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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 , 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 is a , not a quality of a proposition, which means that the 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 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 , 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 . 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. Look at the conclusion, and identify where it, or parts of it, are in the premisses. See what you have to work with in the premisses, and if there are any low hanging fruit like conjunctions that can be simplified. A series of conditionals, and a conditional conclusion might mean a . Watch out for Constructive Dilemmas (a conclusion that is a disjunction of the consequents of two conditionals?) Keep in mind that rules of replacement come in handy in transforming premisses into something you can work with: for conditionals Material Implication and , maybe ; DeMorgan's for turning negated conjuncts or disjuncts into something you can work on; Distribution if you have both disjunction and conjunction in a proposition; and remember that Double Negation can be useful. And finally, if you don't see it, try something, and maybe the solution will become clear. If not, try something else!

What does all this mean? What use is logic? If you remember way back at the beginning of this course, we mentioned truth, but we quickly switched from the truth to propositions, which as you will remember are defined as statements capable of being true or false. How we establish the truth of a proposition is something that lies outside of logic, or more properly outside of DEDUCTIVE logic, so that while SOUNDNESS may be something we talk about, the concern of logic is with the formal quality of VALIDITY. So logic cannot tell us which arguments are sound, but it can tell us which arguments are candidates for soundness, because soundness requires first of all that an argument be valid, and only then would the truth or falsity of the premisses be of interest. And while the truth of premisses may be open to uncertainty, controversy, or probability, the validity of arguments is something we can determine definitively. So even if we cannot prove that the premisses someone is using to prove a conclusion are false, we might be able to prove that even if those premisses were true that the conclusion would not necessarily be true because the argument is invalid. And this is something worth knowing, even if we don't know what is true.

The main difficulty is putting the arguments we encounter in everyday life into a form that we can manipulate with our logical tools. First, we have to be able to recognize arguments and their components, propositions. What are some forms of non-propositional language? Why are explanations not arguments? An ARGUMENT is any group of propositions of which one is claimed to follow from the others, which are regarded as providing support or grounds for the truth of that one. The first task in converting an argument from ordinary language, then is to Leeward Community College (p ~ Philosophy 110  ~ q) S pring 2010 6 identify the conclusion; what is not conclusion is either premiss or irrelevant. We learned some PREMISS AND CONCLUSION INDICATORS, which will help us, but ordinary language can be difficult, with premisses unstated or not in declarative form, conclusions in the form of rhetorical questions, and so forth. A few further hints.

Propositions are defined by their truth claims or meaning, not a particular expression. This means that a single proposition could be expressed in different ways, in different languages, for example. Also it means that the of propositions may not be obvious. Take for example the proposition, "I am happy ". The negation of that could be "I am not happy" or it could be "I am sad." The trick is to determine whether the meaning is the same for the purposes of the argument you are translating. So if we have an argument that says: "If I had ice cream, I would be happy; but I am sad, therefore I don't have ice cream", "sad" means "not happy" and we have your garden variety Modus Tollens. So, advice: don't use any more statement labels than are absolutely necessary.

Quantification (or Predicate Logic) introduces more complexity (or dexterity) into symbolic logic. We get inside of a propositions, using letters now to represent a subject (some thing, an x) and a predicate (a quality or feature about x). These two together give us a propositional function, which is just like a normal proposition (can be true or false), but also can be Quantified , or given a range of things to which the proposition applies. You should know the difference between Universal statements and Existential statements, and the relations between them, especially the Logical Equivalences. ~(x)Mx means not all x are M, so that some x (at least one) is not M: ( ∃x)~Mx ~( ∃x)Mx means there isn't any x that is M, so all x are not M : (x)~Mx And of course there are two more (Remember your square of opposition).

These equivalences are useful because they allow us to convert a quantified proposition with a negation on the quantifier to a normal form predicate proposition with the negation on a predicate. And we need to do that in order to apply our rules of instantiation (substituting a particular individual for the variable “x”) so that we can use our rules of inference to do proofs on arguments with quantified propositions. There are rules for instantiation. A Universal proposition applies to all things, to any “x”, so we can substitute anything for x, and the instantiation will be true (if the universal statement is). But for an Existential proposition, we have to chose an individual that the statement is in fact true of, since it is not true of all things. And we can undo instantiation with the rules of Generalization: if we instantiate a universal proposition with some example, as long as that individual is an arbitrary example, we can say that whatever we manage to prove about that individual is also true of all things, so we can do from that individual back to a universally quantified proposition with an x. If we instantiate an existential, we picked that individual for a reason, so it is not arbitrary, and this means we cannot go from that individual to a universal statement. It also means that if we have picked some arbitrary individual (remember e, Expo?) for a universal Leeward Community College (p ~ Philosophy 110  ~ q) S pring 2010 7 instantiation, we cannot use that same individual to instantiate an existential statement. Or even if we have already instantiated some other existential statement, we cannot use the same individual to instantiate this one. So the rule is, you cannot use an individual to instantiate an existential statement in a proof in quantified logic that has occurred previously in the proof. And so, once we instantiate an existential, we can only generalize to an existentially quantified statement, if something is true of a particular individual, then it is true of some (at least one) thing.

A few things that bear repeating: Arguments are never true or false. Truth is a property of propositions. True premisses and a true conclusion in a deductive argument tells you nothing about its validity. You can only apply the first 9 rules of inference when the appropriate connective is the main one on a line of a proof. (Avoid illicit simplification, especially.) Remember our TLA SLO's: WFF, CCS, and Q.E.D. (lol) Never eat anything bigger than your head.

So keep in mind that you have survived this class so far, and you have seen everything that will be on the final (or something like it) in the weekly quizzes. Knock on wood, raise your knee, and watch out for those ~Cs. Good luck, and may your mushroom eyes serve you well!

Study- guide for the Final Exam: Philosophy 110 @ LCC Spring 2010 w/Stroble