Truth Tables

1. Truth Tables Hurley 6.2 - 6.4

2. Truth Function Truth Table Truth value assignment Tautologous Self-contradictory Contingent Logically equivalent Contradictory Consistent Inconsistent Valid Invalid Concepts

3. How to… • Use truth tables to test • Sentences for tautolgousness, self-contradiction or contingency • Pairs of sentences for equivalence or contradiction • Sets of sentences for consistency or inconsistency • Arguments for validity or invalidity • Determine validity and invalidity from information about premises and conclusion

4. Truth Tables for the Connectives • Given the truth tables for the connectives we can compute the truth value of sentences built out of them if we know the truth values of their parts. • We can do this because the connectives are truth functional!

5. Truth Value Assigment • Each row of a truth table represents a truth value assignment: an assignment of truth values to the sentence letters. • So, in the exercise where you were given truth values for the sentence letters and asked to compute the truth value of the whole sentence the directions gave you a truth value assignment. • We can think of truth value assignments as a possible worlds (or really sets of possible worlds) • And a complete truth table as representing all possible worlds

6. Sentences Tautologous Self-contradictory Contingent Pairs of sentences Equivalent Contradictory Neither Truth Table Tests • Set of sentences • Consistent • Inconsistent • Arguments • Valid • Invalid

7. Sentences • Tautology (tautologous sentence) • Necessarily true • True in every truth value assignment • Self-contradictory sentence • Necessarily false • False in every truth value assignment • Contingent sentence • Neither necessarily true nor necessarily false • True in some truth value assignments, false in others

8. Testing Sentences for Tautologousness • Write the sentence

9. Testing Sentences for Tautologousness • Write the sentence • Determine the number of rows (for n sentence letters, 2n rows)

10. Testing Sentences for Tautologousness • Write the sentence • Determine the number of rows (for n sentence letters, 2n rows) • Identify the main connective and box the column underneath it Now we need to assign truth values to each sentence letter on each row of the column underneath it. We assign these truth values according to a standard pattern.

11. Why? And what pattern? • We want the truth table to display all possible truth value assignments for the sentence letters without duplicating any, so we adopt a convention to guarantee that. • The column under the first sentence letter gets half true, half false; the column under the second sentence letter has half true, half false for rows where the first is true and half true, half false for rows where the first is false; the column under the third subdivides in the same way, and so on.

12. Etc… The column for the first type letter is half T and half F, the second subdivides that, the third subdivides the second, and so on... 1 TF 2 T TTFF TF F 3 T T TT TFT F TT F FF T TF TFF F TF F F 4 T T T TT T TFTT F TTT F FT F T TT F TFT F F TT F F FF T T TF T TFF T F TF T F FF F T TFF TFF F F TF F F F We don’t give yougreat big truth tableson tests because wehave to grade them!

13. Testing Sentences for Tautologousness • Write the sentence • Determine the number of rows (for n sentence letters, 2n rows) • Identify the main connective and box the column underneath it • Assign truth values to sentence letters according to pattern, duplicating columns under same sentence letters. T T F F

14. Testing Sentences for Tautologousness • Write the sentence • Determine the number of rows (for n sentence letters, 2n rows) • Identify the main connective and box the column underneath it • Assign truth values to sentence letters according to pattern, duplicating columns under same sentence letters. T T T T F F F F

15. Testing Sentences for Tautologousness • Write the sentence • Determine the number of rows (for n sentence letters, 2n rows) • Identify the main connective and box the column underneath it • Assign truth values to sentence letters according to pattern, duplicating columns under same sentence letters. T T T T F T F T F F F F Now we’ve assigned truth values to all the sentence letters and are ready to compute truth values for the whole sentence working from smaller to larger subformulas.

16. Testing Sentences for Tautologousness • Write the sentence • Determine the number of rows (for n sentence letters, 2n rows) • Identify the main connective and box the column underneath it • Assign truth values to sentence letters according to pattern, duplicating columns under same sentence letters. • Compute truth values T T T T F T F T F F F F We want truth values for ~ P in the column under its main connective. We’ll compute them from the truth values under P given the truth table for negation.

