Either they (Fred, George, Emily, Donna) will all graduate with honors or none of them will. Either Fred will graduate with honors or Emily and Donna both will. Fred and George will graduate with honors and only Donna and Emily will graduate with honors. Either George or Emily will graduate with honors but they won’t both graduate with honors. George won’t graduate with honors but Fred will, and Donna will graduate with honors if and only if Emily does. If Emily and Donna don’t both graduate with honors then neither George nor Fred will graduate with honors. Construct truth-functional paraphrase of each of the following sentences then provide a symbolization key and use symbols to paraphrase your sentences: 1. IF Felice vacations in Bermuda, so will Clarence. Veronica will vacation in Bermuda only if both Clarence and Robert will also do so. 2. IF either Felice or Veronica vacation in Bermuda, they both will. Clarence will vacation in Bermuda only if Robert does and neither Felice nor Veronica do. 3. If Veronica vacations in Bermuda, then Clarence will but Felice won’t. Robert will vacation in Bermuda if and only if Clarence does, and Veronica will vacation in Bermuda if and only if Felice does. 4. Veronica will vacation in Bermuda only if Clarence doesn’t and Felice will vacation in Bermuda if and only if Robert does. Felice will vacation in Bermuda only if Veronica does and Robert doesn’t. Veronica will vacation in Bermuda only if Robert does and Clarence doesn’t. For each of the following, provide a truth-functional paraphrase and then symbolize your paraphrases in SL indicating which sentence each of the sentence letters symbolizes: 1. Casablanca, The Lion in Winter, Witness for the Prosecution, The Third Man, and Charade will all be shown at this year’s classical film festival. 2. IF Phil sees Casablanca, he will enjoy Bogart’s and Bergman’s performances but he won’t hear Bogart say “Play it again, Sam.” 3. Phil will see The Lion in Winter only if Marion will, and both of them will see Charade. 4. Eric will see The Lion in Winter if and only if Betty does, and if they see it, they will love it. 5. If Witness for the Prosecution and The Lion in Winter are both screened at 8:00 pm, Marion and Phil will see Witness for the Prosecution and Eric and Betty will see The Lion in Winter. 6. Phil will see Charade if and only if Audrey Hepburn and Cary Grant are both in it, and they are. 7. IF it’s the case that if Eric likes Katherine Hepburn, then he’ll see The Lion in Winter, then if Marion likes Eric, she will see The Lion in Winter. 8. If Claude Raines, Sydney Greenstreet, and Peter Lorre were in the movie Betty saw last night, then she saw Casablanca. 9. Neither Betty nor Eric like James Coburn, but they both like Audrey Hepburn, and Audrey Hepburn is in Charade, so they will both see it.
The Correct Answer and Explanation is:
This exercise involves translating natural language sentences into the precise language of sentential logic (SL). The initial paragraph of the prompt regarding Fred, George, Emily, and Donna is a set of statements without a question and is extraneous to the tasks that follow. It will be disregarded in this answer.
The core process for this translation has three steps. First, we create a truth-functional paraphrase, which rephrases the sentence to make its logical structure more explicit. For example, “P only if Q” is paraphrased as “If P, then Q.” Second, we identify the basic, non-compound statements, known as atomic sentences. We then create a symbolization key, assigning a unique capital letter to each atomic sentence. Third, we use the key and the standard logical connectives (¬ for “not,” ∧ for “and,” ∨ for “or,” → for “if…then,” and ↔ for “if and only if”) to symbolize the paraphrased sentence.
This method removes the ambiguity inherent in everyday language. Words like “but” are translated as a simple conjunction (“and”), while phrases like “only if” or “provided that” are translated as conditional statements (“if…then”). Careful use of parentheses is crucial to ensure the scope of each connective is correct, preserving the original meaning of the sentence. This systematic translation allows for the rigorous, formal analysis of arguments and logical relationships.
Bermuda Vacation Sentences
Symbolization Key:
- F: Felice vacations in Bermuda.
- C: Clarence vacations in Bermuda.
- V: Veronica vacations in Bermuda.
- R: Robert vacations in Bermuda.
1. IF Felice vacations in Bermuda, so will Clarence. Veronica will vacation in Bermuda only if both Clarence and Robert will also do so.
- Paraphrase: If Felice vacations in Bermuda, then Clarence will vacation in Bermuda, and, if Veronica vacations in Bermuda, then both Clarence and Robert will vacation in Bermuda.
- Symbolization: (F → C) ∧ (V → (C ∧ R))
2. IF either Felice or Veronica vacation in Bermuda, they both will. Clarence will vacation in Bermuda only if Robert does and neither Felice nor Veronica do.
- Paraphrase: If Felice or Veronica vacations in Bermuda, then Felice and Veronica will both vacation in Bermuda, and, if Clarence vacations in Bermuda, then Robert will vacation in Bermuda and Felice will not and Veronica will not.
- Symbolization: ((F ∨ V) → (F ∧ V)) ∧ (C → (R ∧ (¬F ∧ ¬V)))
3. If Veronica vacations in Bermuda, then Clarence will but Felice won’t. Robert will vacation in Bermuda if and only if Clarence does, and Veronica will vacation in Bermuda if and only if Felice does.
- Paraphrase: If Veronica vacations in Bermuda, then Clarence will vacation in Bermuda and Felice will not, and, Robert will vacation in Bermuda if and only if Clarence does, and, Veronica will vacation in Bermuda if and only if Felice does.
