Logica & Linguaggio: Tablaux

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August Emil Hawkins
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1 Logica & Linguaggio: Tablaux RAFFAELLA BERNARDI UNIVERSITÀ DI TRENTO P.ZZA VENEZIA, ROOM: 2.05,

2 Contents 1 Heuristics Heuristic (II) Exercises Tableaux method: Refutation tree Method Exercise: Entailment among given FOL formulae Exercise: Validity of an argument Animal problem Solution: Animal Problem Fallacies of reasoning Fallacies of Relevances Circular Reasoning Semantic Fallacies Inductive Fallacies Formal Fallacies Formal Fallacies: example Formal Fallacies: example

3 7.5.3 Formal Fallacies: example Formal Fallacies: example

4 1. Heuristics Prove x.( y.(r(x, y))) R(a, a) is valid. 1. ( x.( y.r(x,y))) R(a,a)) 2. x.( y.(r(x,y))) Rule (8) applied to R(a,a) Rule (8) applied to y.r(a,y) Rule (11) applied to R(a,a) Rule (11) applied to 4. Can we conclude that the x.( y.(r(x,y))) R(a,a) is valid? Let us interpret the above theorem as following: Let the natural numbers be our domain of interpretation. Let R(x,y) stand for x < y Then, x.( y.(r(x,y))) is satisfiable. E.g. 2 < 4. From this it follows that R(a,a) should be true as well, but it is not. Hence, x.( y.(r(x,y))) R(a,a) is not satisfiable in this interpretation, much less valid.

5 On line 5. Rule (11) has violeted the constraint: the term a was already used. Similarly, a could have not been used in line 4. either. Hence we don t get a contraddiction and the tableau is not closed. Note : The same term can be used many times for universal instantiation. Heuristic : When developing a semantic tableau in FOL use the existential instantiation rule (Rule 11) before the universal instantiation rule (Rule 10).

6 1.1. Heuristic (II) 1. ( x.(a(x) B(x)) x.a(x)) 2. x.(a(x) B(x)) Rule (8) applied to ( x.a(x)) Rule (8) applied to x. A(x) Rule (12) applied to A(a) Rule (11) applied to A(a) B(a) Rule (10) applied to A(a) Rule (1) applied to B(a) Rule (1) applied to 6. Closed Note: if we had used Rule 10 before we would have not been able to apply Rule 11.

7 2. Exercises Prove each of the following using semantic tableaux: 1. x.(a(x) (B(x) A(x))) 2. x.a(x) A(a) 3. ( x.a(x) x.b(x)) x.(a(x) B(x))

8 x.(a(x) (B(x) A(x))) 2. x.(a(x)) A(a) 1. ( x.(a(x) (B(x) A(x)))) 2. x.( (A(x) (B(x) A(x)))) Rule (12) applied to (A(a) (B(a) A(a))) Rule (8) applied to A(a) Rule (12) applied to (B(a) A(a)) Rule (8) applied to B(a) Rule (8) applied to A(a) Rule (8) applied to 5. Closed We cannot close the branch. 1. ( x.(a(x)) A(a)) 2. x.a(x) Rule (8) to line 1 3. A(a) Rule (8) to line 1 4. A(b) Rule (11) to line 2

9 3. ( xa(x) xb(x)) x(a(x) B(x)) 1. (( xa(x) xb(x)) x(a(x) B(x))) 2. xa(x) xb(x) Rule (8) to line 1 3. ( x(a(x) B(x))) Rule (8) to line 1 4. x (A(x) B(x)) Rule (13) to line 3 5. xa(x) Rule (1) to line 2 6. xb(x) Rule (1) to line 2 7. A(a) Rule (11) to line 5 8. B(b) Rule (11) to line 6 9. (A(a) B(a)) Rule (10) to line A(a) B(a) Rule (6) to line 9 We cannot close the right branch. We could have used b in line 9 to give (A(b) B(b)), but then the left branch could have not been closed.

10 3. Tableaux method: Refutation tree Recall: KB = α iff M(KB) M(α) in words, Knowledge Base (KB) entails sentence α if and only if α is true in all models of the KB (i.e. for all interpretations where KB is true). We use the tableaux to search for invalidity: we negate the conclusion of the entailment and try to close the tree. I.e. check whether is unsatisfiable. KB { α} If it is unsatisfiable, it means that there is no interpretation I s.t. I = KB and I = α; in other words, for all interpretations the conclusion must be true when the premises are true. Hence, the entailment is valid.

11 4. Method You have to prove: KB = φ. You check whether KB { φ} is unsatisfiable. 1. Try to close the branch as soon as you can. 2. Once you have found a branch that is completed and does not close, you have found an interpretation that falsifies the entailment, it s a counter-example.hence, you have proved that the entailment is not valid. You are done!

12 5. Exercise: Entailment among given FOL formulae Prove whether the following entailments are valid and give a counter-example (a domain and an interpretation) if they are not. x(f(x) G(x)), x(g(x)) = F(a) x(f(x) G(x)) = F(a) x(f(x)) x(g(x)), xg(x) = x F(x) 5.1. Exercise: Validity of an argument a) Check by means of tableaux method whether the argument below is valid. y(suspect(y)), x(murder(x)) = x( y(suspect(x) Murder(y))). b) Build a counterexample if the argumentation is not valid.

