- Some S are not P: The X indicates that at least one member lies within the zone that represents members of S but not of P. Other members of S may or may not lie within P.
Substitution is a powerful technique for recognizing valid and invalid deductive arguments. Validity depends only on the form of the argument. Therefore, we can replace any arcane or confusing terms in a deductive argument with familiar terms, then decide whether or not the argument is valid. For example, the following four arguments all have the same invalid form:
If a star is not a quasar, then it is theoretically impossible for it to be any type of star other than a neutron star. This follows from the fact that no neutron stars are quasars. No neutron stars are quasars. Therefore, no non-quasars are non-neutron stars. No S are P. ∴ no non-P are non-S No cats are dogs. Therefore, no non-dogs are non-cats.
Recognizing that the first three arguments are invalid is easy for some readers and difficult for others. Some of us experience mind-glaze when faced with arguments involving unfamiliar and highly technical terms; others find abstract, symbolic notation even more obscure. Some can analyze arguments easier when the argument is in a standard notation; others prefer their arguments to be couched in everyday language. Everyone can immediately recognize the fallacy of the cats-anddogs argument, for obviously the world is full of objects that are neither cat nor dog. If this catsand-dogs argument is invalid, then the other three arguments must be invalid because they have the same form.
Substitution relies on four principles that we have encountered in this chapter:
- Validity or invalidity of a deductive argument depends only on the form of the argument, not on its topic (note: this is not true for inductive arguments).
- A valid deductive argument is one in which the conclusion is necessarily true if the premises are true (note: this is not true for inductive arguments).
- If we know that the premises of an argument are true and yet the conclusion is false, then the argument must be invalid.
- Validity or invalidity is much easier to recognize for arguments about familiar objects than for abstract arguments.
To employ substitution, simply identify the elements of the argument and replace each element with a familiar term. In the examples above, the elements are neutron stars and quasars, or S and P, or cats and dogs, and the structural equivalents are S=neutron stars=cats and P=quasars=dogs. Formal logic assumes that the premises are true, so it is easiest if one picks substitutions that yield a true initial statement. Then, an absurd result can be attributed correctly to invalid logic.
Substitution may be the main way that most people (logicians excluded) evaluate deductions, but this method seldom is employed consciously. Instead, we unconsciously perceive that an argument is familiar, because it is similar in form to arguments that we use almost every day. Conversely, we may recognize that an argument sounds dubious, because it seems like a distortion of a familiar argument form. With that recognition, we then can deliberately employ substitution to test the argument.
