Solutions to Chapter 12 Semantic concepts

A. Revisit your answers to exercise 11A. Determine which sentences were tautologies, which were contradictions, and which were neither tautologies nor contradictions.

1. 1.
   ​

   $A\to A$ Tautology

2. 2.
   ​

   $C\to \neg C$ Neither

3. 3.
   ​

   $(A\leftrightarrow B)\leftrightarrow \neg (A\leftrightarrow \neg B)$ Tautology

4. 4.
   ​

   $(A\to B)\vee (B\to A)$ Tautology

5. 5.
   ​

   $(A\wedge B)\to (B\vee A)$ Tautology

6. 6.
   ​

   $\neg (A\vee B)\leftrightarrow (\neg A\wedge \neg B)$ Tautology

7. 7.
   ​

   $\left[(A\wedge B)\wedge \neg (A\wedge B)\right]\wedge C$ Contradiction

8. 8.
   ​

   $[(A\wedge B)\wedge C]\to B$ Tautology

9. 9.
   ​

   $\neg \left[(C\vee A)\vee B\right]$ Neither

B. Use truth tables to determine whether these sentences are jointly satisfiable, or jointly unsatisfiable:

1. 1.
   ​

   $A\to A$, $\neg A\to \neg A$, $A\wedge A$, $A\vee A$ Jointly satisfiable (see line 1)

   $A$

   $A$

   $\to$

   $A$

   $\neg$

   $A$

   $\to$

   $\neg$

   $A$

   $A$

   $\wedge$

   $A$

   $A$

   $\vee$

   $A$

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2. 2.
   ​

   $A\vee B$, $A\to C$, $B\to C$ Jointly satisfiable (see line 1)

   $A$	$B$	$C$	$A$	$\vee$	$B$	$A$	$\to$	$C$	$B$	$\to$	$C$
   T	T	T	T	T	T	T	T	T	T	T	T
   T	T	F	T	T	T	T	F	F	T	F	F
   T	F	T	T	T	T	T	T	T	F	T	T
   T	F	F	T	T	F	T	F	F	F	T	F
   F	T	T	F	T	F	F	T	T	T	T	T
   F	T	F	F	T	T	F	T	F	T	F	F
   F	F	T	F	F	F	F	T	T	F	T	T
   F	F	F	F	F	F	F	T	F	F	T	F

3. 3.
   ​

   $B\wedge (C\vee A)$, $A\to B$, $\neg (B\vee C)$ Jointly unsatisfiable

   $A$

   $B$

   $C$

   $B$

   $\wedge$

   $(C$

   $\vee$

   $A)$

   $A$

   $\to$

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   $\neg$

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4. 4.
   ​

   $A\leftrightarrow (B\vee C)$, $C\to \neg A$, $A\to \neg B$ Jointly satisfiable (see line 8)

   $A$

   $B$

   $C$

   $A$

   $\leftrightarrow$

   $(B$

   $\vee$

   $C)$

   $C$

   $\to$

   $\neg$

   $A$

   $A$

   $\to$

   $\neg$

   $B$

   T

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C. Use truth tables to determine whether each argument is valid or invalid.

1. 1.
   ​

   $A\to A\therefore A$ Invalid (see line 2)

   $A$	$A$	$\to$	$A$	$A$
   T	T	T	T	T
   F	F	T	F	F

2. 2.
   ​

   $A\to (A\wedge \neg A)\therefore \neg A$ Valid

   $A$	$A$	$\to$	$(A$	$\wedge$	$\neg$	$A)$	$\neg$	$A$
   T	T	F	T	F	F	T	F	T
   F	F	T	F	F	T	F	T	F

3. 3.
   ​

   $A\vee (B\to A)\therefore \neg A\to \neg B$ Valid

   $A$	$B$	$A$	$\vee$	$(B$	$\to$	$A)$	$\neg$	$A$	$\to$	$\neg$	$B$
   T	T	T	T	T	T	T	F	T	T	F	T
   T	F	T	T	F	T	T	F	T	T	T	F
   F	T	F	F	T	F	F	T	F	F	F	T
   F	F	F	T	F	T	F	T	F	T	T	F

4. 4.
   ​

   $A\vee B,B\vee C,\neg A\therefore B\wedge C$ Invalid (see line 6)

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5. 5.
   ​

   $(B\wedge A)\to C,(C\wedge A)\to B\therefore (C\wedge B)\to A$ Invalid (see line 5)

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   $(C$

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   $(C$

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D. Determine whether each sentence is a tautology, a contradiction, or a contingent sentence, using a complete truth table.

