Semantic concepts

Revisit your answers to exercise 3A. Determine which sentences were tautologies, which were contradictions, and which were neither tautologies nor contradictions.

1. $A\rightarrow A$

2. $C\rightarrow\neg C$

3. $(A\leftrightarrow B)\leftrightarrow\neg(A\leftrightarrow\neg B)$

4. $(A\rightarrow B)\vee(B\rightarrow A)$

5. $(A\wedge B)\rightarrow(B\vee A)$

6. $\neg(A\vee B)\leftrightarrow(\neg A\wedge\neg B)$

7. $\bigl[(A\wedge B)\wedge\neg(A\wedge B)\bigr]\wedge C$

8. $[(A\wedge B)\wedge C]\rightarrow B$

9. $\neg\bigl[(C\vee A)\vee B\bigr]$

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

1. $A\rightarrow A$, $\neg A\rightarrow\neg A$, $A\wedge A$, $A\vee A$

2. $A\vee B$, $A\rightarrow C$, $B\rightarrow C$

3. $B\wedge(C\vee A)$, $A\rightarrow B$, $\neg(B\vee C)$

4. $A\leftrightarrow(B\vee C)$, $C\rightarrow\neg A$, $A\rightarrow\neg B$

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

1. $A\rightarrow A\therefore A$

2. $A\rightarrow(A\wedge\neg A)\therefore\neg A$

3. $A\vee(B\rightarrow A)\therefore\neg A\rightarrow\neg B$

4. $A\vee B,B\vee C,\neg A\therefore B\wedge C$

5. $(B\wedge A)\rightarrow C,(C\wedge A)\rightarrow B\therefore(C\wedge B)\rightarrow A$

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

1. $\neg B\wedge B$

2. $\neg D\vee D$

3. $(A\wedge B)\vee(B\wedge A)$

4. $\neg[A\rightarrow(B\rightarrow A)]$

5. $A\leftrightarrow[A\rightarrow(B\wedge\neg B)]$

6. $[(A\wedge B)\leftrightarrow B]\rightarrow(A\rightarrow B)$

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

1. $A\wedge\neg B$, $\neg(A\rightarrow B)$, $B\rightarrow A$

2. $A\vee B$, $A\rightarrow\neg A$, $B\rightarrow\neg B$

3. $\neg(\neg A\vee B)$, $A\rightarrow\neg C$, $A\rightarrow(B\rightarrow C)$

4. $A\rightarrow B$, $A\wedge\neg B$

5. $A\rightarrow(B\rightarrow C)$, $(A\rightarrow B)\rightarrow C$, $A\rightarrow C$

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

1. $\neg B$, $A\rightarrow B$, $A$

2. $\neg(A\vee B)$, $A\leftrightarrow B$, $B\rightarrow A$

3. $A\vee B$, $\neg B$, $\neg B\rightarrow\neg A$

4. $A\leftrightarrow B$, $\neg B\vee\neg A$, $A\rightarrow B$

5. $(A\vee B)\vee C$, $\neg A\vee\neg B$, $\neg C\vee\neg B$

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

1. $A\rightarrow B$, $B\therefore A$

2. $A\leftrightarrow B$, $B\leftrightarrow C\therefore A\leftrightarrow C$

3. $A\rightarrow B$, $A\rightarrow C\therefore B\rightarrow C$

   .

4. $A\rightarrow B$, $B\rightarrow A\therefore A\leftrightarrow B$

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

1. $A\vee\bigl[A\rightarrow(A\leftrightarrow A)\bigr]\therefore A$

2. $A\vee B$, $B\vee C$, $\neg B\therefore A\wedge C$

3. $A\rightarrow B$, $\neg A\therefore\neg B$

4. $A$, $B\therefore\neg(A\rightarrow\neg B)$

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

Answer each of the questions below and justify your answer.

1. Suppose that $\mathcal{A}$ and $\mathcal{B}$ are logically equivalent. What can you say about $\mathcal{A}\leftrightarrow\mathcal{B}$?

2. Suppose that $(\mathcal{A}\wedge\mathcal{B})\rightarrow\mathcal{C}$ is neither a tautology nor a contradiction. What can you say about whether $\mathcal{A},\mathcal{B}\therefore\mathcal{C}$ is valid?

3. Suppose that $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ are jointly unsatisfiable. What can you say about $(\mathcal{A}\wedge\mathcal{B}\wedge\mathcal{C})$?

4. Suppose that $\mathcal{A}$ is a contradiction. What can you say about whether $\mathcal{A},\mathcal{B}\vDash\mathcal{C}$?

5. Suppose t…

Consider the following principle:

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

Is this principle correct? Explain your answer.