Discrete Mathematics | Propositional Logic | Test 2 | Question: 12

Question

Two sentences are said to be $\textit{contradictory}$ if one is negation of the other. A $\textit{contradiction}$ is a conjunction of two contradictory sentences i.e. it is a conjunction of the form $S \wedge \neg S.$     
A set of premises is $\textit{inconsistent}$ if a contradiction can be logically derived.       
If a true sentential interpretation of the conjunction of the premises can be found, then the premises are $\textit{consistent}.$        
Consider the following two statements:    
i. The set of four premises $S_1:$ $\{V \rightarrow L ; L \rightarrow B; M \rightarrow \neg B; V \wedge M \}$ is $\textit{inconsistent}$ because we can logically derive the contradiction $B \wedge \neg B.$     
ii.The set of three premises $S_2:$ $\{W \rightarrow A; A \rightarrow L; \neg W \wedge L\}$ is $\textit{consistent}$ because there exists an interpretation of $W,A$ and $L$ such that under this interpretation all the three premises will be true.  

Which one of the following statements is correct?   
    

• A. Only $(i)$ is correct    
       
• B. Only $(ii)$ is correct    
      
• C. Both $(i)$ and $(ii)$ are correct    
       
• D. None of the above

Answer 1

$(i)$
Given premises,  $S1: \{V \rightarrow L ; L \rightarrow B; M \rightarrow \neg B; V \land M \}.$

Let’s see if we can logically derive the contradiction $ B \land \neg B$ from the $S1$.

$\begin{align}
V \land M \vDash M \tag*{(Simplification)} \\
\{M,\ M \rightarrow \neg B\} \vDash \neg B &\text{ ...(1)} \tag*{(Modus ponens)} \\
\\
V \land M \vDash V \tag*{(Simplification)} \\
\{V \rightarrow L,\ L \rightarrow B \} \vDash V \rightarrow B \tag*{(Hypothetical syllogism)} \\
\{V,\ V \rightarrow B\} \vDash B &\text{ ...(2)} \tag*{(Modus ponens)} \\
\end{align}$

So, from $(1)$ and $(2)$ we can derive,
$\begin{align} B \land \neg B \tag*{(Conjunction)} \end{align}$

We have arrived at a contradiction,  $\therefore$ the set of premises $S1$ is inconsistent.
Hence, $(i)$ is correct.
 
$(ii)$

(thanks to ankitgupta.1729’s comment explaining sentential interpretation)

A sentence P is a sentential interpretation of sentence Q iff P can be obtained from Q by replacing the component atomic sentences of Q by other (not necessarily distinct) sentences.

A sentential interpretation of a sentence must preserve its sentential form.

So, here if we replace W with $4+4=9$, A with $5+5=10$ and L with $6+6=12$, then the conjunction of the premises, ie. $(W \rightarrow A) \land (A \rightarrow L) \land (\neg W \land L)$ becomes true.

We have found a true sentential interpretation of the conjunction of the set of premises, $S2$.
$\therefore S2$ is consistent.

Hence, $(ii)$ is correct.

Both $(i)$ and $(ii)$ are correct.

Answer: C

Comments

what is the meaning of the given statement in the question “If a true sentential interpretation of the conjunction of the premises can be found, then the premises are consistent.” ?

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‘true sentential interpretation’ means a true interpretation in propositional logic.
So the given statement, “If a true sentential interpretation of the conjunction of the premises can be found, then the premises are consistent” essentially means that if the conjunction of the premises is satisfiable then the premises are consistent.

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According to definition,

A sentence P is a sentential interpretation of sentence Q iff P can be obtained from Q by replacing the component atomic sentences of Q by other (not necessarily distinct) sentences.

A sentential interpretation of a sentence must preserve its sentential form.

For example,

$P_1: $ If SKY is playing well, then Rohit is happy.

$P_2:$ If 2+2=4 or 3+3=6 then 4+4=8

Here $P_2$ is a sentential interpretation of $P_1$ but $P_1$ is not a sentential interpretation of $P_2.$

For this question, you can assume sentential interpretation of W as “4+4=9”, A as “5+5=10” and L as “6+6=12” (in base 10), so that all the premises becomes true

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Oh, I wasn’t aware of this definition. Learned something new today.
Could you point me to some more resources about this?
Thanks!

Answer 2

 Any contradiction = False

conjunction of premises → false

 if this argument is valid then set of premises is inconsistent.

 As only one case to make it valid,  False → False

 i)

1. V→ L
2. L→ B
3. M → B’
4. V $\wedge$ M

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$\therefore$  False

premise 4: V $\wedge$ M = true at V= true and M=true

premise 1: V→ L= true as V=true and L=true

premise 2: L→ B = true as L= true and B= true

premise 3: M→ B’ = False as M=true and B’=false

On conclusion to be false can’t make all premises as true. Therefore argument is valid. Inconsistent

ii)statement is correct. If there exists at least one interpretation where all premises are true then consistent.

Interpretation means a row in truth table.

W’ $\wedge$ L= true at W’=True and L=true

A→ L= true as L= true

W→ A= true as W= false

Able to find atleast one case where conjunction of these premises is true.

Ans: C

Answer 3

1. The set of four premises $S_{1}$: $\{V → L; L→ B; M → \neg B; V \wedge M\}$ is $inconsistent$ because we can logically derive the contradiction $B \wedge \neg B$.  

Given, Set of premises: $S_{1}: \{V → L; L→ B; M → \neg B; V \wedge M\}$ 

$$\begin{array}{c} 1. V \to L \\ 2. L \to B \\3. M \to \neg B \\ 4.  V \wedge M \\ \hline \therefore B \wedge \neg B \\ \hline \end{array}$$

Conclusion: $B \wedge \neg B = F$; Let’s say, $B = T$.

So, we try to make all premises $true$ :     

Premises $4. V \wedge M \equiv T$, So, $V \equiv M \equiv T$

Premises $3. M \to \neg B \equiv T \to F \equiv F$

Premises $2. L \to B \equiv T \to T \equiv T$

Premises $1. V \to L \equiv T \to T \equiv T$

So, we can’t make all premises $True$ as premises $3.$ is $false.$

So, we can logically infer that conclusion and the given statement is $true$.

2. The set of three premises $S_{2}$: $\{W \to A; A \to L; \neg W \wedge L\}$ is $consistent$ because there exists an interpretation of $W,A$ and $L$ such that under this interpretation all the three premises will be true.

 If we can find true sentential interpretation by conjunction of given premises then they will be $consistent.$

$\therefore (W \to A)\wedge (A \to L) \wedge (\neg W \wedge L)$

$=> (W’ + A)(A’ + L)(W’L)$

$=> (W’A’ + W’L + AL)(W’L)$

$=> W’A’L + W’L + W’AL$

$=> W’L + W’AL$

$=> W’L \equiv \neg W \wedge L$

If $W = T$ then $\neg W \wedge L = F \wedge L = F$

If $W = F$ then $\neg W \wedge L = T \wedge L = L$ which can be $T$ also and $F$ also. So, it is $satisfiable.$

So, given set is $consistent.$

$Ans: C$