Tony asked:

Try as I might I find myself unable to apply the 'valid' and 'invalid' deductive forms to syllogisms that
go beyond if p, then q, p. therefore q to syllogisms that introduce another factor. For example All
mammals have legs/ My cat is a mammal/ Therefore my cat has four legs. This example includes
another factor, i.e., how do I interpret the second statement? It goes something like if p, then q, BUT
q is r. Then what happens next?

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It is not clear just what your difficulty is, for I do not understand why you think you must "interpret" the
second premise. Perhaps you mean that you think you have to decide whether the second premise is
true. But you do not. For instance, suppose the second premise was "My snake is a mammal" That
statement is, of course, false. No snake is a mammal. But, nevertheless, the conclusion, "My snake
has four legs" would follow necessarily from those premises, although, of course, that conclusion
would be false, since no snakes have four legs. The central issue in logic is whether the premises
support the conclusion. But not whether the premises or conclusion are true or false. That is a
separate issue.

Ken Stern

"Valid" and "invalid" forms? What you have above is totally straightforward. Here are the forms:

Roughly speaking, according to C.S. Peirce, there are three basic types of logic, derived from the
three-part syllogism. This syllogism consists of:

R, a rule: (the beans in this bag are white), C, a case of the rule: (these beans are from the bag), E, a
result: (these beans are white).

By altering the order of the elements in this expression, Peirce realized that one could symbolize
entirely different types of thinking. Thus, deduction consists of statements in the above order: (1) R,
C, E; induction in the order (2) C, E, R; and hypothesis construction (also termed "abduction") the
order (3) R, E, C.

Ok? Now, given deduction (RCE), you can say: R: all mammals have legs; C: my cat is a mammal; E:
therefore my cat has legs.If you want to introduce the number of legs, you have to say something
like: R: "all mammals have four legs". Or you could introduce another syllogism similar to this one: All
mammals have legs; (my cat is a mammal) AND (normal cats have four legs) AND (my cat is a
normal cat); therefore my cat has four legs. You just concatenate (haha) conditions.

If you want invalidforms, you use the inverse or the converse, which do notlogically follow: my cat
has legs; my cat is a mammal; therefore all mammals have legs. No. Or, all mammals have legs; my
cat is nota mammal; therefore my cat does nothave legs. No again.

Steven Ravett Brown