1. By Mike Monge

2. 1.If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid. 2.Ceres is a planet only if it has cleared its orbital path. Ceres has not cleared its orbital path, which means that Ceres is not a planet. 3.My brother being a devout Jehovah’s Witness would imply that he doesn’t celebrate birthdays. My brother celebrates birthdays. So, my brother is not a devout Jehovah’s witness. Using letters and symbols to represent arguments can help reveal the skeletal structure of an argument, and examining its structure allows us to determine the validity of the argument. With that in mind consider the following arguments:

3. 1.If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid. 2.Ceres is a planet only if it has cleared its orbital path. Ceres has not cleared its orbital path, which means that Ceres is not a planet. 3.My brother being a devout Jehovah’s Witness would imply that he doesn’t celebrate birthdays. My brother celebrates birthdays. So, my brother is not a devout Jehovah’s witness. Each of these arguments deals with very different subject matters, but each one has the same form or structure. The validity of the argument is solely a function of its structure, and not the content of the argument.

4. 1.If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid. 2.Ceres is a planet only if it has cleared its orbital path. Ceres has not cleared its orbital path, which means that Ceres is not a planet. 3.My brother being a devout Jehovah’s Witness would imply that he doesn’t celebrate birthdays. My brother celebrates birthdays. So, my brother is not a devout Jehovah’s witness. Let’s concentrate on the first argument.

5. 1.If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid. Let’s concentrate on the first argument.

6. 1.If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid. First, we will highlight each of the simple statements.

7. the sample were an acid 1.If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid. First, we will highlight each of the simple statements.

8. the sample were an acid the sample is not an acid 1.If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid. First, we will highlight each of the simple statements.

9. the sample were an acidthe litmus paper would have turned redthe sample is not an acid 1.If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid. First, we will highlight each of the simple statements.

10. the sample were an acidthe litmus paper would have turned redThe litmus paper did not turn redthe sample is not an acid 1.If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid. First, we will highlight each of the simple statements.

11. the sample were an acidthe litmus paper would have turned redThe litmus paper did not turn redthe sample is not an acid 1.If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid. Then, we will replace the simple statements with letters.

12. Then, we will replace the simple statements with letters. SL L S

13. the sample were an acidthe litmus paper would have turned redThe litmus paper did not turn redthe sample is not an acid 1.If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid. Next, we replace the logical words with symbols. SL L S

14. Next, we replace the logical words with symbols. SL L S

15. the sample were an acidthe litmus paper would have turned redThe litmus paper did not turn redthe sample is not an acid 1.If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid. Now, with all replacements made, we can move everything into position. SL L S

16. Now, with all replacements made, we can move everything into position. SL L S

17. the sample were an acidthe litmus paper would have turned redThe litmus paper did not turn redthe sample is not an acid 1.If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid. Now, with all replacements made, we can move everything into position. SL L S

18. the sample were an acidthe litmus paper would have turned redThe litmus paper did not turn redthe sample is not an acid 1.If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid. This basic pattern is so common and useful that is has its own name. It is called modus tollens. Any argument which exhibits the pattern of modus tollens is a valid argument. SL L S

19. Let’s go back to the other arguments, one at a time. SL L S

20. 2.Ceres is a planet only if it has cleared its orbital path. Ceres has not cleared its orbital path, which means that Ceres is not a planet. Let’s go back to the other arguments, one at a time. SL L S

21. 2.Ceres is a planet only if it has cleared its orbital path. Ceres has not cleared its orbital path, which means that Ceres is not a planet. We need to find the simple statements again. SL L S

22. 2.Ceres is a planet 2.Ceres is a planet only if it has cleared its orbital path. Ceres has not cleared its orbital path, which means that Ceres is not a planet. We need to find the simple statements again. SL L S

23. 2.Ceres is a planet Ceres is not a planet 2.Ceres is a planet only if it has cleared its orbital path. Ceres has not cleared its orbital path, which means that Ceres is not a planet. We need to find the simple statements again. SL L S

24. 2.Ceres is a planet it has cleared its orbital path Ceres is not a planet 2.Ceres is a planet only if it has cleared its orbital path. Ceres has not cleared its orbital path, which means that Ceres is not a planet. We need to find the simple statements again. SL L S

25. 2.Ceres is a planet it has cleared its orbital pathCeres has not cleared its orbital pathCeres is not a planet 2.Ceres is a planet only if it has cleared its orbital path. Ceres has not cleared its orbital path, which means that Ceres is not a planet. We need to find the simple statements again. SL L S