1. 1.
   ​

   $\neg B\wedge B$ Contradiction

2. 2.
   ​

   $\neg D\vee D$ Tautology

3. 3.
   ​

   $(A\wedge B)\vee (B\wedge A)$ Contingent

4. 4.
   ​

   $\neg [A\to (B\to A)]$ Contradiction

5. 5.
   ​

   $A\leftrightarrow [A\to (B\wedge \neg B)]$ Contradiction

6. 6.
   ​

   $[(A\wedge B)\leftrightarrow B]\to (A\to B)$ Contingent

E. Determine whether each the following sentences are logically equivalent using complete truth tables. If the two sentences really are logically equivalent, write “equivalent.” Otherwise write, “Not equivalent.”

1. 1.
   ​

   $A$ and $\neg A$

2. 2.
   ​

   $A\wedge \neg A$ and $\neg B\leftrightarrow B$

3. 3.
   ​

   $[(A\vee B)\vee C]$ and $[A\vee (B\vee C)]$

4. 4.
   ​

   $A\vee (B\wedge C)$ and $(A\vee B)\wedge (A\vee C)$

5. 5.
   ​

   $[A\wedge (A\vee B)]\to B$ and $A\to B$

F. Determine whether each the following sentences are logically equivalent using complete truth tables. If the two sentences really are equivalent, write “equivalent.” Otherwise write, “not equivalent.”

1. 1.
   ​

   $A\to A$ and $A\leftrightarrow A$

2. 2.
   ​

   $\neg (A\to B)$ and $\neg A\to \neg B$

3. 3.
   ​

   $A\vee B$ and $\neg A\to B$

4. 4.
   ​

   $(A\to B)\to C$ and $A\to (B\to C)$

5. 5.
   ​

   $A\leftrightarrow (B\leftrightarrow C)$ and $A\wedge (B\wedge C)$

G. Determine whether each collection of sentences is jointly satisfiable or jointly unsatisfiable using a complete truth table.

H. Determine whether each collection of sentences is jointly satisfiable or jointly unsatisfiable, using a complete truth table.

1. 1.
   ​

   $\neg B$, $A\to B$, $A$ Unsatisfiable

2. 2.
   ​

   $\neg (A\vee B)$, $A\leftrightarrow B$, $B\to A$ Consistent

3. 3.
   ​

   $A\vee B$, $\neg B$, $\neg B\to \neg A$ Unsatisfiable

4. 4.
   ​

   $A\leftrightarrow B$, $\neg B\vee \neg A$, $A\to B$ Consistent

5. 5.
   ​

   $(A\vee B)\vee C$, $\neg A\vee \neg B$, $\neg C\vee \neg B$ Consistent

I. Determine whether each argument is valid or invalid, using a complete truth table.

1. 1.
   ​

   $A\to B$, $B\therefore A$ Invalid

2. 2.
   ​

   $A\leftrightarrow B$, $B\leftrightarrow C\therefore A\leftrightarrow C$ Valid

3. 3.
   ​

   $A\to B$, $A\to C\therefore B\to C$ Invalid.

4. 4.
   ​

   $A\to B$, $B\to A\therefore A\leftrightarrow B$ Valid

J. Determine whether each argument is valid or invalid, using a complete truth table.

1. 1.
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   $A\vee [A\to (A\leftrightarrow A)]\therefore A$ Invalid

2. 2.
   ​

   $A\vee B$, $B\vee C$, $\neg B\therefore A\wedge C$ Valid

3. 3.
   ​

   $A\to B$, $\neg A\therefore \neg B$ Invalid

4. 4.
   ​

   $A$, $B\therefore \neg (A\to \neg B)$ Valid

5. 5.
   ​

   $\neg (A\wedge B)$, $A\vee B$, $A\leftrightarrow B\therefore C$ Valid

K. Answer each of the questions below and justify your answer.

1. 1.
   ​

   Suppose that $\mathrm{𝒜︁}$ and $\mathrm{ℬ︁}$ are logically equivalent. What can you say about $\mathrm{𝒜︁}\leftrightarrow \mathrm{ℬ︁}$?

2. $\mathrm{𝒜︁}$ and $\mathrm{ℬ︁}$ have the same truth value on every line of a complete truth table, so $\mathrm{𝒜︁}\leftrightarrow \mathrm{ℬ︁}$ is true on every line. It is a tautology.

