Truth Functional Logic Exercises

David Lu
July 10, 2018

Lots more exercises!

.	Shown by considering all possibilities	Shown with one example
Argument	Valid if for every possibility, if the premises are true, the conclusion is also true	Invalid if in at least one case the premises are true and the conclusion is false
Set of sentences	Jointly impossible if in every possibility at least one sentence is false	Jointly possible if in at least one case all are true
Pair of sentences	Equivalent if in every possibility they have the same truth value	Not equivalent if they differ in truth value in at least one cases
Sentence	Necessarily true if true in every possibility and necessarily false if false in every possibility	Contingent if true in at least one case and false in at least one case (two examples)

(Recall that each row of the truth table represents one possibility – one possible way the truth values can be assigned to the atomic sentences – and the entire column represents all possibilities for a sentence.)

For each of these argument forms, determine whether it is valid or invalid.

1. $P \lor Q$, $\neg Q$ $\therefore P$
    
2. $P$ $\therefore P \lor Q$
    
3. $P \lor Q$, $P$ $\therefore \neg Q$
    
4. $P \land Q$, $\neg P \lor R$ $\therefore Q \land R$
    
5. $P \lor Q$, $\neg Q \lor R$ $\therefore P \lor R$
    
6. $P \lor Q$, $\neg P \lor R$ $\therefore P \land R$
    
7. $\neg (P \land Q)$, $P \lor R$ $\therefore \neg Q \lor R$
    
8. $P \lor Q$, $\neg P$ $\therefore Q \land \neg R$
    
9. $\neg (P \land Q)$, $\neg P$ $\therefore Q$
    
10. $P \lor Q$, $R \lor S$ $\therefore Q \to Q$
     
11. $P \to (Q \lor R)$, $Q$ $\therefore P$
     
12. $P \to (Q \to R)$ $\therefore (P \to Q) \to R$
     
13. $P \to (Q \land R)$ $\therefore (P \to Q) \land R$

Using the following abbreviations, symbolize each sentence, using conjunction, disjunction, and negation.

$A$ - Alice has the car	$B$ - Boris has the car	$C$ - Carol has the car
$D$ - Dan drives	$E$ - Emily drives	$F$ - Fred drives

1. Boris and Alice don’t both have the car.
    
2. Either Boris has the car or Carol has it.
    
3. Dan and Emily both drive.
    
4. Either Dan doesn’t drive or Boris has the car.
    
5. Neither Boris nor Carol has the car.
    
6. Boris doesn’t have the car, and neither does Carol.
    
7. Dan doesn’t drive, although Boris has the car.
    
8. It’s not true that neither Boris nor Alice has the car.
    
9. It’s not true that Emily and Fred both drive.
    
10. Boris has the car and either Emily or Fred drives.
     
11. Alice has the car, and neither Emily nor Fred drives.
     
12. Neither Dan nor Fred drives, but Emily does.
     
13. Dan and Emily don’t drive, but Fred does.
     
14. Either Emily drives or else Fred or Dan drives.
     

Harder translations involving conditionals.

$I$ - Interest rates rise
$T$ - Taxes rise
$D$ - The dollar strengths on foreign markets
$B$ The trade balance improves

(Note: ’$P$ unless $Q$’ can be translated as $\neg Q \to P$, ’$P$ if $Q$’ should be translated as $Q \to P$)

1. The dollar will strengthen on foreign markets only if interest rates and taxes rise.
    
2. The dollar will not strengthen on foreign markets without both a rise in interest rates and an improvement in the trade balance.
    
3. If neither interest rates nor taxes rise, then the trade balance will improve unless the dollar strengthens on foreign markets.
    
4. Either taxes will rise without a rise in interest rates or there will be a rise in taxes with a rise in interest rates but with a dollar that is stronger on foreign markets.
    
5. Neither interest rates nor taxes will rise if the trade balance improves.
    
6. If the dollar strengthens on foreign markets, then the trade balance will not improve unless interest rates and taxes rise.
    
7. If the dollar doesn’t strengthen on foreign markets, then the trade balance won’t improve if there is no rise in taxes.
    
8. The trade balance will improve if and only if there is a rise in taxes and a strengthening of the dollar on foreign markets.

Determine whether the following sentence forms are jointly possible or jointly impossible.

