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+2 votes
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“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?
in Mathematical Logic by Veteran (117k points) | 154 views
0

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  

+2
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.
0
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

+3 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 (20.7k points)
selected by
0
please correct me if wrong
0
No. Your first and second English statements are not equal.
+1

"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

0
Please check now.
0

why "if....then"??

"is" represents by $"\rightarrow" ??$

if I say like this

"every logic satisfiable AND valid"

then where is error?

+1

"every logic satisfiable AND valid" will not hold.

Because valid is depending on satisfiable.

AND means no dependency.

 

 

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