Ikasgai honetan predikatuen logika lantzen da. Horretarako lehendabizi proposizioen logikarne hizkuntza zergatik zabaldu behar dugun azaltzen da. Hasiera horren ondoren, predikatuen hizkuntza formala aurkezten da, lehenengo haren sintaxia eta gero haren semantika. Logikan zentrala den ondoriotasunaren erlazioa (sintaktikoa zein semantikoa) aurkeztu eta lantzen da ondoren eta harekin lotutako beste kontzeptu metalogiko batzuk ere. Kontzeptu horiek gai-zerrendan islatzen dira.

Knowledge of the first order predicate logic, in its semantics and in its calculus (axiomatic systems and rules systems). Main metatheoretical properties of that logic. Introduction to the basic notions of second order logic.

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

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

1. From propositional logic to predicate logic. Analysis of a sentence in its components. Quantifiers. Elementary logic: first order predicate logic.

2. Languages in first order predicate logic. Logical and non-logical constants. Type of similarity. Terms. Well-formed expressions or formulas. Scope of a quantifier. Free and bound variables. Free term for a variable in a formula. Sentences (closed formulas).

3. Set-theoretic notions for the semantics of first order predicate logic: Sets, relations, applications. Equivalence relation, order relation. Cardinals and ordinals. Number sets. Basic algebraic structures. Boolean algebra.

4. Semantics of first order predicate logic. Relational structure. Homological structures. Substructures. Assignment of values to the variables. Value of a term for an assignment. Satisfaction of a formula and of a set of formulas by an assignment in a relational structure. Truth and model. Logical validity.

5. Theory of a structure. Logically equivalent structures. Homomorphisms and isomorphisms for homological structures. Semantic consequence. Logically equivalent formulas.

6. Languages of pure predicate logic (LQp). Semantics for languages of LQp. Languages of pure monadic predicate logic (LQpm). Semantics for languages of LQpm. Foundations of syllogistics.

7. Semantic consequence in LQpm. Decision procedures. Consequences of the theorem of Löwenheim-Skolem. Semantic tableaux: logical trees.

8. Predicate logic with identity: Identity as a logical symbol. Normal interpretation of identity. Truth and validity in predicate logic with identity. Generalized quantifiers. Numerical quantifiers. Proper names and definite descriptions. Formalization and semantics of definite descriptions.

9. Axiomatic systems for first order predicate logic. The system SLQ (Church 1956). The system SLQ= for first order predicate logic with identity. Other axiomatic systems. Prenex normal forms. Skolem prenex normal forms.

10. Gentzen´s natural system for first order predicate logic. Primitive and derived rules. Derivations in the system.

11. Introduction to the metatheory of first order logic. The soundness theorem. The consistency theorem. The theorem of deduction in a formal system.

12. First order formal theory. Enumeration Lemma. Lindenbaum's Lemma. Gödel-Henkin's Lemma. The theorem of semantic completeness of first order logic. The compactness theorem. The Löwenheim-Skolem theorem. Categoricity and k-categoricity.

13. Computability. Turing Machine. Recursive functions. Undecidability of first order logic. Decidable fragments of first order logic.

14. Languages of second order predicate logic. Semantics of second order languages. Skolenm functions. Metalogical properties of second order 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. L. Wittgenstein, Tractatus logico-philosophicus.

2. A. Tarski, Introduction to logic and to the methodology of deductive sciences (chapters I, III and VI)



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.

• Continuous Assessment System
• Final Assessment System
• Tools and qualification percentages:
  • Written test to be taken (%): 50
  • Realization of Practical Work (exercises, cases or problems) (%): 50

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 Tarski's chapters and another one from Wittgenstein's Tractatus; 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 comments on Tarski's and Wittgenstein's works mentioned above and 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.

Kurtsoan zehar entregatu ez badira eskatutako ariketak eta lanak orduan azterketa idatzia izango du %100ko pisua deialdi honetan.

Bestela aurreko deialdian bezala %50

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

Basic bibliography

Copi, I & C. Cohen, Introduction to logic.

Gamut, L.T.F. (1991), Logic, Language and Meaning. Vol. I: Introduction to Logic. Chicago: The University of Chicago Press.

Mates, B., Elementary Logic

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

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

In-depth bibliography

Detlefsen, M. et al. (1999), Logic from A to Z. London: Routledge.
Enderton, H.B. (1972), A Mathematical Introduction to Logic. London: Academic Press.
Goldrei, D. (2005), Propositional and Predicate Calculus: A Model of Argument. London: Springer.
Hamilton, A.G. (1978), Logic for Mathematicians. Cambridge: Cambridge University Press.
Hamilton, A.G. (1982), Numbers, sets and axioms. Cambridge: Cambridge University Press.
Hedmann, S., (2004), A First Course in Logic. Oxford: Oxford University Press.
Machover, M. (1996), Set theory, logic and their limitations. Cambridge: Cambridge University Press.
Mendelson, E. (1979), Introduction to Mathematical Logic. Second Edition. New York: Van Nostrand.
Partee, B.H. et al. (1993), Mathematical Methods in Linguistics. Dordrecht: Kluwer.
Quine, W.V. (1981), Mathematical Logic. Revised edition. Cambridge, Mass.: Harvard University

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.
7. The Review of Symbolic Logic.

Web addresses

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