26. 2.Ceres is a planet it has cleared its orbital pathCeres has not cleared its orbital pathCeres is not a planet 2.Ceres is a planet only if it has cleared its orbital path. Ceres has not cleared its orbital path, which means that Ceres is not a planet. Notice that the words “only if” are replaced by the horseshoe, and that “which means that” is a conclusion indicator. SL L S

27. Notice that the words “only if” are replaced by the horseshoe, and that “which means that” is a conclusion indicator. SL L S

28. 2.Ceres is a planet it has cleared its orbital pathCeres has not cleared its orbital pathCeres is not a planet 2.Ceres is a planet only if it has cleared its orbital path. Ceres has not cleared its orbital path, which means that Ceres is not a planet. Let’s look at the last argument. There is one wrinkle we haven’t yet discussed. SL L S

29. 3.My brother being a devout Jehovah’s Witness would imply that he doesn’t celebrate birthdays. My brother celebrates birthdays. So, my brother is not a devout Jehovah’s witness. Let’s look at the last argument. There is one wrinkle we haven’t yet discussed. SL L S

30. 3.My brother being a devout Jehovah’s Witness would imply that he doesn’t celebrate birthdays. My brother celebrates birthdays. So, my brother is not a devout Jehovah’s witness. As before, we will identify the simple statements. SL L S

31. 3.My brother being a devout Jehovah’s Witness 3.My brother being a devout Jehovah’s Witness would imply that he doesn’t celebrate birthdays. My brother celebrates birthdays. So, my brother is not a devout Jehovah’s witness. As before, we will identify the simple statements. SL L S

32. 3.My brother being a devout Jehovah’s Witness my brother is not a devout Jehovah’s witness 3.My brother being a devout Jehovah’s Witness would imply that he doesn’t celebrate birthdays. My brother celebrates birthdays. So, my brother is not a devout Jehovah’s witness. As before, we will identify the simple statements. SL L S

33. 3.My brother being a devout Jehovah’s Witness he doesn’t celebrate birthdays my brother is not a devout Jehovah’s witness 3.My brother being a devout Jehovah’s Witness would imply that he doesn’t celebrate birthdays. My brother celebrates birthdays. So, my brother is not a devout Jehovah’s witness. As before, we will identify the simple statements. SL L S

34. 3.My brother being a devout Jehovah’s Witness he doesn’t celebrate birthdaysMy brother celebrates birthdaysmy brother is not a devout Jehovah’s witness 3.My brother being a devout Jehovah’s Witness would imply that he doesn’t celebrate birthdays. My brother celebrates birthdays. So, my brother is not a devout Jehovah’s witness. As before, we will identify the simple statements. SL L S

35. 3.My brother being a devout Jehovah’s Witness he doesn’t celebrate birthdaysMy brother celebrates birthdaysmy brother is not a devout Jehovah’s witness 3.My brother being a devout Jehovah’s Witness would imply that he doesn’t celebrate birthdays. My brother celebrates birthdays. So, my brother is not a devout Jehovah’s witness. We can also fill in the symbols. SL L S

36. We can also fill in the symbols. SL L S

37. 3.My brother being a devout Jehovah’s Witness he doesn’t celebrate birthdaysMy brother celebrates birthdaysmy brother is not a devout Jehovah’s witness 3.My brother being a devout Jehovah’s Witness would imply that he doesn’t celebrate birthdays. My brother celebrates birthdays. So, my brother is not a devout Jehovah’s witness. There is something very different about this argument. In this case, the second part of the conditional, the consequent, is being negated, and the second premise is affirmative. We need to adjust our structure… SL L S

38. There is something very different about this argument. In this case, the second part of the conditional, the consequent, is being negated, and the second premise is affirmative. We need to adjust our structure… S L L S

39. 3.My brother being a devout Jehovah’s Witness he doesn’t celebrate birthdaysMy brother celebrates birthdaysmy brother is not a devout Jehovah’s witness 3.My brother being a devout Jehovah’s Witness would imply that he doesn’t celebrate birthdays. My brother celebrates birthdays. So, my brother is not a devout Jehovah’s witness. This argument still fits the pattern of modus tollens. The second premise asserts the opposite of the consequent, while the conclusion asserts opposite of the antecedent. S L L S

40. Click on the circle below an argument if the argument is an example of modus tollens. p  q ~q  ~p (A  B)  ~C ~C  A  B That’s right! That’s not right. Look closely. A  (~B  C) ~(~B  C)  ~A That’s right! (A  B)  ~C A  B  ~C That’s not right. Look closely.

41. For those of you who don’t trust me yet, here is a truth table to prove that modus tollens is a valid argument pattern. AB A  B ~B~A TTTFF TFFTF FTTFT FFTTT