Notes for the Logic Course 2010/2011

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1 Notes for the Logic Course 2010/2011 Propositional Logic Rosella Gennari Computer Science Free University of Bozen-Bolzano February 2, 2011

2 Disclaimer. The course notes are meant as complementary material for the students of the course of Logic at the Free University of Bozen-Bolzano. They are by no means exhaustive and students need refer to the course textbook whenever indicated. The notes and exercises are periodically updated and made available in the course web page: logic.

3 1 Contents 1 Formulae and Parsing Trees Recognition Generation Formalisation Program specification Natural language Chemical reactions Subformulae. 3 4 Truth tables Formulae Chemical reactions Validity and satisfiability Formulae and interpretations Interpretations and truth tables Satisfiability of Sets of Formulae Satisfiability of Sets of Formulae via Interpretations and Truth Tables. 4 7 Entailment Entailment via Interpretations and Truth Tables From truth tables to formulae Interpretations Biimplication and Equivalence Formalisation and Reasoning Entailment Satisfiability

4 February 2, 2011 Contents 1 Formulae and Parsing Trees 1.1 Recognition Detect which of the following expressions over P = {p, q} is a propositional formulae by building their parsing trees. Justify your answers. 1. p p 2. pq 3. p pq 4. p q p 5. p p q 1.2 Generation. Consider the parsing trees in Figure 1. For each of them, 1 specify the smallest signature, i.e., P, so that the formula is in PLP, 2 generate the formulae of the parsing trees start from the leaves, and left label the nodes as in Example of [Chiswell and Hodges, 2007] by walking upwards. 2 Formalisation 2.1 Program specification Consider the following program and its informal specifications in italics:

5 2 FORMALISATION 2 if x > 0 then y:=1 else y:=2; S fi; z:=y; // In case x 0 then y = 2. If the S program terminates then z = 2. // Formalise the two specifications in a suitable propositional language. 2.2 Natural language Formalise the following natural-language sentences in a suitable propositional languages. To sneeze or not to sneeze, that yields the question. Careful with that. It is necessary to be a bird in order to fly. It is sufficient to be a bird in order to fly. It is necessary and sufficient to be a bird in order to fly. If x + y = 2 then x = 2 y. If Italy is close to France and France is close to the Netherlands, then Italy is close to the Netherlands. If Italy is close to France and if France is close to the Netherlands, then Italy is close to the Netherlands. p p p p p Figure 1: Parsing Trees. p pq p

6 3 SUBFORMULAE Chemical reactions Under certain conditions, the following chemical reactions are possible: HCl + NaOH NaCl + H 2 O, C + O 2 CO 2, CO 2 + H 2 O H 2 CO 3. Formalise the above set of chemical reactions in a suitable propositional language. 3 Subformulae. Consider the parsing trees in 1.2. For each of them, list the subformulae of the associated formula. 4 Truth tables 4.1 Formulae Build the truth tables of the following formulae: 1. p; 2. p q p; 3. p q q q. 4.2 Chemical reactions Consider the set of propositional formulae formalising 2.3. Using truth tables, check whether the set augmented with the negation of the propositional formula for H 2 CO 3 is satisfiable. 5 Validity and satisfiability 5.1 Formulae and interpretations Find an interpretation and a formula such that the formula is true in that interpretation or: the interpretation satisfies the formula.

7 6 SATISFIABILITY OF SETS OF FORMULAE 4 Find an interpretation and a formula such that the formula is not true in that interpretation or: the interpretation does not satisfy the formula. Find a formula that cannot be true in any interpretation or: no interpretation can satisfy the formula. 5.2 Interpretations and truth tables Which of the following formulae is a tautology? Which is only satisfiable? Decide on it by means of interpretations and truth tables. 1. p p; 2. p q p; 3. q p p; 4. p p; 5. p p; 6. p q q p. 6 Satisfiability of Sets of Formulae 6.1 Satisfiability of Sets of Formulae via Interpretations and Truth Tables. Find a finite set of formulae that is satisfiable. Verify this by means of interpretations and truth tables. Find a finite set of formulae that is unsatisfiable. Verify this by means of interpretations and truth tables. 7 Entailment 7.1 Entailment via Interpretations and Truth Tables. Find a finite set of formulae Θ {φ} so that Θ = φ. Verify this by means of interpretations and truth tables.

