Exercises: Translation practice in propositional logic (with answers)

Pick a capital letter to represent each simple statement, and represent the following statements symbolically, using the tilde, dot, wedge, horseshoe and triple bar.  The answers are printed below.

 

1. Mark Twain wrote Huckleberry Finn as well as Letters from the Earth.

2. If Sam Clemens called himself “Mark Twain,” he should have put it in quotation marks.

3. Twain wrote Letters from the Earth but he did not write Ecce Homo.

4. Either the author of Joan of Arc and Letters from the Earth wrote Tom Sawyer, or else he wrote A Connecticut Yankee in King Arthur’s Court.

5. In Letters, Satan, Gabriel and Michael all wonder if creating natural law was such a good idea.

6. Satan says something sarcastic and has to leave Heaven for a while; he goes to find Earth.

7. He finds that humans believe that God spends nights sitting up watching over them, but he thinks that either they are just wrong or else they are insane.

8. If they are just wrong, it’s because they don’t think logically.

9. If they are insane, it’s because it’s part of their God-given nature.

10. God says you should forgive, but he forgave neither Adam nor Eve, and he punishes their descendants to this very day.

 

 

 

11. I think, therefore I am. (Descartes)

12. If “I think, therefore I am” is true, then I am a thinking thing.

13. If I am a thinking thing, then I am not a material thing.

14. If I am not a material thing, then I am a pure spirit or mind.

 

15. If arguments are always made up of multiple statements, then no single statement can ever be an argument.

16. If “if” always indicates the beginning of a single statement, then all “if” statements are just statements, and none of them are arguments.

17. Either a valid argument is sound or it is unsound, but no valid arguments are cogent.

18. If premises are either true or false, then arguments can’t be either true or false.

19. If an argument is weak, it’s inductive.

20. An argument is strong only if it’s inductive.

21. A necessary condition of an argument being valid is that it be deductive.

22. A sufficient condition of an argument being valid is that it be sound.

23. An argument commits the fallacy of Appeal to Authority if and only if it invokes the expertise of some person and that person is not really qualified.

24. A passage is an illustration only if it makes a claim and then provides an example to make it clear.

25. If you are the Vice President, then if your aide is found guilty of lying to federal investigators, then if you don’t go on the record to distance yourself from him, then you are going to be “under a cloud” on the cover of Time.

26. If you are the Vice President and your aide is found guilt of lying to federal investigators, then if you don’t go on the record to distance yourself from him, you will be under a cloud on Time’s cover.

27. If you are the Vice President and your aide is found guilty and you don’t go on the record, then you will be under a cloud.

28. Neither rain nor snow nor sleet nor hail will keep me from putting this letter in your mailbox.

29. If neither rain nor snow nor sleet nor hail will prevent me from putting this letter in your mailbox, then if I am neither a postal worker nor a sociopath, then I must just be a good friend.

30. If the set of all sets that are not members of themselves is a member of itself, then it is not a member of itself.

 

31. Existence is not a predicate

32. “Existence” is not a predicate.

33. If all words have both a sense and a reference, then “Alice” has to have both.

34. If Wittgenstein invented truth tables, then Hume critiqued the Argument from Design only if Kant pointed out that “existence” is not a predicate and Leibnitz called identical things “indiscernibles.”

 

Answers

 

1. Mark Twain wrote Huckleberry Finn as well as Letters from the Earth.

H ∙ L

2. If Sam Clemens called himself “Mark Twain,” he should have put it in quotation marks.

S ⊃ Q

3. Twain wrote Letters from the Earth but he did not write Ecce Homo.

L ∙ ~E

4. Either the author of Joan of Arc and Letters from the Earth wrote Tom Sawyer, or else he wrote A Connecticut Yankee in King Arthur’s Court.

(J ∙ L) v C

J = the author of Joan of Arc wrote Tom Sawyer

L = the author of  Letters from the Earth wrote Tom Sawyer

C = the author of both Joan of Arc and Letters from the Earth wrote A Connecticut Yankee in King Arthur’s Court

But this could be simplified to

T v C, where

T = The author of both Joan of Arc and Letters from the Earth wrote Tom Sawyer

C= The author of both Joan of Arc and Letters from the Earth wrote A Connecticut Yankee in King Arthur’s Court

 

5. In Letters, Satan, Gabriel and Michael all wonder if creating natural law was such a good idea.

(S ∙ G) ∙ M        or          S ∙ (G ∙ M)

6. Satan says something sarcastic and has to leave Heaven for a while; he goes to find Earth.

(S ∙ L) ∙ F

7. He finds that humans believe that God spends nights sitting up watching over them, but he thinks that either they are just wrong or else they are insane.

