29345 - Logic (LM)

Learning outcomes

At the end of the course, students are supposed to become acquainted with the metatheory of some selected formal system: modal logic, intuitionistic logic, Peano's arithmetic or Goedel's incompleteness theorems.

Course contents

Quantified Modal Logic

 

The course presupposes the knowledge of the basics of first-order logic as thought in a standard course of logic in the triennium

First lecture: 1st october 2018
Time 2018-19:
1° semester
Mon 17-19 room C via Zamboni 34
Tue 17-19 room C via Zamboni 34
The 15-17 room IV via Zamboni 38

1. Introduction to philosophical questions related to the modal discourse. Quine 1943, Carnap, Ruth Barcan. Quine 1947 Kripke 1959 e 1963.

2. Propositional modal languages

3. Basic semantical concepts of Kripke semantics . Frame, models based on a frame. Truth at a point of a model.Tryth in a model, validity.

4. We consider the formulas T, D, 4, E, B, Triv, Ver from a semantical point of view.

5. First order modal language, FOML.Termine. Terms and formulas. Definition of substitution of a term for a variable in terms and formulas. Examples. Alphabetic variant of a given formula.

6. Modal logics classically quantified.Axioms and rules. Definition of proof and of derivation. Admissible rules. Examples of proofs. Proofs of CBF and GF.

7. Tarski-Kripke frames. Models based on Tarski-Kripke frames. Definition of satisfaction, truth, validity.  Lemmas on the relation between satisfaction and substitution. Soundness for Q=.K.

8. Kripke frames with double domains (internal and external): valid and not valid formulas on Kripkean models with internal domains either variant or increasing or decreasing.

9.Free quantified modal logics. Axioms and rules. Proofs of relevant theorems and admissible rules.

10 Preliminaries for the completeness theorems. We first consider the logic Q.K (classically quantified K without identity). Lemma on constants. Lindenbaum-Henkin lemma and the diamond's lemma.

11. Completeness theorem for Q°=.K (free quantified K with identity)

12. Identity. Equivalence classes.Refinement of the diamond's lemma in presence of identity

13.Logics with the Barcan formulas.

14 Languages with the existence predicate. Garson's approach.

15 Indexed languages and counterpart semantics. The operation of substitution as the key mechanism responsible of many so called anomalies of quantified modal logics.

18. Epistemic logics and  term modal logics

Readings/Bibliography

G.Hughes & M.Cresswell, A New Introduction to Modal Logic, New York : Routledge, 1996

J. M. Garson, Modal logic for philosophers, Cambridge UP, 2006.

Lecture notes of the teacher

Teaching methods

Lectures and discussions.

Assessment methods

The final exam will consist in an oral discussion, in which students are supposed to prove their correct comprehension of basic notions and results related to the syntax and semantics of quantified modal logics.

Teaching tools

Slides

Office hours

See the website of Giovanna Corsi