Logic & Truth: Testable Propositions & Conditional Statements - Prof. Joe A. Oppenheimer, Study notes of Political Science

Download Logic & Truth: Testable Propositions & Conditional Statements - Prof. Joe A. Oppenheimer and more Study notes Political Science in PDF only on Docsity! 700 initial notes on logic (class 3): (also see handouts for 700) 1. The purpose here is to come to understand the glue that relates what it takes to A. to insure one has testable propositions; B. understand what it means to test a conjecture or proposition; C. justify one’s arguments; and D. insure that we make consistent, perhaps even correctable claims. 2. I am saying that a big part of that glue is logic and we will go over enough of it for you to understand how these items relate together. A. logic -- define. Has to do w the rules of creating complex arguments from simple statements each of which can take on one of 2 truth values, T or F. Thus, we need to relate these rules to an understanding of truth. a. T, F, recall, has to do with the relation betw what a statement describes and “things as they really are.” [correspondence theory of T] b. Consider, for the moment a statement: X. Table 1 - Truth (& Falsehood) of X – The Correspondence Way Things as they are X correspond T do not correspond F B. Consider Table 1 above: it gives us a way of portraying the conditions under which a statement is true, or false. We will call such a display a truth table - and it can be used to show the conditions for more complicated cases as well, as we will show. 3. Some definitions of the basic logical concepts and their interpretation given this theory of truth. A. Define NOT. It is noted as “~” or “-“ in logic. What would its truth table (see table 2) look like? Table 2 - Not: ~ Things as they are X ~X correspond T F do not correspond F T B. Defining a few other notions and thinking about their truth tables and properties: a. OR: a disjunction is T iff at least one of the things connected by it (disjuncts) is true. 1) Example: (After a long interrogation, in a tense hearing, a member of a congressional investigating committee declares, where he thinks the witness is trying to obfuscate matters: X: “Listen, either Gingrich violated the rules of congress, or he didn’t.” (Table 3, 7, 7, 7, 7, 7) Introductory Notes on Logic Page 2 Table 3 - Or: V Things as they are G ~G G V ~G correspond w G T F T do not correspond w G F T T 2) So the statement by the member of Congress is always true: it can’t be false. The truth value of such a statement does not depend on its correspondence with anything external! Regardless of how one defines or judges truth the last column would be true. It is a tautology. b. Defining AND: a conjunction is T iff all of the things connected by it (conjuncts) is true. 1) Example: (After a long interrogation, a journalist moderating a TV show on political fund raising asks a member of Congress, sarcastically, summing up: G: “Listen, so are you trying to say that Gingrich both violated the rules and didn’t violate the rules?” Table 4 - And: & Things as they are G ~G G & ~G correspond w G T F F do not correspond w G F T F 2) So the implicit statement by the journalist is that what the member of congress said isn’t true: it can’t be. The truth value of such a statement also does not depend on its correspondence with anything external! Regardless of how one judges truth the last column would be false. It is a contradiction. 3) Hence we can see the importance of CONSISTENCY. We have said one objective was consistency and now we can define it in terms of SATISFIABILITY - the property that a set of conditions in the real world could exist which would make the statement true. C. So we now have a a sense of a CORRESPONDENCE THEORY OF TRUTH, a method of displaying the analysis of the truth of statements, and have a few other things to boot. a. We have discovered that some (compound) statements will take on their truth value not from our definition of the nature of truth but rather from the properties by which their simple components relate to one another. This is also going to be found to be true of sets of statements as a whole. They can take their truth from how they relate to other statements. b. Simple (and other) statements can relate to one another as impossibly, possibly, or necessarily true (as a set) c. We might wish to claim that for purposes of simple reporting about empirical matters, any 2 statements which have identical truth tables have the same MEANING (when one is true, the other is and similarly they are false at the same time). 1) This can also be thought of as one statement is true iff the other is true. Introductory Notes on Logic Page 5 b. When would you say that they are telling the truth, and when not? 1) If there is no defection, (lines 3 & 4) then the threat remains, but isn’t actualized. We certainly would not wish to say it is an “ideal” or “vacuous” threat merely because the antecedent conditions are not filled: things correspond with ~D. 2) If I kill you (K) (cases 1 & 3) - for what ever reason - then also, we certainly have no evidence to assert that it was a “vacuous” threat - even though it is possible that they killed for another reason. c. So what we have in the conditional D 6 K, is a situation such that the statement is false only if the defection occurs and there is no killing, is the statement false. This is indicated by the shaded block in ?. d. Note some EQUIVALENCE now: 1) the similarity (logical equivalence) between the D 6 K statement and the disjunction: ~D V K [compare the structure of ? and the last column in ?]. 2) But this lets us see that D 6 K is the same as ~K 6 ~D. H. Consider then the conditional: D 6 K then is a. a statement giving a sufficient condition (D) for the occurrence of another (K). [K could have occurred anyhow. b. or K is a necessary condition for D since [D necessarily triggers K]. 