17. Testing Sentences for Tautologousness • Write the sentence • Determine the number of rows (for n sentence letters, 2n rows) • Identify the main connective and box the column underneath it • Assign truth values to sentence letters according to pattern, duplicating columns under same sentence letters. • Compute truth values F T T T F T F T T F T F T F F F Got it! The truth values for ~ P are in the column under its main connective.

18. Testing Sentences for Tautologousness • Write the sentence • Determine the number of rows (for n sentence letters, 2n rows) • Identify the main connective and box the column underneath it • Assign truth values to sentence letters according to pattern, duplicating columns under same sentence letters. • Compute truth values F T T T F T F T T F T F T F F F Now we want to computer truth values for Q  P so we’ll look at the truth values for its antecedent and consequent.

19. Testing Sentences for Tautologousness • Write the sentence • Determine the number of rows (for n sentence letters, 2n rows) • Identify the main connective and box the column underneath it • Assign truth values to sentence letters according to pattern, duplicating columns under same sentence letters. • Compute truth values F T T T T F T F T T T F T F F T F F T F We’ve computed truth values for Q ⊃ P and now have what we need to compute truth values for the whole sentence we’re testing

20. Testing Sentences for Tautologousness • Write the sentence • Determine the number of rows (for n sentence letters, 2n rows) • Identify the main connective and box the column underneath it • Assign truth values to sentence letters according to pattern, duplicating columns under same sentence letters. • Compute truth values F T T T T F T F T T T F T F F T F F T F At last we can compute truth values for ~P ∨ (Q ⊃ P)! To do that we look at the truth values for ~P and Q ⊃ P, which are under their main connectives.

21. Testing Sentences for Tautologousness • Write the sentence • Determine the number of rows (for n sentence letters, 2n rows) • Identify the main connective and box the column underneath it • Assign truth values to sentence letters according to pattern, duplicating columns under same sentence letters. • Compute truth values F T T T T T F T T F T T T F T T F F T F T F T F Now we have truth values for ~P∨(Q ⊃ P)in the main column of the truth table--the boxed column under the main connective.

22. Testing Sentences for Tautologousness • Write the sentence • Determine the number of rows (for n sentence letters, 2n rows) • Identify the main connective and box the column underneath it • Assign truth values to sentence letters according to pattern, duplicating columns under same sentence letters. • Compute truth values • Read down the main column F T T T T T F T T F T T T F T T F F T F T F T F The truth table is complete! Now we just have to read down the main column to determine whether the sentence is tautologous, self-contradictory or contingent.

23. Testing Sentences for Tautologousness • Write the sentence • Determine the number of rows (for n sentence letters, 2n rows) • Identify the main connective and box the column underneath it • Assign truth values to sentence letters according to pattern, duplicating columns under same sentence letters. • Compute truth values • Read down main column: • Tautologous: all T • Self-contradictory: all F • Contingent: neither all T nor all F F T T T T T F T T F T T T F T T F F T F T F T F

24. Done! It’s a tautology! • Write the sentence • Determine the number of rows (for n sentence letters, 2n rows) • Identify the main connective and box the column underneath it • Assign truth values to sentence letters according to pattern, duplicating columns under same sentence letters. • Compute truth values • Read down main column: • Tautologous: all T • Self-contradictory: all F • Contingent: neither all T nor all F F T T T T T F T T F T T T F T T F F T F T F T F Tautologous, self-contradictory or contingent? Tautologous

25. Pairs of Sentences • Equivalent • Necessarily have same truth value • Have same truth value in every truth value assignment • Contradictory • Necessarily have opposite truth value • Have opposite truth value in every truth value assignment • Neither • Neither equivalent nor contradictory

26. Testing Pairs of Sentences for Equivalence ~ (P  Q) / ~ P  ~ Q • Write the sentences side by side with a slash between them.

27. Testing Pairs of Sentences for Equivalence ~ ( P  Q ) / ~ P  ~ Q • Write the sentences side by side with a slash between them. • Determine the number of rows for number of sentence letters in both sentences.

