if the water is filtered, then it does not contain lead.
the water does not contain lead.
therefore, the water is filtered.
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this argument is not valid.
if is not valid by the fallacy of the .....
here's a reference that contains all the valid argument types and the fallacy types.
http://ux.brookdalecc.edu/fac/tlt/math/136/logic.pdf
your answer is on page 7 of this document.
it's called "fallacy of the converse"
the symbolic form would be:
premise 1: p->q
premise 2: q
conclusion: therefore p
to understand why this is a fallacy, you need to go to the truth tables for the implies statement.
that truth table is shown below:
p q p->q
T T T
T F F
F T T
F F T
you can see from this table that the only time q if false is when p is true and q is false.
if p is false, then q can be true or false and the statment is still true.
the fallacy of the converse states that the following logic is invalid.
p->q
q
therefore p
it is saying that because q is true, then p must be true.
but you can see from the truth table for the implied statement that when q is true, p can either be false or true, therefore, you can't make that statement shown in the fallacy of the converse.
just because p implies q doesn't mean that q implies p.
the converse of a statement is shown below in symbolic form.
statement: p->q
converse: q->p
this can be true some of the time and can be false some of the time. it is not always true.
an example of when it is not true would be:
statement: all cats are mammals
converse: all mammals are cats
while the statement is true, its converse is clearly not true.
that's why your problem statement is invalid by the fallacy of the converse.
your statement:
statement: if the water is filtered, it does not contain lead.
converse: if the water does not contain lead, it is filtered.
nowhere in the statement does it implied the converse is true.
if the water does not contain lead, it may or may not be filtered.
not all water contains lead.