Ikasgai honetan Logikako hizkuntza eta sistemetarako sarrera egiten da. Horretarako lehendabizi hizkuntza formal bat eta hizkuntza naturalen arteko ezberdintasunak azaltzen dira eta gero proposizioen hizkuntza logikoa eta hizkuntza honi lotutako kontzeptuak aurkeztuko dira. Kontzeptu horiek gai-zerrendan islatuta azaltzen dira. Kontzeptu eta metodo logikoak lantzeko ariketa praktikoak egitea beharrezkoa da.

Natural language and formal language. Logical form. The notion of logical consequence. Argumentation and Logic. Fallacies. Syntax and semantics of propositional logic. Formal systems for propositional logic. Metalogic: basic concepts and results.

Compulsory course belonging to the module of 'Logic'. Necessary to reach competence in that field and abilities in its applications in philosophical reasoning.

The coordinator of the course will be in charge of the horizontal coordination. The person responsible for vertical coordination is the coordinator of the degree in philosophy.

1. Natural language and formal language. Basic notions on formal languages. Language and metalanguage: use and mention. Semantics and Calculus. From formal languages to formal systems. Formal propositional logic and formal first order predicate logic. Logic and metalogic. Validity and demonstration. Semantic consequence and syntactic consequence.

2. Propositional logic: The formal language PL. Systems of notation. Semantics: Truth-values, interpretation. n-adic connectives in a m-valued propositional logic.

3. Main connectives: Truth-values. Evaluation of formulas by means of truth-tables. Semantic equivalence among formulas. Tautology. Contradiction. Satisfaction of a formula. Neutral (contingent) formula. Satisfaction of a set of formulas.

4. Reduction of connectives. Functional completeness: Complete sets of connectives. Other decision procedures: Reductio ad absurdum, conjunctive and disjunctive normal forms.

5. Model of a formula and of a set of formulas. Semantic deduction: definition and properties. The 'theorem' of deduction. Semantic tables: Truth-trees.

6. Axiomatic systems for propositional logic. The system of Principia Mathematica (PM). Syntactic consequence in PM. The system of Church 1956 (SL). Syntactic consequence in SL. The 'theorem' of deduction: Applications. Other formal systems.

7. Gentzen-type natural formal system for propositional logic. Primitive rules and derived rules. Demonstration and derivation (syntactic consequence).

8. Introduction to non-classical propositional logic. Deviant logic and extended logic. Many-valued logic: the semantics of Lukasiewicz, Kleene and Post. The fundamentals of intuitionistic logic.

9. Modal propositional logic. Main axiomatic systems. Semantics: Modal frames, Kripkean models, evaluation of formulas (satisfaction and truth).

10. Introduction to metalogic. The SL system: soundness, simple and absolute consistency. The theorem of deduction. The semantic completeness of SL. Decidability. Independence of axioms and rules in formal systems for propositional logic.

The whole syllabus will be taught by the professor. Some sessions will be devoted to the discussion of the following texts (which must be previously read and analyzed by all students:

1. A. J. Ayer, Language, Truth and Logic(chapters 1-3).

2. S. Haack, Philosophy of Logics (chapters 10-12).



In the teaching hours for exercises in the room solutions will be given to the exercises in the Notebook, which will be distributed at the beginning of the course, and also to the exercises given by the professor at the end of each topic.

• Final Assessment System
• Tools and qualification percentages:
  • Written test to be taken (%): 50
  • Realization of Practical Work (exercises, cases or problems) (%): 40
  • Exhibition of works, readings ... (%): 10

Students will be graded on the basis of three components: 50% for the final written examination, which will be composed by exercises, theoretical questions and discussions of one short text from Ayer's chapters and another one from Haack's chapters; 25% for finding the solutions to the exercises in the Notebook, in the frame of the teaching hours for exercises in the room; 25% for the solutions to the exercises in small groups, given differently to each group by the professor and presented in the teaching hours for exercises in the room.

Klaserako eskatutako lanak eginda badaude, dauden portzentaiekin kontatuko dira notarako. Kasu honetan azterketa %50 balioko du.

Ariketa horiek ez badira egin edo aprobatzeko moduan ez badaude orduan azterketaren nota %100ekoa izango da.

1. Notebook for Exercises.
2. All teaching materials distributed as a supplement for each topic.

Basic bibliography

Copi, I., Introduction to Logic. New York: Macmillan, 1953.

Lemmon, E.J., Beginning Logic. 7th printing. Indianapolis: Hackett, 1988.

Mates, B., Elementary Logic. Oxford: Oxford University Press, 1972.

Newton-Smith, W.H., Logic. An Introductory Course. London: Routledge, 1985.

Restall, G., Logic. An Introduction. London: Routledge, 2006.

Tomassi, P., Logic. London: Routledge, 1999.

In-depth bibliography

Church, A., Introduction to Mathematical Logic (revised ed.). Princeton: P.U.P., 1956.
Detlefsen, M. et al., Logic from A to Z. London: Routledge, 1999.
Enderton, H.B., A Mathematical Introduction to Logic. London: Academic Press, 1972.
Gabbay, D. and F. Guenthner (eds.), Handdbook of Philosophical Logic, Vols. II and III. new edition. Dordrecht: Kluwer, 1994.
Goldrei, D., Propositional and Predicate Calculus. A Model of Argument. London: Springer-Verlag, 2005.
Haack, S., Deviant Logic, Fuzzy Logic. Chicago: University of Chicago Press, 1996.
Hamilton, A.G., Logic for Mathematicians, revised edition. Cambridge: Cambridge University Press, 1978.
Hilbert, D. and W. Ackermann, Principles of Mathematical Logic. New York: Chelsea, 1950.
Honderich, T. (ed.), The Oxford Companion to Philosophy. Oxford: Oxford University Press, 1995.
Hughes, G. and M. Cresswell, An Introduction to Modal Logic, London: Methuen, 1968.
Hughes, G. and M. Cresswell, A Companion to Modal Logic, London: Methuen. 1984
Hughes, G. and M. Cresswell, A New Introduction to Modal Logic. London: Routledge, 1996.
Kleene, S., Introduction to Metamathematics. Princeton: Van Nostrand, 1952.
Mendelson, E., Introduction to Mathematical Logic, 2nd edition. New York: Van Nostrand, 1979.
Read, S., Thinking about Logic. Oxford: Oxford University Press, 1995.
Smullyan, R., First-Order Logic. London: Constable, 1968.

Journals

1. Journal of Philosophical Logic.
2. The Bulletin of Symbolic Logic.
3. History and Philosophy of Logic.
4. Notre Dame Journal of Formal Logic.
5. Journal of Applied Logic.
6. Journal of Logic, Language, and Information.

Web addresses

http://plato.stanford.edu
http://www.iep.utm.edu