- Symbolization: (V → (C ∧ ¬F)) ∧ (R ↔ C) ∧ (V ↔ F)
4. Veronica will vacation in Bermuda only if Clarence doesn’t and Felice will vacation in Bermuda if and only if Robert does. Felice will vacation in Bermuda only if Veronica does and Robert doesn’t. Veronica will vacation in Bermuda only if Robert does and Clarence doesn’t.
- Paraphrase: (If Veronica vacations in Bermuda, then Clarence will not) and (Felice will vacation in Bermuda if and only if Robert does) and (If Felice vacations in Bermuda, then Veronica does and Robert does not) and (If Veronica vacations in Bermuda, then Robert does and Clarence does not).
- Symbolization: ((V → ¬C) ∧ (F ↔ R)) ∧ (F → (V ∧ ¬R)) ∧ (V → (R ∧ ¬C))
Classical Film Festival Sentences
1. Casablanca, The Lion in Winter, Witness for the Prosecution, The Third Man, and Charade will all be shown at this year’s classical film festival.
- Paraphrase: Casablanca will be shown and The Lion in Winter will be shown and Witness for the Prosecution will be shown and The Third Man will be shown and Charade will be shown.
- Key: C: Casablanca will be shown. L: The Lion in Winter will be shown. W: Witness for the Prosecution will be shown. T: The Third Man will be shown. A: Charade will be shown.
- Symbolization: C ∧ L ∧ W ∧ T ∧ A
2. IF Phil sees Casablanca, he will enjoy Bogart’s and Bergman’s performances but he won’t hear Bogart say “Play it again, Sam.”
- Paraphrase: If Phil sees Casablanca, then he will enjoy Bogart’s and Bergman’s performances and he will not hear Bogart say “Play it again, Sam.”
- Key: P: Phil sees Casablanca. E: Phil will enjoy Bogart’s and Bergman’s performances. H: Phil will hear Bogart say “Play it again, Sam.”
- Symbolization: P → (E ∧ ¬H)
3. Phil will see The Lion in Winter only if Marion will, and both of them will see Charade.
- Paraphrase: If Phil sees The Lion in Winter, then Marion will see The Lion in Winter, and Phil will see Charade and Marion will see Charade.
- Key: P: Phil will see The Lion in Winter. M: Marion will see The Lion in Winter. PC: Phil will see Charade. MC: Marion will see Charade.
- Symbolization: (P → M) ∧ (PC ∧ MC)
4. Eric will see The Lion in Winter if and only if Betty does, and if they see it, they will love it.
- Paraphrase: Eric will see The Lion in Winter if and only if Betty does, and if Eric and Betty both see The Lion in Winter, then they will love it.
- Key: E: Eric will see The Lion in Winter. B: Betty will see The Lion in Winter. L: Eric and Betty will love The Lion in Winter.
- Symbolization: (E ↔ B) ∧ ((E ∧ B) → L)
5. If Witness for the Prosecution and The Lion in Winter are both screened at 8:00 pm, Marion and Phil will see Witness for the Prosecution and Eric and Betty will see The Lion in Winter.
- Paraphrase: If Witness for the Prosecution is screened at 8:00 pm and The Lion in Winter is screened at 8:00 pm, then Marion and Phil will see Witness for the Prosecution and Eric and Betty will see The Lion in Winter.
- Key: W: Witness for the Prosecution is screened at 8:00 pm. L: The Lion in Winter is screened at 8:00 pm. M: Marion and Phil will see Witness for the Prosecution. E: Eric and Betty will see The Lion in Winter.
- Symbolization: (W ∧ L) → (M ∧ E)
6. Phil will see Charade if and only if Audrey Hepburn and Cary Grant are both in it, and they are.
- Paraphrase: Phil will see Charade if and only if Audrey Hepburn is in it and Cary Grant is in it, and Audrey Hepburn is in it and Cary Grant is in it.
- Key: P: Phil will see Charade. A: Audrey Hepburn is in Charade. C: Cary Grant is in Charade.
- Symbolization: (P ↔ (A ∧ C)) ∧ (A ∧ C)
7. IF it’s the case that if Eric likes Katherine Hepburn, then he’ll see The Lion in Winter, then if Marion likes Eric, she will see The Lion in Winter.
- Paraphrase: If (if Eric likes Katherine Hepburn, then he will see The Lion in Winter), then (if Marion likes Eric, she will see The Lion in Winter).
- Key: K: Eric likes Katherine Hepburn. E: Eric will see The Lion in Winter. M: Marion likes Eric. S: Marion will see The Lion in Winter.
- Symbolization: (K → E) → (M → S)
8. If Claude Raines, Sydney Greenstreet, and Peter Lorre were in the movie Betty saw last night, then she saw Casablanca.
- Paraphrase: If Claude Raines was in the movie and Sydney Greenstreet was in the movie and Peter Lorre was in the movie, then Betty saw Casablanca.
- Key: R: Claude Raines was in the movie. S: Sydney Greenstreet was in the movie. L: Peter Lorre was in the movie. C: Betty saw Casablanca.
- Symbolization: (R ∧ S ∧ L) → C
9. Neither Betty nor Eric like James Coburn, but they both like Audrey Hepburn, and Audrey Hepburn is in Charade, so they will both see it.
- Paraphrase: If (Betty does not like James Coburn and Eric does not like James Coburn, and they both like Audrey Hepburn, and Audrey Hepburn is in Charade), then they will both see Charade.
- Key: B: Betty likes James Coburn. E: Eric likes James Coburn. A: Betty and Eric both like Audrey Hepburn. C: Audrey Hepburn is in Charade. S: Betty and Eric will both see Charade.
- Symbolization: (((¬B ∧ ¬E) ∧ A ∧ C) → S)
