**Benchmark
9E: The Mathematical World : Reasoning**
**(grades 9-12, page 234)**
To be convincing, an argument needs to have both true statements and valid
connections among them. Formal logic is mostly about connections among
statements, not about whether they are true. People sometimes use poor
logic even if they begin with true statements. (1 of 5)
Standard 3-4 page 143, Grades 9-12

Judge the validity of arguments
Standard 3-5 page 143, Grades 9-12

Construct simple valid arguments

Logic requires a clear distinction among reasons: A reason may be sufficient
to get a result, but perhaps is not the only way to get there; or, a reason
may be necessary to get the result, but it may not be enough by itself;
some reasons may be both sufficient and necessary. (2 of 5)
Standard 3-1 page 81, grades 5-8

Recognize and apply deductive and inductive reasoning
Standard 3-4 page 143, Grades 9-12

Judge the validity of arguments

Standard 3-5 page 143, Grades 9-12

Construct simple valid arguments

Wherever a general rule comes from, logic an be used in testing how well
it works. Proving a generalization to be false (just one exception will
do) is easier than proving it to be true (for all possible cases). Logic
may be of limited help in finding solutions to problems if one isn't sure
that general rules always hod or that particular information is correct;
most often, one has to deal with probabilities rather than certainties.
(3 of 5)
Standard 3-2 page 143, Grades 9-12

Formulate counterexamples
Once a person believes a general rule, he or she may be more likely to
notice cases that agree with it and to ignore cases that don't. To avoid
biased observations, scientific studies sometimes use observers who don't
know hat the results are "supposed" to be. (4 of 5)
Standard 3-4 page 143, Grades 9-12

Judge the validity of arguments
Very complex logical arguments can be made form a lot of small logical
steps. Computers are particularly good at working with complex logic but
not all logical problems can be solved by computers. High-speed computers
can examine the validity of some logical propositions for a very large
number of cases, although that may not be a perfect proof. (5 of 5)
Standard 3-3 page 81, Grades 5-8

Make and evaluate mathematical conjectures and arguments
Standard 3-3 page 143, Grades 9-12

Follow logical arguments