3 votes 3 votes “Not every satisfiable logic is valid” Representation of it will be $1)\sim \left ( \forall S(x)\rightarrow V(x) \right )$ or $2)\sim \left ( \forall S(x)\vee V(x) \right )$ Among $1)$ and $2)$, which one is correct? and why? Mathematical Logic discrete-mathematics mathematical-logic + – srestha asked Jun 4, 2019 srestha 956 views answer comment Share Follow See all 3 Comments See all 3 3 Comments reply Hirak commented Jun 4, 2019 reply Follow Share I think 2 because the expression of 2 can be simplified as --> there exist(S(x)) ∧ ~V(x)) where as 1 states--> ~ (if a logic is satisfiable then it is valid)..this means if a logic is satisfiable then it is not valid..but this is not the case, it may be valid may not be valid 0 votes 0 votes Arjun commented Jun 4, 2019 reply Follow Share Isn't the same example given in Rosen? Before asking please see reference books - you'll learn more rather than seeing someone's comment/answer. Only when you have genuine doubts or complex stuffs which are not in standard books, you should ask -- that's how people become GATE toppers. 3 votes 3 votes srestha commented Jun 4, 2019 reply Follow Share Actually, we blindly know, $\rightarrow$ work with $\forall$, but no meaning behind it. I want to know, if there is really any meaning behind this. 0 votes 0 votes Please log in or register to add a comment.
Best answer 5 votes 5 votes Not = ~ Every = $\forall$ logic = $x$ Satisfiable = $S( )$ Valid = $V()$ “Not every satisfiable logic is valid” ( It means that the underline statement is not true) = Not (every satisfiable logic is valid) =Not( For all logic if a logic is satisfiable then it will be valid) = $\sim ( \forall(x) S(x) \rightarrow V(x) )$ Satbir answered Jun 4, 2019 selected Jun 5, 2019 by srestha Satbir comment Share Follow See all 6 Comments See all 6 6 Comments reply Show 3 previous comments Satbir commented Jun 5, 2019 reply Follow Share Please check now. 0 votes 0 votes srestha commented Jun 5, 2019 reply Follow Share why "if....then"?? "is" represents by $"\rightarrow" ??$ if I say like this "every logic satisfiable AND valid" then where is error? 0 votes 0 votes Satbir commented Jun 5, 2019 reply Follow Share "every logic satisfiable AND valid" will not hold. Because valid is depending on satisfiable. AND means no dependency. 2 votes 2 votes Please log in or register to add a comment.