3. 2.
   ​

   Suppose that $(\mathrm{𝒜︁}\wedge \mathrm{ℬ︁})\to \mathrm{𝒞︁}$ is neither a tautology nor a contradiction. What can you say about whether $\mathrm{𝒜︁},\mathrm{ℬ︁}\therefore \mathrm{𝒞︁}$ is valid?

4. Since the sentence $(\mathrm{𝒜︁}\wedge \mathrm{ℬ︁})\to \mathrm{𝒞︁}$ is not a tautology, there is some line on which it is false. Since it is a conditional, on that line, $\mathrm{𝒜︁}$ and $\mathrm{ℬ︁}$ are true and $\mathrm{𝒞︁}$ is false. So the argument is invalid.

5. 3.
   ​

   Suppose that $\mathrm{𝒜︁}$, $\mathrm{ℬ︁}$ and $\mathrm{𝒞︁}$ are jointly unsatisfiable. What can you say about $(\mathrm{𝒜︁}\wedge \mathrm{ℬ︁}\wedge \mathrm{𝒞︁})$?

6. Since the sentences are jointly unsatisfiable, there is no valuation on which they are all true. So their conjunction is false on every valuation. It is a contradiction

7. 4.
   ​

   Suppose that $\mathrm{𝒜︁}$ is a contradiction. What can you say about whether $\mathrm{𝒜︁},\mathrm{ℬ︁}\models \mathrm{𝒞︁}$?

8. Since $\mathrm{𝒜︁}$ is false on every line of a complete truth table, there is no line on which $\mathrm{𝒜︁}$ and $\mathrm{ℬ︁}$ are true and $\mathrm{𝒞︁}$ is false. So the entailment holds.

9. 5.
   ​

   Suppose that $\mathrm{𝒞︁}$ is a tautology. What can you say about whether $\mathrm{𝒜︁},\mathrm{ℬ︁}\models \mathrm{𝒞︁}$?

10. Since $\mathrm{𝒞︁}$ is true on every line of a complete truth table, there is no line on which $\mathrm{𝒜︁}$ and $\mathrm{ℬ︁}$ are true and $\mathrm{𝒞︁}$ is false. So the entailment holds.

11. 6.
    ​

    Suppose that $\mathrm{𝒜︁}$ and $\mathrm{ℬ︁}$ are logically equivalent. What can you say about $(\mathrm{𝒜︁}\vee \mathrm{ℬ︁})$?

12. Not much. Since $\mathrm{𝒜︁}$ and $\mathrm{ℬ︁}$ are true on exactly the same lines of the truth table, their disjunction is true on exactly the same lines. So, their disjunction is logically equivalent to them.

13. 7.
    ​

    Suppose that $\mathrm{𝒜︁}$ and $\mathrm{ℬ︁}$ are not logically equivalent. What can you say about $(\mathrm{𝒜︁}\vee \mathrm{ℬ︁})$?

14. $\mathrm{𝒜︁}$ and $\mathrm{ℬ︁}$ have different truth values on at least one line of a complete truth table, and $(\mathrm{𝒜︁}\vee \mathrm{ℬ︁})$ will be true on that line. On other lines, it might be true or false. So $(\mathrm{𝒜︁}\vee \mathrm{ℬ︁})$ is either a tautology or it is contingent; it is not a contradiction.

L. Consider the following principle:

Suppose $\mathrm{𝒜︁}$ and $\mathrm{ℬ︁}$ are logically equivalent. Suppose an argument contains $\mathrm{𝒜︁}$ (either as a premise, or as the conclusion). The validity of the argument would be unaffected, if we replaced $\mathrm{𝒜︁}$ with $\mathrm{ℬ︁}$.

Is this principle correct? Explain your answer.
The principle is correct. Since $\mathrm{𝒜︁}$ and $\mathrm{ℬ︁}$ are logically equivalent, they have the same truth table. So every valuation that makes $\mathrm{𝒜︁}$ true also makes $\mathrm{ℬ︁}$ true, and every valuation that makes $\mathrm{𝒜︁}$ false also makes $\mathrm{ℬ︁}$ false. So if no valuation makes all the premises true and the conclusion false, when $\mathrm{𝒜︁}$ was among the premises or the conclusion, then no valuation makes all the premises true and the conclusion false, when we replace $\mathrm{𝒜︁}$ with $\mathrm{ℬ︁}$.