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+3 votes
“Not every satisfiable logic is valid”

 Representation of it will be $1)\sim \left ( \forall S(x)\rightarrow V(x) \right )$


$2)\sim \left ( \forall S(x)\vee V(x) \right )$

Among $1)$ and $2)$, which one is correct? and why?
in Mathematical Logic by Veteran | 230 views

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  

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.
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.

1 Answer

+5 votes
Best answer

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)

= $\sim ( \forall(x) S(x) \rightarrow V(x) )$

by Boss
selected by
please correct me if wrong
No. Your first and second English statements are not equal.

"Every satisfiable logic is not valid" (Every student is not AIR-1)

if u do like this, isnot there chances of mistake??

U are working with meaning of this statement. But sometime logic work right rather  meaning of statement

Please check now.

why "if....then"??

"is" represents by $"\rightarrow" ??$

if I say like this

"every logic satisfiable AND valid"

then where is error?


"every logic satisfiable AND valid" will not hold.

Because valid is depending on satisfiable.

AND means no dependency.



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