1. $P \lor Q$, $P \land \neg Q$, $\neg (P \land Q)$
    
2. $P \lor Q$, $\neg (P \land Q)$, $P \land \neg Q$
    
3. $P \lor (Q \land R)$, $\neg Q \land \neg P$
    
4. $P \lor Q$, $\neg Q \lor R$, $\neg (P \lor R)$
    
5. $P \to Q$, $\neg Q \land P$
    
6. $P \to Q$, $Q \lor R$, $P \land \neg R$
    
7. $P \leftrightarrow Q$, $Q \leftrightarrow R$, $P \lor \neg R$
    
8. $P \leftrightarrow Q$, $\neg Q \lor R$, $P \land \neg R$
    
9. $P \lor Q$, $P \to R$, $Q \to R$, $\neg R$

Determine if each pair of sentence forms is equivalent or not.

1. $P \lor Q$ :: $Q \lor P$
    
2. $\neg P \lor Q$ :: $P \lor \neg Q$
    
3. $\neg (P \land Q)$ :: $\neg P \lor Q$
    
4. $\neg P \land Q$ :: $\neg (P \land Q)$
    
5. $P \lor (Q \land R)$ :: $(P \lor Q) \land R$
    
6. $\neg \neg (P \lor R)$ :: $\neg \neg P \land \neg \neg R$
    
7. $P \to Q$ :: $Q \to P$
    
8. $P \to (Q \land R)$ :: $P \to (Q \to R)$
    
9. $P \to Q$ :: $\neg Q \to \neg P$
    
10. $P \to \neg Q$ :: $Q \to \neg P$
     
11. $(P \land Q) \to R$ :: $P \to (Q \to R)$
     
12. $P \leftrightarrow Q$ :: $Q \leftrightarrow P$
     
13. $\neg (P \leftrightarrow Q)$ :: $P \leftrightarrow \neg Q$

Determine whether each of the following sentence forms is necessarily true, necessarily false, or contingent.

1. $P \lor Q$
    
2. $\neg (P \land Q) \lor P$
    
3. $(P \land Q) \lor \neg P$
    
4. $(P \lor Q) \land \neg (P \land Q)$
    
5. $(P \lor P) \land \neg P$
    
6. $(P \land Q) \lor (P \lor R)$
    
7. $(P \land Q) \land \neg (P \lor Q)$
    
8. $\neg (P \land Q) \lor ((R \lor P) \land (R \lor Q))$
    
9. $P \lor (Q \lor \neg P)$
    
10. $(P \lor \neg P) \lor Q$
     
11. $(P \land \neg P) \lor (Q \land \neg Q)$
     
12. $(P \lor Q) \lor \neg (P \lor Q)$
     
13. $P \to Q$
     
14. $P \to \neg P$
     
15. $(P \land Q) \to (Q \land P)$
     
16. $P \to (P \lor Q)$
     
17. $(P \lor \neg P) \to (P \land \neg P)$
     
18. $P \to (P \to Q)$
     
19. $P \to (Q \to P)$
     
20. $(P \land Q) \to \neg(\neg P \lor \neg Q)$

Translate the argument and show whether valid or invalid.

(’$P$ only if $Q$’ can be translated as $\neg Q \to \neg P$)

1. Either they won’t lower interest rates or they won’t rise taxes. This is because if they lower interest rates, then there will be no ferderal revenue problem, and they will raise taxes only if there is a federal revenue problem. (L - lower interest, R - raise taxes, F - federal revenue problem)

 

1. If they lower interest rates and raise taxes, then there will be no federal revenue problem. But they won’t lower taxes, so there will be a federal revenue problem. (L - lower interest, R - raise taxes, F - federal revenue problem)

Translate the sentences and determine whether they are jointly possible or not.

1. If God exists, he is omnipotent and beneficent.
   If God is beneficent, he is willing to prevent evil.
   If God is omnipotent, he is able to prevent evil.
   If there is evil, either God is not willing to prevent evil or God is not able to prevent evil.
   There is evil.
   God exists.
   (G - God exists, O - God is omnipotent, B - God is beneficent, W - God is willing to prevent evil, A - God is able to prevent evil, E - evil exists.)
    

2. Jack can’t cram for logic and go to the party.
   Jack will be unhappy unless he goes to the party.
   If Jack hasn’t been studying regularly, then if he doesn’t cram he will fail.
   If Jack fails, he will be unhappy.
   Jack has no been studying regularly, but he will not be unhappy.
   (C - Jack crams, P - Jack parties, U - Jack will be unhappy, S - Jack studies regularly, F - Jack fails)