28. Testing Pairs of Sentences for Equivalence ~ ( P  Q ) / ~ P ~ Q • Write the sentences side by side with a slash between them. • Determine the number of rows for sentence letters in both sentences. • Identify the main connectives and box the columns underneath them. Be careful about identifying main connectives! The main connective of “~(PQ)” is “~”, not “”!

29. Testing Pairs of Sentences for Equivalence ~ ( P  Q ) / ~ P ~ Q • Write the sentences side by side with a slash between them. • Determine the number of rows for number of sentence letters in both sentences. • Identify the main connectives and box the columns underneath them. • Assign truth values according to pattern, duplicating columns under same sentence letters for both sentences. T T T T T F T F F T F T F F F F

30. Testing Pairs of Sentences for Equivalence ~ ( P  Q ) / ~ P  ~ Q • Write the sentences side by side with a slash between them. • Determine the number of rows for number of sentence letters in both sentences. • Identify the main connectives and box the columns underneath them. • Assign truth values according to pattern, duplicating columns under same sentence letters for both sentences. • Compute. T T T T T T T F T F F T T F T F F F F F

31. Testing Pairs of Sentences for Equivalence ~ ( P  Q ) / ~ P  ~ Q • Write the sentences side by side with a slash between them. • Determine the number of rows for number of sentence letters in both sentences. • Identify the main connectives and the columns underneath them. • Assign truth values according to pattern, duplicating columns under same sentence letters for both sentences. • Compute. F T T T T T F T T F T F F F T T F T T F F F F F

32. Testing Pairs of Sentences for Equivalence ~ ( P  Q ) / ~ P ~ Q • Write the sentences side by side with a slash between them. • Determine the number of rows for number of sentence letters in both sentences. • Identify the main connectives and the columns underneath them. • Assign truth values according to pattern, duplicating columns under same sentence letters for both sentences. • Compute. F T T T F T T F T T F F T F F F T T T F T T F F F T F F

33. Testing Pairs of Sentences for Equivalence ~ ( P  Q ) / ~ P ~ Q • Write the sentences side by side with a slash between them. • Determine the number of rows for number of sentence letters in both sentences. • Identify the main connectives and the columns underneath them. • Assign truth values according to pattern, duplicating columns under same sentence letters for both sentences. • Compute. F T T T F T F T F T T F F T T F F F T T T F F T T F F F T F T F

34. Testing Pairs of Sentences for Equivalence ~ ( P  Q ) / ~ P ~ Q • Write the sentences side by side with a slash between them. • Determine the number of rows for number of sentence letters in both sentences. • Identify the main connectives and the columns underneath them. • Assign truth values according to pattern, duplicating columns under same sentence letters for both sentences. • Compute. F T T T F T F F T F T T F F T F T F F F T T T F F F T T F F F T F T T F

35. Testing Pairs of Sentences for Equivalence ~ ( P  Q ) / ~ P  ~ Q • Write the sentences side by side with a slash between them. • Determine the number of rows for number of sentence letters in both sentences. • Identify the main connectives and the columns underneath them. • Assign truth values according to pattern, duplicating columns under same sentence letters for both sentences. • Compute. • Now read down the main columns row by row to determine whether the sentences are equivalent, contradictory or neither. F T T T F T F F T F T T F F T F T F F F T T T F F F T T F F F T F T T F The truth table is complete and we’re ready to read it to see what it tells us.