H ∙ (W v I)

 

8. If they are just wrong, it’s because they don’t think logically.

W ⊃ ~T

9. If they are insane, it’s because it’s part of their God-given nature.

I ⊃ G

10. God says you should forgive, but he forgave neither Adam nor Eve, and he punishes their descendants to this very day.

 

[F ∙ ~(A v E)] ∙ P      or        [F ∙ (~A ∙ ~E)] ∙ P

Remember? a negative disjunction is logically equivalent to a conjunction of negatives

 

 

 

(Descartes)

11. I think, therefore I am.

T / A

This is an argument, not a conditional, so we’ll use “/” to set the conclusion off from the premise.

12. If “I think, therefore I am” is true, then I am a thinking thing.

T ⊃ I

13. If I am a thinking thing, then I am not a material thing.

T ⊃ ~M

14. If I am not a material thing, then I am a pure spirit or mind.

~M ⊃ S

Don’t write “I am a pure spirit or mind” as a disjunction since the two words (“spirit” and “mind”) are meant to be synonyms here.

 

 

 

15. If arguments are always made up of multiple statements, then no single statement can ever be an argument.

A ⊃ ~S

16. If “if” always indicates the beginning of a single statement, then all “if” statements are just statements, and none of them are arguments.

I ⊃ (S ∙ ~A)

17. Either a valid argument is sound or it is unsound, but no valid arguments are cogent.

(S v ~S) ∙ ~C

18. If premises are either true or false, then arguments can’t be either true or false.

(T v ~T) ⊃ ~(A v ~A) tricky: “T” stands for “Premises are true” and “A” stands for “Arguments are true.”

Literally this reads “If either Premises are true or Premises are not true, then it is false that either Arguments are true or Arguments are false.”

19. If an argument is weak, it’s inductive.

W ⊃ I

20. An argument is strong only if it’s inductive.

S ⊃ I

21. A necessary condition of an argument being valid is that it be deductive.

V ⊃ D

22. A sufficient condition of an argument being valid is that it be sound.

S ⊃ V

23. An argument commits the fallacy of Appeal to Authority if and only if it invokes the expertise of some person and that person is not really qualified.

A ≡ (I ∙ ~Q)

24. A passage is an illustration only if it makes a claim and then provides an example to make it clear.

I ⊃ (C ∙ P)

25. If you are the Vice President, then if your aide is found guilty of lying to federal investigators, then if you don’t go on the record to distance yourself from him, then you are going to be “under a cloud” on the cover of Time.

V ⊃ (G ⊃ (~R ⊃ U))

26. If you are the Vice President and your aide is found guilt of lying to federal investigators, then if you don’t go on the record to distance yourself from him, you will be under a cloud on Time’s cover.

(V ∙ G) ⊃ (~R ⊃ U)

27. If you are the Vice President and your aide is found guilty and you don’t go on the record, then you will be under a cloud.

((V ∙ G) ∙ ~R) ⊃ U

28. Neither rain nor snow nor sleet nor hail will keep me from putting this letter in your mailbox.

(~R ∙ ~S) ∙ (~L ∙ ~H)

29. If neither rain nor snow nor sleet nor hail will prevent me from putting this letter in your mailbox, then if I am neither a postal worker nor a sociopath, then I must just be a good friend.

[(~R ∙ ~S) ∙ (~L ∙ ~H)] ⊃ [~(P v O) ⊃ F ]

 

30. If the set of all sets that are not members of themselves is a member of itself, then it is not a member of itself.

M ⊃ ~M

 

31. Existence is not a predicate

~P

32. “Existence” is not a predicate.

~P

 

33. If all words have both a sense and a reference, then “Alice” has to have both.

(S ∙ R) ⊃ (A ∙ I)

34. If Wittgenstein invented truth tables, then Hume critiqued the Argument from Design only if Kant pointed out that “existence” is not a predicate and Leibnitz called identical things “indiscernibles.”

W ⊃ (H ⊃ (~K ∙ L))

4 Responses to Exercises: Translation practice in propositional logic (with answers)

  1. SVEN says:

    SVEN PA MORE HANS!!!!! XD

  2. ALDIN ANG PROF SA GENPHIL SA DLSL WOOOOO!!! says:

    LEMMOR = CHIGGA
    BUNO= PERVERT
    HANS= PERVERT GOD
    JHERALD = PAA

  3. marc says:

    thank you very much for this activity. it helped me a lot to understand more this topic :))

  4. Lynn says:

    Wouldn’t this be better for #33; If (S&R), then [A&(S&R)] ?

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