Leeward Community College (p ~ Philosophy 110 ~ q) S pring 2010 1 Introduction to Logic Study Guide The final exam is meant to assess your knowledge and skill you have acquired throughout the semester. The first thing to realize is that you have acquired some knowledge and skill, and preparing for the exam is just recognizing that fact. The downside is that like any test, the scale must be steep enough that everyone registers at a definite point, which means the test must be difficult enough that not everyone (or anyone) will score 100%. And as well, a test is only a sample of everything that you have learned, and thus is somewhat random. This means you should review everything to be ready for whatever it is that appears on the exam. But this is a good thing too, since reviewing is the best well to acquire that knowledge and skill you will be tested on, even if it is not on the test! What have we learned? What logic is: What an argument is: It isn't just saying “no it isn't.” Most importantly we learned what PROPOSITIONS are, that they are statements that can be affirmed or denied, or can be true or false, and that they can be premisses or conclusions in an argument. We learned the difference between DEDUCTIVE ARGUMENTS, where the conclusion follows necessarily, and INDUCTIVE ARGUMENTS, that only establish a conclusion with probability. We learned all about VALIDITY , and good thing too because there are lots of arguments out there that don’t have it. Arguments that are invalid (or otherwise flawed) but slip by are called FALLACIES, and we now know their names and where they live. (How many? What are their names?) In symbolic logic we learned about TRUTH FUNCTIONALITY, which is a really cool thing. It means that the truth-value of an complex proposition is a function of the truth values of the simple statements in the proposition and the truth functional CONNECTIVES which link those statements together. SIMPLE STATEMENTS for symbolic logic are propositions, that is, statements that are capable of being true or false (even if some of them are only ever one or the other), that are not analyzable into any more basic propositions. NEGATION: negation is a truth function, not a quality of a proposition, which means that the truth value of the negation depends on the truth value of the simple statement which it connects (to itself, sort of), as in ~p. Leeward Community College (p ~ Philosophy 110 ~ q) S pring 2010 2 DOUBLE NEGATION: if the value of the negation depends on the truth value of the simple statement, then the negation of negation depends on the value of the negation which depends on the value of the simple statement! OK, its not that great of a discovery, but it works. The rest of the LOGICAL CONNECTIVES require two simple statements, and they define a truth value that depend upon the truth value of both of the simple statements (usually). The connectives are CONJUNCTION, DISJUNCTION, CONDITIONAL, AND BI-CONDITIONAL We could repeat the complete definitions of these, but that would be pointless since you already have them memorized, right? But we might remind ourselves of the significant parts: p • q CONJUNCTION: only true it both conjuncts are true. p ∨ q DISJUNCTION: only false if both disjuncts are false. p q CONDITIONAL: only false if antecedent true and consequent false. p ≡ q BI-CONDITIONAL: only true is value of both simple statements are the same, or only false if value of simple statements is different. And the most wondrous thing, we have mechanism come home to help us! Now it appears in the form of a truth-table that exhaustively lists all the possible combinations of the simple statements, and so the values of any connectives, and ultimately the value of any proposition. So we can use a TRUTH-TABLE to determine, precisely, the truth value of any proposition, and so the value of any group of propositions. And if we have the truth values of a group of propositions, which just, shall we say, have the relation of premisses to a conclusion, we can test for the validity of the argument. TRUTH-TABLE TEST FOR VALIDITY: identify all the basic or simple statements, label them, put them into a matrix (remember, 2 n where n=the number of simple statements, first column is half true/half false, second 1/4, and so on until the last column (of a simple statement) should alternate T/F. Got it?). Then we lay out the propositions and determine their truth values in each row, and finally identify the premisses and the conclusion, look for lines where ALL the premisses might be true and the conclusion false, which would show the argument is invalid. If we find one, the argument is invalid. If we find none, the argument is valid. The BACKWARD METHOD is a short-cut way to use the truth tables, isolating out those lines of a possible truth table by assigning values that either make that premisses true or the conclusion false, and then seeing if it is still possible to make the conclusion false or the premisses true (respectively). Once we have the truth tables, we can prove the validity of arguments. BUT WAIT! If we can prove the validity of an argument, and validity is a structural rather than a semantic relations (concerns the form of the argument, not it's content), then if we show an argument is valid every other argument that has the same form must as well also be valid! This means that we can Leeward Community College (p ~ Philosophy 110 ~ q) S pring 2010 3 identify the FORM of an argument by substituting STATEMENT VARIABLES for the simple statements in the argument; and we can reverse this by substituting statements for statement variables, which gives us a SUBSTITUTION INSTANCE of an ARGUMENT FORM. But the difficulty is that we do not have to substitute only simple statements to produce an argument form; we could substitute a complex proposition for a statement variable, and vice versa (the other way around). If we want to capture a specific argument, we have to use the finest net available, and be sure to substitute statement variables only for simple statements so that we can obtain the SPECIFIC FORM of that argument. And then we can test for validity? And once we can show that an ARGUMENT FORM is valid, we can use it in any circumstance, and be assured of the correctness of our reasoning. In fact, we can use these VALID ARGUMENT FORMS in the process of evaluating the validity of longer, more complex arguments!! Oh Goody! We have nine (9) such argument forms, which we will then henceforth refer to as RULES OF INFERENCE, and they are: A further tool that is necessary for us to do deductions is logical equivalence, which allows use to replace propositions with their logically equivalent counterparts. A LOGICAL EQUIVALENCE is a material equivalence that is a TAUTOLOGY. This means that not only will the statement of material equivalence (which is double material implication) be true when both statements have the same truth value, but that both statements will always have the same truth value, which means that they must have the same meaning and so can be substituted!! Thus we can add to our nine RULES OF INFERENCE another ten RULES OF Leeward Community College (p ~ Philosophy 110 ~ q) S pring 2010 4 REPLACEMENT. Now we can take any argument, and if we can apply our rules of inference to the premisses, and any statements thus derived from the premises, to arrive at the conclusion of the argument, then the argument must be valid. This is because VALIDITY means that IF THE PREMISSES ARE TRUE, THE CONCLUSION NECESSARILY IS TRUE. And validity means that if the conclusion can be false when the premisses are true, the argument is invalid. Or that if the premisses are incapable of all being true (INCONSISTENT), the argument is necessarily valid (but not sound), and conversely, if the conclusion is incapable of being false (TAUTOLOGY), the argument must be valid as well. These results are related to the PARADOX OF MATERIAL IMPLICATION we encountered with the conditional, only more so since it is the PARADOX OF STRICT (or Logical) IMPLICATION. But we are not bothered by this, since if everything follows from inconsistent premisses, that is as good as nothing in particular following from them; and as well if a tautology follows from any premisses, if follows from none in particular, so we might as well deduce it from itself! Enough of that. Reductio ad absurdum If we can, using our rules of inference, derive a contradiction from a given set of premisses, if Leeward Community College (p ~ Philosophy 110 ~ q) S pring 2010 5 those premises are true, that contradiction must be true, but contradictions cannot be true! So we know the premisses are inconsistent, and the argument is necessarily unsound, and can't prove the conclusion to be true. But as well, we can use this to do an Indirect Proof . We can assume something, another proposition in addition to the premisses of the argument, and see if we can reach a contradiction with our rules. If we can, we know that that assumption cannot be true, so we are justified in deriving its negation. Usually our assumption is the negation of the conclusion, so we can get the conclusion itself by indirect proof. Review the strategy for doing deductive proofs, since it is not mechanical like the truth-table method.