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Which is appropriate for the statement and why?

asked in Mathematical Logic by Active (4.7k points)
edited by | 50 views

2 Answers

+1 vote
Best answer

No Gold is Silver.

Let $G(x) :$ $x$ is Gold

$S(x) :$ $x$ is Silver

Interpretation 1 :  For all $x$, If $x$ is Gold then it is Not silver.

$\forall x(G(x) \rightarrow \sim S(x))$

Interpretation 2 : There does not exist any God which is Silver. Or It is Not the case that there exists some $x$ which is Gold and Silver.

$\sim \exists x(G(x) \wedge S(x))$

Your Second Formula :  

$ \sim \forall x(G(x) \rightarrow  S(x))$ : It is Not the case that All the Gold are Silver. Or There exists some $x$ which is Gold but Not Silver. 

So, This is Not a Valid formula for "No Gold is Silver".

answered by Boss (23.4k points)
selected by
+1 vote
No gold is silver

Okay, considering the domain to be set of all items in this world

Let me frame the statement that there is at least(may be in the worst case the only one) one gold that is silver and it is

$\exists x(Gold(x) \land Silver(x))$

Now if I negate the above statement, it would mean that No Gold is silver

$\lnot \exists x(Gold(x) \land Silver(x))$

Following demorgan law I rewrite above expression as

$\forall x(\lnot Gold(x) \lor \lnot Silver(x))$

Using the fact that $p \rightarrow q \equiv \lnot p \lor q$

$\forall x(Gold(x) \rightarrow \lnot Silver(x))$
answered by Boss (24.9k points)
Can I say like case 2 fails when x is gold but not silver making for all False and entire expression true. But it is possible in this statements those says some gold are silver which contradicts our given statement that no Gold should be silver.

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