13 6. Animal problem Consider the following problem 1. The only animals in this house are cats. 2. Every animal that loves to look at the moon is suitable for a pet. 3. When I detest an animal, I avoid it. 4. All animals that don t prowl at night are carnivorous. 5. No cat fails to kill mice. 6. No animals ever take to me, except the ones in this house. 7. Kangaroos are not suitable for pets. 8. All animals that are carnivorous kill mice. 9. I detest animals that do not take to me. 10. Animals that prowl at night always love to look at the moon.

14 Is it true that I always avoid kangaroo? (a) Represent the facts above as FOL sentences and formalize the problem. Remember to give the keys of your formalization and add the extra information you might need in order to answer the question. (b) Give the proof of your answer by means of tableaux.

15 6.1. Solution: Animal Problem Animal(x): Cat(x): Inhouse(x): Detest(x,y): x detests y Avoid(x,y): x avoids y x is an animal x is a cat x is in this house Take(x,y): x takes to y Kan(x): x is a kangaroo x.(animal(x) Inhouse(x)) Cat(x) [1.] x.detest(r, x) Avoid(r, x) x.(animal(x) Inhouse(x)) Take(x, r) [6.] x.(animal(x) Take(x, r)) Detest(r, x) [9.] x.kan(x) Animal(x) x.cat(x) Kan(x) ( x.kan(x) Avoid(r, x)) [3.] [extra] [extra] [negation of Con.]

16 7. Fallacies of reasoning Fallacies are mistakes that occur in arguments and that superficially appear to be good arguments. There are many kinds of fallacies, and philosophers and logicians have identified patterns of bad reasoning habits. Here I give a few examples. See John Nolt, Dennis Rohatyn and Achille Varzi Logic. for a detailed description.

17 7.1. Fallacies of Relevances They occur when the premises of an argument have no bearing upon its conclusion. There are a number of these fallacies, e.g. ad hominem abusive (i.e. arguments that attack a person s age, character, family, etc. when there is no reason to take the person s views seriously). Jones advocates fluoridation of the city water supply. Jones is a convicted thief. Therefore, we should not fluoridate the city water supply. Remark Even if Jones is a convicted thief, this has no bearing on whether the water supply should be fluoridated.

18 7.2. Circular Reasoning They occur when an argument assumes its own conclusion. Such an argument is always true, but it is useless as a means of proving its conclusion. 1. Example Eating ice cream in public is immoral because it is just plain wrong. Remark The premise Eating ice cream in public is plain wrong and the conclusion Eating ice cream in public is immoral. These two sentences say basically the same think. 2. Example 1. Capital punishment is justified. 2. For our country is full of criminals who commit barbarous acts of murder. Therefore, it is perfectly legitimate to punish such inhuman people by putting them to death. Remark The conclusion and the first premise say the same thing.

19 7.3. Semantic Fallacies They occur when the language employed to express an argument has multiple meanings or is vague. E.g. 1. It is silly to fight over mere words. 2. Discrimination is just a word Therefore, it is silly to fight over discrimination

20 7.4. Inductive Fallacies They occur when an inductive probability of an argument (i.e. the probability of its conclusion given its premises) is low. 1. The patient became violently ill immediately after eating lunch. There were no sign of illness prior to eating, and she was in good spirits during the meal. 2. She is in good health overall, and her medical history shows no record of physical problems. Therefore, she was the victim of food poisoning.

21 7.5. Formal Fallacies They occur when 1.) we misapply a valid rule of inference or else 2.) follow a rule which is demonstrably invalid. If a formal fallacy is suspected, it is important to ascertain both that 1.) the rule on which the reasoning seems to be based is invalid, 2.) the argument itself is invalid (by means of giving a counter-example)

22 Formal Fallacies: example For instance: If it rains heavily tomorrow, the game will be postponed. It will not rain heavily tomorrow Therefore, the game will not be postponed. Remark 1. The argument has an invalid form: R P R P. the invalidity of the rule can be verified by e.g. truth tables. 2. The counter-example (premises both true and conclusion false.) to show the invalidity of the argument, could be, e.g., The game will be postponed because it will snow heavily, or the visiting team misses the flight etc.

23 Formal Fallacies: example For instance: If Smith inherited a fortune, then she is rich. She is reach. Therefore, she inherited a fortune. Remark This is a fallacy of affirming the consequent. 1. The argument has an invalid form: P Q Q P. The invalidity of the rule can be verified by e.g. truth tables. 2. The counter-example (premises both true and conclusion false.) E.g., Smith made the fortune by creating a software corporation.

24 Formal Fallacies: example For instance: Every sentence in this slide is well written. Therefore, this slide is well written Remark This is a typical fallacy of composition. p 1,..., p n are parts of w. p 1,..., p n have property F. Therefore, w has property F

25 Formal Fallacies: example For instance: This slide is written in English. Therefore, every sentence in this book is in English. Remark This is a typical fallacy of division. w has property F p 1,..., p n are parts of w. Therefore, p 1,..., p n have property F.

Some Basic Concepts. Highlights of Chapter 1, 2, 3.

Some Basic Concepts. Highlights of Chapter 1, 2, 3. Some Basic Concepts Highlights of Chapter 1, 2, 3. What is Critical Thinking? Not Critical as in judging severely to find fault. Critical as in careful, exact evaluation and judgment. Critical Thinking