the axiomatization of Jan Łukasiewicz [3]. The innovation in the
current work is the construction of a new modal logic, named modal
syllogistic, that is an extension of Aristotle’s non-modal syllogistic with the
modal operators “necessity” and “possibility”.
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3. Contents
1 Introduction
2 Syllogistic Structures
3 Syllogistic
4 Modal Syllogistic
5 Extended Modal Syllogistics
6 Future Work
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4. Section 1
Introduction
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5. Contents
1 Introduction
Origin of Syllogistic. Retrospection
Thesis Summary
2 Syllogistic Structures
3 Syllogistic
4 Modal Syllogistic
5 Extended Modal Syllogistics
6 Future Work
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6. Retrospection
384 BC Aristotle’s Syllogistic
1879 Gottlob Frege’s Predicate Logic
1951 Jan Lukasiewicz’s axiomatization of Aristotle’s Syllogistic
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7. Retrospection. Aristotle’s Syllogistic
Syllogism
• A syllogism is a logical argument that applies deductive reasoning
from two premises to reach a conclusion.
• A syllogism consists of three propositions: major premise, minor
premise and conclusion.
• A three-line form of syllogism:
Major Premise
Minor Premise
Conclusion
• The three syllogism propositions could be any of the following four
types:
◦ All a is b.
◦ Some a is b.
◦ All a is not b.
◦ Some a is not b.
where a, b are set variables.
These four types of propositions have a corresponding notation
A(a, b), I(a, b), E(a, b), O(a, b) introduced by the medieval Latin
schools.
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8. Retrospection. Predicate Logic. Lukasiewicz’s Syllogistic System I
384 BC Aristotle’s Syllogistic
1879 Gottlob Frege’s Predicate Logic
1951 Jan Lukasiewicz’s axiomatization of Aristotle’s Syllogistic
• Aristotelian syllogistic was the leading concept among the Western
medieval logical thought and its domination continued until the
invention of modern predicate logic in the late nineteenth century.
The introduced predicate logic by Frege forced the development of
propositional logic and first-order predicate logic which replaced the
use of Aristotelian logic.
• In his pioneering work(1951), Jan Łukasiewicz describes Aristotle’s
syllogistic from the standpoint of modern formal logic. He presents
the Aristotle’s syllogistic as a quantifier-free extension of propositional
logic which has as atomic formulas the propositions A(a, b) (All a is b)
and I(a, b) (Some a is b) and their counterpart negations
E(a, b)
def
= ¬I(a, b) and O(a, b)
def
= ¬A(a, b), where a, b are set
variables.
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9. Retrospection. Lukasiewicz’s Syllogistic System II
• Łukasiewicz accepts A and I as primitive relations. After that, using
them and the propositional negation ¬, he defines the other two
relations E(a, b)
def
= ¬I(a, b) and O(a, b)
def
= ¬A(a, b).
• Łukasiewicz defines an axiomatic system consisting of the following
axioms and inference rules:
Axioms:
(Ł1) A(a, a)
(Ł2) I(a, a)
(Ł3) A(b, c) ∧ A(a, b) =⇒ A(a, c)
(Ł4) A(b, c) ∧ I(b, a) =⇒ I(a, c)
Inference rules: Modus Ponens and substitution of set variables.
• The standard semantic of this language is set-theoretical: set variables
are interpreted as arbitrary non-empty sets; the atomic formula A(a, b)
- as set inclusion a ⊆ b; the atomic formula I(a, b) - as the overlap
relation between sets a ∩ b = ∅.
• In 1956 Shepherdson gives a Henkin-style proof of completeness and
decidability for the Lukasiewicz’s and some other syllogistic systems.
He considers not only systems for Aristotelian categories (i.e. only
non-empty sets) but also and systems for unrestricted general
categories (i.e. sets which could be the empty set).
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10. Retrospection. Aristotle’s Modal Syllogistic
• In his works “Prior Analytics” and “De Interpretatione”, Aristotle
extended the theory of syllogisms by introducing three modal
operators in the logical system: necessity, possibility and contingency.
• The third modality “contingency” in Aristotle’s works has too vague
definition: in the “De Interpretatione” it means the same as
“possibility”, while in the “Prior Analytics” it has another more
complicated meaning [3, p. 134]. Due to inconsistencies and faults like
these, Aristotle’s modal syllogistic is unclear, incomprehensive and
inconsistent[3, p. 133].
• In the Middle Ages, the logicians discussed and tried to fill in the
found gaps in Aristotle’s modal theory.
• Nowadays, Jan Łukasiewicz constructed a modal system following the
Aristotle’s ideas [3]. However, this Łukasiewicz’s modal logic is
controversial too [8].
• In our work on modal syllogistic we use a standard syllogistic system
based on Kripke semantic.
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11. Thesis Summary. General Results
This work represents 2 general results:
1) Introduction of quantifier-free first-order syllogistic system Syll,
following the ideas of Lukasiewicz and Shepherdson.
2) Introduction of modal syllogistic ModSyll, which is a combination of
the propositional syllogistic Syll with the Kripke modal system. In this
sense the modal syllogistic is a dynamic version of syllogistic.
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12. Thesis Summary. Syllogistic
We present the Aristotle’s syllogistic as a quantifier-free extension of
propositional logic with the primitive relations All and Some between set
variables. Our syllogistic works with unrestricted set variables.
• Language: (1) a set Var of set variables; (2) the atomic relations All
and Some; (3) the standard propositional connectives from
propositional logic.
