Discrete Mathematics | Propositional Logic | Test 1 | Question: 9

Question

Consider the following statements:    
     

1. "Ralph is a dog if he’s not a puppet" can be formalized as $\neg$ (Ralph is a puppet) $\rightarrow$ (Ralph is a dog)     
      
2. "Ralph is not a dog because he’s a puppet" can't be formalizable and the reason is that ‘because’ is not truth-functional.    
        
3. "Suppose $\{s_n\}$ is monotonic. Then $\{s_n\}$ converges iff it is bounded" can be formalized as $\{s_n\}$ is monotonic $\rightarrow$ $(\{s_n\}$ converges $\leftrightarrow \{s_n\}$ is bounded) or it can also be formalized as $\{s_n\}$ is monotonic. \textit{Therefore,} $(\{s_n\}$ converges $\leftrightarrow \{s_n\}$ is bounded)     

       
Which one of the following is correct?     
     

• A. Only $(i)$ is correct     
       
• B. Only $(iii)$ is correct     
       
• C. $(i)$ and $(iii)$ are correct     
        
• D. $(i),(ii)$ and $(iii)$ are correct

Comments

@ankitgupta.1729

Sir, Is this wrong? Please explain.

(A because B) $\equiv$ (A is true because B is true) $\equiv$ (If B is true, then A is true (here B feels like the sufficient condition for A being true)).

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@KG

A logical operator or logical connective is called truth-functional if the truth-value of a whole sentence depends on the truth-values of its atomic sentences.

”and”, “or”,”not” are truth-functional. Here, “Truth-functional” comes from “truth function” which is also called as boolean function.

Now, here “because” is not truth-functional and reason is that for a particular combination of truth values of atomic sentences, the truth value of a whole sentence can’t be decided. 

Here, we have a sentence “Ralph is not a dog because he’s a puppet”.

After getting rid of indexical “he”, we can rewrite the proposition as “Ralph is not a dog because Ralph is a puppet”.

Now, if “because” is truth functional then we can decide the truth-value of “Ralph is not a dog because Ralph is a puppet” when both atomic sentences are true.

Now, if we assume, “Ralph is not a dog” and “Ralph is a puppet” both are true then you can say that being Ralph is a puppet is responsible for Ralph not being a dog though we don’t know the reason but let’s assume this is the reason.

So, True because True $\equiv$ True.

Now, if you consider a statement  “Ralph is not a dog because 2+2=4”.

Now, assuming again “Ralph is not a dog” and “2+2=4” both are true then you can say that 2+2=4 is not responsible for Ralph being not a dog.

So, True because True $\equiv$ False.

That’s why ‘because’ is not truth-functional. It is used for “causality”.

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Why isn’t ‘because’ a logical connective in propositional logic? - Mathematics Stack Exchange – refer for simple and detailed explaination on because is not a truth-functional