A light at the end of the tunnel Petr Cintula 1 and Carles Noguera 2 - - PowerPoint PPT Presentation

A light at the end of the tunnel

Petr Cintula1 and Carles Noguera2

1Institute of Computer Science,

Czech Academy of Sciences, Prague, Czech Republic

2Institute of Information Theory and Automation,

Czech Academy of Sciences, Prague, Czech Republic

Petr Cintula and Carles Noguera (CAS) The future of MFL II 1 / 15

Vague vs. graded properties

After years of discussion with philosophers, it seems that MFL cannot hope to give a complete explanation of the whole vagueness phenomenon. Fuzzy logics are about grades. (Hájek: logics of comparative truth) Graded properties give rise to sentences that admit mutual comparison (mathematically, they can be modelled by a function from a universe of discourse to a set of ordered grades). Vague properties are characterized by the existence of borderline cases and the fact that they generate sorites paradox. Many vague properties (but not all?) are graded: red, old, tall, rich. There are also graded that are not vague, e.g. acute angle. Graded properties are epistemologically necessary in order to pack infinitely many similar notions in just one notion.

Petr Cintula and Carles Noguera (CAS) The future of MFL II 2 / 15

Graded properties in Computer Science

Weighted structures: combinatorial optimization weighted graphs approximation algorithms weighted automata ... Research in these CS areas and research in MFL have so far completely ignored each other. First-order MFL is, actually, the logic of weighted structures!

Petr Cintula and Carles Noguera (CAS) The future of MFL II 3 / 15

What should the role of Logic be?

Logic was born as the study of valid arguments, i.e. correct reasoning. It became Mathematical Logic: a part of Mathematics that tried to provide foundations for the whole discipline. It has also become a tool for Computer Science. In all these roles, the classical paradigm imposes a violence to all properties, forcing them to be crisp, well-defined, without borderline cases.

Petr Cintula and Carles Noguera (CAS) The future of MFL II 4 / 15

Reconsidering our cathedral

Everyday reasoning (and Computer Science!) is full of graded properties and predicates.

(Old) New cathedral

MFL can be the study of reasoning with graded properties with applications to weighted structures in CS.

Petr Cintula and Carles Noguera (CAS) The future of MFL II 5 / 15

A three-layered cathedral

1

Natural language and natural reasoning scenarios: Interdisciplinary research relating Logic to Cognitive Science, Psychology, Linguistics, and Philosophy, to understand correct reasoning in natural language with graded properties.

2

Formal interpreted languages and artificial reasoning scenarios: Logical systems with specific interpretations for their formal syntax, coping with graded properties in specific contexts. This has a potential for applications to Computer Science (weighted structures).

3

Formal abstract languages and Mathematical Logic: Continuation

• f the mathematical study of general logical systems with a

graded semantics.

Petr Cintula and Carles Noguera (CAS) The future of MFL II 6 / 15

Some ideas for the first layer – 1

An example of this kind of interdisciplinary research: Keith Stenning, Michiel Van Lambalgen. Human Reasoning and Cognitive Science, MIT Press, 2008.

Petr Cintula and Carles Noguera (CAS) The future of MFL II 7 / 15

Some ideas for the first layer – 2

What is rationality? A definition from an era oriented toward theory: In rational discourse one strives to arrive at justified true believe. A definition from the current pragmatically oriented age: The agent must have a means-end competence to fit its actions or decisions, according to its beliefs or knowledge representations, to its desires or goal-structure. Are there optimal rules for conducting such activities?

Petr Cintula and Carles Noguera (CAS) The future of MFL II 8 / 15

Some ideas for the first layer – 3

Stanovich discusses rules governing reasoning using the following distinction: Normative rules: reasoning as it should be, ideally. Descriptive rules: reasoning as it is actually practiced. Prescriptive rules: norms that result from taking into account our bounded rationality with its computational limitations. There are several points view about their roles and relationships (panglossian, apologist, meliorist, eliminativist).

Petr Cintula and Carles Noguera (CAS) The future of MFL II 9 / 15

Some ideas for the first layer – 4

Cognition:

1

Cognitive explanations must refer to models, conceived of as representational mechanisms

2

which function in the same way as the phenomena being represented

3

and which are capable of generating behavior and thoughts of various kinds. The role of logic in this scheme was twofold:

1

a formal, symbolic, representation language (which is very expressive)

2

an inference mechanism generation behavior and thoughts.

Petr Cintula and Carles Noguera (CAS) The future of MFL II 10 / 15

Wason’s selection task (1966)

Proposed rule: If there is a vowel on one side, then there is an even number on the other side Which cards must you turn in order to decide if this rule is true? Correct answer: turn the cards E and 7. A majority of the subjects in the experiment chose to turn E and 4!

Petr Cintula and Carles Noguera (CAS) The future of MFL II 11 / 15

Some ideas for the first layer – 5

Blows against this view: Wason’s experiments seemingly undermined the role of logic as an inference mechanism. Slowness of logical inference mechanisms, especially when search is involved. The advent of neural network theory challenged the symbolic representational format given by logic. Result: from the position of being absolutely central in the cognitive revolution, which was founded on conceptions of reasoning, computation, and the analysis of language, the psychology of deduction has gone to being the deadbeat of cognitive psychology, pursued in a ghetto.

Petr Cintula and Carles Noguera (CAS) The future of MFL II 12 / 15

Some ideas for the first layer – 6

A marriage of Logic and Psychology: Aristotle provided the first rules of reasoning. Kant believed that logical laws constitute the very fabric of thought, thinking which does not proceed according to these laws is not properly thinking. John Stuart Mill developed a psychologism, which holds that all of thinking and knowledge are psychological phenomena and that therefore logical laws are psychological laws. An example: the law

• f noncontradiction ¬(A ∧ ¬A) represents the impossibility of

thinking contradictory thoughts at the same time. Normative and descriptive rules coincide.

Petr Cintula and Carles Noguera (CAS) The future of MFL II 13 / 15

Some ideas for the first layer – 7

A divorce: Frege is the driving force of the reaction, bringing a new emphasis

• n normativity.

Psychologism makes logic pertain to ideas only, and as a consequence it lacks resources to explain why logic is applicable to the real world. Logical and mathematical knowledge are objective, and this

• bjectivity cannot be safeguarded if logical laws are properties of

individual minds.

Petr Cintula and Carles Noguera (CAS) The future of MFL II 14 / 15

Some ideas for the first layer – 8

Let us try to reconcile them: Frege and his followers considered only one logical system: classical logic. Wason selection task only proves that people don’t reason according to the material implication of classical logic. But we are not bound to this old-fashioned conception of logic. We can apply the availability of a multiplicity of logics back onto the subject matter of discourse and psychology. People may be reasoning correctly using non-classical logics. The use of one logic or another depends on the domain in which

• ne reasons.

Stenning and Van Lambalgen explain many reasoning scenarios by means of non-monotonic logics (default reasoning).

Graded logics have not been applied yet, but they should!

Petr Cintula and Carles Noguera (CAS) The future of MFL II 15 / 15