The 19 Rules
At this point in our study the competent critical thinker realizes the distinction between the notions of VALIDITY and SOUNDNESS. Validity is a property or attribute that an argument either has or does not have. Validity, like pregnancy, is an all or nothing state of affairs. Whether the critical thinker chooses to frame arguments in Categorical syllogisms as learned in Chapters 3, 4 and 5 or propositional (symbolic) syllogisms learned in Chapter 5, guaranteeing the validity of any DEDUCTIVE argument is merely a matter of choosing a known VALID DEDUCTIVE FORM. That done, simply plug in the ‘variable’ terms of the argument and validity is the result through logical necessity. A deductive argument is said to be valid if and only if it takes a form that makes it impossible for the premises to be factually true and the conclusion factually false at the same time. If this is not the case then we call that deductive argument invalid (from the Latin invalidus weak).
Soundness is a far more difficult matter to determine as the competent critical thinker realizes. A deductive argument is sound, if and only if, it is first valid, and second all of its premises are factually true. Any valid deductive argument that has questionable, unproven, or even improvable premises is said to be unsound. (We will deal with the issues of proving soundness by induction, probability and scientific method in Chapter 9: Induction.)
Chapter 7 of our text first sets forth the 9 elementary valid argument forms that provide the basis for logically necessary inferences in ‘sentential’ logic, i.e. the logic of sentences. Then the 10 rules of replacement for logically equivalent expressions are added to provide what is known as the 19 rules of inference. These are used, or are supposed to be used, throughout the academic disciplines whenever formal arguments are presented for public review or debate. When they are not used in academic disputations, confusion, disorder, and serious errors are the likely results from invalid reasoning.
Therefore, serious critical thinking students will take the time to learn these 19 rules. Together with the 15 valid forms of categorical syllogisms, they constitute the necessary and sufficient weaponry for any warrior entering the battle of ideas. Without them, expect to become a roadside casualty and simply left behind.
Please refer to the INTERNET ENCYCLOPEDIA OF PHILOSOPHY (#5 - a,b,c) for a detailed explication of 19 truth-functional propositional rules.