• Semantic: The semantic of our syllogistic Syll is based on
Shepherdson’s idea for interpretation of Aristotelian syllogistic [11].
Following this paper, we introduce a syllogistic structure
W = (W, ≤, O) where the domain W is a set of elements; ≤ and O
are the binary relations “part-of” and “overlap” over W which satisfy a
fixed list of axioms.
• Axiomatic System: The axiomatic system for Syll is an extension of
the Hilbert-style axiomatic system for propositional logic with a list of
axioms for the relations All and Some (see [11, p. 144]).
• Completeness: using the well-known canonical model construction.
• Decidability: showing that Syll has the finite model property.
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13. Thesis Summary. Modal Syllogistic
Combining the propositional syllogistic Syll with the standard modal logic,
we obtain a dynamic version of syllogistic: the modal syllogistic ModSyll.
• Language: an extension of the language of propositional syllogistic
with the modal operators and ♦.
• Semantic: The semantic for modal syllogistic is a combination of the
semantic for propositional syllogistic with the standard Kripke
semantic for modal logics. We introduce a modal syllogistic
structure(frame) Q = (Q, {W(x) | x ∈ Q}, R) such that: the domain
Q is a set of nodes(worlds), R is an accessibility relation in Q and
W(x) = (W(x), ≤x, Ox) is a defined syllogistic structure for each
x ∈ Q.
• Axiomatic System: an extension of the axiomatic system for
propositional syllogistic Syll with the well-known modal distribution
axiom (K ): (α ⇒ β) ⇒ ( α ⇒ β) and the inference rule for
necessitation (N).
• Completeness: following again the canonical model construction.
• Decidability: showing that ModSyll has the finite model property.
The proof goes through a suitable modification of filtration method
from modal logic(see [12, p. 182]).
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14. Thesis Structure
• In Section 2, we introduce the notions of (abstract) syllogistic
structure and standard syllogistic structure. Then we prove a
representation theorem that shows that every (abstract) syllogistic
structure can be represented as a standard syllogistic structure.
• In Section 3, we introduce a propositional logic related to syllogistic
structure, which we call a (propositional) syllogistic.
• In Section 4, we introduce a modal syllogistic as a combination of
propositional syllogistic with the Kripke modal system.
• In Section 5, we consider the connection between modal formulas
and classes of frames in modal syllogistic. We introduce the notion
extension of modal syllogistic ( notated ModSyll+Ax).
• In Section 6, we sketch without proofs some variations and ideas for
future work on modal syllogistic like: (1) temporalization of syllogistic;
(2) replacement the usage of unrestricted set variables with
Aristotelian(i.e. non-null) set variables.
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15. The End
Thank you for your attention!
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16. Bibliography I
[1] Burgess, J. (2002), Basic tense logic, Handbook of Philosophical
Logic, 2nd edition, vol.7, pp. 1-–42 Kluwer Academic Publishers.
[2] Łukasiewicz, J. (1951), Aristotle’s Syllogistic from the standpoint of
modern formal logic, Clarendon Press, Oxford, 1st edition.
[3] Łukasiewicz, J. (1957), Aristotle’s Syllogistic from the standpoint of
modern formal logic, Clarendon Press, Oxford, 2nd enlarged edition.
[4] McAllester & Givan, R. (1992), Natural language syntax and
first-order inference, Artificial Intelligence, vol.56, pp.1–20.
[5] Moss, L. (2010), Logics for Natural Language Inference (expanded
version of lecture notes from a course at ESSLLI 2010), available at
http://www.indiana.edu/~iulg/moss/notes.pdf.
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17. Bibliography II
[6] Nishihara, N., Morita, K. & Iwata. S. (1990), An extended syllogistic
system with verbs and proper nouns, and its completeness proof,
Systems and Computers in Japan, vol.21, pp.760–771.
[7] Pratt-Hartmann, I. (2004), Fragments of language, Journal of Logic,
Language and Information, vol.13, pp.207–223.
[8] Rescher, N. (1964), Aristotle’s theory of modal syllogisms and its
interpretation, The Critical Approach to Science and Philosophy,
Collier-MacMillan Limited, pp.152–177.
[9] Sahlqvist, H. (1975), Completeness and correspondence in the first-
and second- order semantics for modal logic, Proceedings of the 3rd
Scandinavian Logic Symposium. North-Holland, Amsterdam,
pp.110–143.
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18. Bibliography III
[10] Sambin, G. and Vaccaro,V.(1989), A new proof of Sahlqvist’s theorem
on modal definability and completeness, Journal of Symbolic Logic,
vol.54, pp.992–999.
[11] Shepherdson, J. (1956), On the interpretation of Aristotelian
syllogistic, Journal of Symbolic Logic, vol.21, pp.137–147.
[12] Vakarelov, D. (2008), A modal approach to dynamic ontology: modal
mereotopology, Logic and Logical Philosophy, vol.17, pp.167–187.
[13] Venema, Y. (2001), Temporal logic, The Blackwell Guide to
Philosophical Logic, Blackwell Publishers, Malden, USA, 2001, pp
203–223, available at https:
//staff.science.uva.nl/y.venema/papers/TempLog.pdf.
[14] Wedberg, A. (1948), The Aristotelian theory of classes, Ajatus
(Helsinki), vol.15, pp.299–314.
Tsvetan Vasilev (Sofia University) Modal Syllogistic May 2016 54 / 55

19. Bibliography IV
[15] Westerståhl, D. (1989), Aristotelian syllogisms and generalized
quantifiers, Studia Logica, vol.48, pp.577–585.
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