36. Testing Pairs of Sentences for Equivalence ~ ( P  Q ) / ~ P ~ Q • We compare the truth values in the main columns row by row to see whether they’re same or opposite • We determine whether the sentences are equivalent, contradictory or neither as follows: • Equivalent: same in every row • Contradictory: opposite in every row • Neither: neither equivalent nor contradictory F T T T F T F F T F T T F F T F T F F F T T T F F F T T F F F T F T T F Equivalent, contradictory or neither?Equivalent

37. Sets of Sentences • Consistent • They can all be true together • There is some truth value assignment that makes all of the sentences true • Inconsistent • Not consistent: they can’t all be true together • There is no truth value assignment that makes all of the sentences true

38. Testing Sets of Sentences for Consistency Any one of them could win…but they can’t all win.

39. Testing Sets of Sentences for Consistency P  Q / ~ P  Q / Q T T T F T F T T • Do the truth table for the sentences in the usual way • We want to see whether there’s a truth value assignment that makes all the sentences true • If there is, the set of sentences is consistent. • If there isn’t, the set of sentences is inconsistent. T F F F T F F F F T T T F T T T F T F T F F F F

40. Testing Sets of Sentences for Consistency P  Q / ~ P  Q / Q T T T F T F T T • We read across the main columns row by row • Each row represents a truth value assignment • If there’s a row in which all main columns have T the set of sentences is consistent. • If there’s no row in which all main columns have T the set of sentences is inconsistent. T F F F T F F F F T T T F T T T F T F T F F F F

41. Testing Sets of Sentences for Consistency P  Q / ~ P  Q / Q T T T F T F T T This rowdoesn’tshowanything! T F F F T F F F This rowshowsconsistency F T T T F T T T F T F T F F F F Consistent or inconsistent?Consistent We talk about sets of sentences being consistent or inconsistent. We don’t talk about rows of a truth table being consistent or inconsistent--that makes no sense!

42. Testing Sets of Sentences for Consistency P  Q / P ~ Q / Q T T T T F T T F T F F T T F F T F T T F F F T T F T F F T F F F Consistent or inconsistent?Inconsistent Suppose things were a little different… Now there’s no row where all main columns get T so this set of sentences is inconsistent!

43. Arguments • Valid • It’s not logically possible for all the premises to be true and the conclusion false • There is no truth value assignment that makes all the premises true and the conclusion false. • Invalid • Not valid. • There is some truth value assignment that makes all the premises true and the conclusion false

44. Testing Arguments for Validity P ⊃ Q / ~ Q // P ∨ Q T T T F T T T T • Do the truth table with slashes between premises and a double slash between the last premise and the conclusion • We want to see whether there’s a truth value assignment that makes all the premises true and the conclusion false • If there is, the argument is invalid. • If there isn’t, the argument is valid. T F F T F T T F F T T F T F T T F T F T F F F F

45. Testing Arguments for Validity P ⊃ Q / ~ Q // P ∨ Q T T T F T T T T • We read across the main columns row by row • Each row represents a truth value assignment • If there’s a row in which the main columns of all premises have T and the main column of the conclusion has F the argument is invalid. • If there’s no row in which the main columns of all premises have T and the main column of the conclusion has F the argument is valid. T F F T F T T F F T T F T F T T F T F T F F F F

46. Testing Arguments for Validity P ⊃ Q / ~ Q // P ∨ Q T T T T F T T T • The argument is invalid because there’s a row in which all premises get T and the conclusion gets F. • That shows that it’s possible for all the premises to be true and the conclusion false--which means the argument is invalid. T F F T F T T F F T T F T F T T This rowshows invalidity F T F T F F F F Valid or Invalid? invalid

47. Validity • Given certain information about premises and conclusion we can sometimes determine whether an argument is valid or invalid. • Suppose the conclusion of an argument is a tautology: does this show the argument is valid, is invalid or is this not enough information to determine whether it’s valid or invalid?

48. Conclusion is a tautology P1 / P2 / . . . Pn // C T T T T… Conclusion is true in every row Must be valid Must be invalid Can be valid or invalid

49. Conclusion is a tautology P1 / P2 / . . . Pn // C T T T T… Conclusion is true in every row There’s no row in which the conclusion is false so There’s no row in which all the premises are true and the conclusion is false so The argument must be valid. Must be valid Must be invalid Can be valid or invalid

50. A tautology follows from anything • We can prove a tautology from any set of premises—even if they have nothing to do with the tautology and • Even from the empty set of premises, i.e. a tautology can be proved from nothing at all! • And in doing proofs, that’s how we’ll prove a sentence is tautologous!