8 7 ENTAILMENT 5 Find a finite set of formulae Θ {φ} so that Θ = φ. Verify this by means of interpretations and truth tables. 7.2 From truth tables to formulae Let P = {p, q, r}. Let φ, ψ, θ, χ be formulae of PLP with the following truth table: p q r φ ψ θ χ Is {φ ψ, θ, χ} satisfiable? Explain your answer. 2. Determine whether {φ, ψ, θ} = χ. 3. Determine whether = φ ψ θ χ.

9 7 ENTAILMENT Interpretations. Prove or refute i.e., show a counterexample each of the following assertions using interpretations i.e., Θ = φ iff every model of Θ is a model of φ or truth tables: 1. Θ { φ} is unsatisfiable iff Θ = φ. 2. if Θ = φ φ and then Θ = φ Θ = φ ; 3. if Θ = φ φ then Θ = φ; 4. if Θ = φ then Θ = φ φ. 7.4 Biimplication and Equivalence. Prove that the following formulae are valid building only the necessary rows of their truth table: p p p q r p q p r p q r p q p r Then use the above and the substitution theorem to justify the following equivalences: φ φ φ ψ χ φ ψ φ χ φ ψ χ φ ψ φ χ

10 8 FORMALISATION AND REASONING 7 8 Formalisation and Reasoning 8.1 Entailment 1. Consider the following argument. If Paul lives in Dublin, he lives in Ireland. Paul lives in Dublin. Therefore Paul live in Ireland. i Formalise the argument using a minimal propositional language; ii use the semantic argument e.g., use interpretations, truth tables and tableau to decide on the resulting entailment problem. 2. Consider the following argument. If Spain reached the World Cup finals, then either Ireland did not slip up or Denmark played very well. If Spain reached the World Cup finals then Ireland slipped up. Denmark did not play very well. Therefore, Spain reached the World Cup finals if and only if Ireland slipped up. i Formalise the argument as an entailment using a minimal propositional language; ii use the semantic argument e.g., use interpretations, truth tables or tableaux to decide on the resulting entailment. 4. Consider the following situation. The satellite of Asimovland is inhabited by exactly one man and two robots, namely, Al and Bob. Both robots are subject to the following laws: 1. a robot protects an inhabitant of Asimovland if and only if the robot does not harm the inhabitant; 2. if Al harms the man or Bob then Bob protects the man; 3. it is necessary that Al protects itself for Bob to harm Al; 4. it is sufficient that Bob protects the man for Al to harm himself; Therefore Bob does not protect Al.

11 8 FORMALISATION AND REASONING 8 i Formalise the laws and entailed sentence Bob does not protect Al using a suitable propositional language. ii Use the semantic argument e.g., use interpretations, truth tables or tableaux to decide on the resulting entailment. Hint: first rewrite the first law taking care that Asimovland has three inhabitants Al, Bob, the man and two robots Al, Bob. The resulting formalisation is as follows. 1 ApM AhM, BpM BhM, ApB AhB, BpA BhA, ApA AhA, BpB BhB; 2 AhM AhB BpM; 3 BhA ApA; 4 BpM AhA; 5 BpA. 8.2 Set Satisfiability 1. Consider the following argument and question. 1 If Paul lives in Dublin, he lives in Ireland. 2 Paul lives in Dublin. 3 Paul lives in Ireland. i Formalise 1, 2 and 3 using a minimal propositional language; ii build a truth table or use interpretations to decide on the satisfiability of the resulting set of formulae. 2. Suppose that an island is inhabited by exactly two persons, Angelo and Roberto. Given this, consider the following argument. 1 If Angelo shaves an inhabitant then this shaves Roberto. 2 Moreover, if Roberto shaves an inhabitant then this does not shave Angelo. 3 Roberto shaves himself. i Formalise 1, 2 and 3 using a suitable propositional language and ii build a truth table or use interpretations to decide on the satisfiability of the resulting set of formulae.

12 REFERENCES 9 References [Chiswell and Hodges, 2007] Chiswell, I. and Hodges, W Mathematical Logic. Oxford University Press.

Logic: Propositional Logic Truth Tables

Logic: Propositional Logic Truth Tables Raffaella Bernardi bernardi@inf.unibz.it P.zza Domenicani 3, Room 2.28 Faculty of Computer Science, Free University of Bolzano-Bozen http://www.inf.unibz.it/~bernardi/courses/logic06