5. Now we have enough tools to understand what it takes to justify an argument or belief: A. What we want to justify are conclusions which are knowledge claims a. they can be on the basis of simple correspondence statements: I know this pipe is copper because of: 1) its color 2) it passed the chem test for copper 3) the plumber’s bill 4) -- where do we stop? What does it mean to find that the justification was insufficient in each case: color alone doesn’t determine copper, wasn’t the right test; the plumber is a liar. b. what is the form of the justification: take case 1 as an example: P1) copper pipe has a certain color (x)(xC 6 xK) P2) this pipe has that color (aK) (T) C) (thus, I know) this pipe is copper (aC) (F) Now we can see the fallacy right off – C is not a necessary condition for K. B. What does it take to justify conclusions? a. proper connections with the premises b. premises which are not known to be false 1) leads to conditional or contingent knowledge claims. c. premises which are known to be true - sound arguments Introductory Notes on Logic Page 6 C. Consider an argument: from a set of premises, P, we justify a conclusion, or set of conclusions, C. What does this mean? It means that if the argument is correct, the conclusions follow, or if P is true, then we expect C to be true. a. How do we get this? By insuring that the structure of the argument reflects the structure of the conditional (see paragraph 4.F) b. This would mean that we would want the relationship between P and C to be such that if P is true, then C must be true. Or we would want the P 6 C statement (which has a truth table which corresponds precisely with this) to be always true (i.e. a tautology: see paragraph 3.B.a.2)). Putting it differently, let’s say we had as premises, P1: Joe’s pet is a dog. P2: Woof is Joe’s pet. We should be able to conclude: C: Woof is a dog. A) Validity: valid arguments (p. 39-40) have to do with the relations betw combinations of the premises and conclusions of arguments. B) Also need that the premises are true c. Possible, impossible, necessary are the values we are looking for in the relations Table 9 - Showing impossibility of false conclusions, given true premises Things as they are P C P6C 1. correspond w P and C T T T 2. correspond w P but not w C F Ruled out by relation betw meanings of P & C 3. correspond w C but not w P F T T4. correspond w neither P or C F 1) If P then C: (/ ~P or C / If ~C then ~P): A) P is sufficent for C (if P occurs, you’ll get C) B) ~C is sufficent for ~P C) C is necessary for P (you cant have P without C) D) ~P is necessary for ~C 2) What does it mean to explain something: to have a theory of it? Say a theory of revolution? R 6 V?? or V 6 R! D. Why deduction is powerful: a. leads to testability: frm if not C then not P is nec. bec not C leads to the P being F b. If this conclusion were to prove wrong, it should be the case that at least one of the two premises were wrong. Hence testing C is an indirect test of the set of premises. Justifying the premises with valid arguments means that one is testing the conclusion. Introductory Notes on Logic Page 7 E. Now of course, not all arguments which are deductively valid are useful. Some aren’t SOUND. This is because we would like our premises to be true, or at least not known to be false. a. What if P not true? 1) Especially if it known to be F? Worthless? Or does it depend, if it is a good approximation (smoothness of reality?) 2) If it is unknown? Can test! 3) What is the purpose of a false premise in a deductive argument? Show what this generates: If the conclusions are false, a premise must be. Table 10 - POSSIBLE COMBINATIONS in A VALID ARGUMENT conclusion true conclusion false Premises true Sound IMPOSSIBLE Some premise false Unsound Unsound 6. One final step. Most premises in scientific arguments don’t look like the example in paragraph 5.C.b. Rather they are of a more universal, open ended form. A. So for example, they take the form: in all democracies all outcomes can be destable destabilized by coalitional changes. Or, for purposes of simplicity, all democracies are unstable. a. How to think about this (it is a conjunction with an infinite number of conjuncts), all of which must be true, for the whole to be true. 1) a truth table method is now difficult: it would need infinite columns, and hence rows. 2) It is still a conditional statement (e.g. all dogs have fleas means if an object has the property of being a dog, then it has the property of having fleas). 3) notation is úx {Dx6Fx}, or (x) {Dx6Fx} 4) What is required is that if a, b, c, ... n are objects they all conform: either ~Da or Fa. b. So this means that to test, one needs to find the one (class of) case(s) which don’t conform. Then you have a false universal premise! 1) So you look through all the cases: a, b, c, ... n as objects to see if they all conform: either ~Da V Fa, . Or, specifically, one looks for the existence of any (class of) object(s), call it x such that Dx & Fx 2) Notationally, úx {Dx & Fx}. This requires an infinite search, and one can only get a probabilistic response on the positive side (i.e. rejecting the null). Introductory Notes on Logic Page 10 b. Were the arguments valid? c. Were the concclusions justified? d. Were the premises justified? e. How were the hypotheses justified? Motivated? f. discovery? g. finding truth? h. Is there a trade off betw applicability and rigor? i. Asserting and arguing for political values or policy positions C. Other questions to think about - a. In these articles, there probably is no criteria put forward for claims of knowledge within our field. Pick two articles you read, one from a journal from my list and one from a journal of your choice. Tell me the criteria that seems implicit in the arguments. How can you tell? Is there a difference by journal or area of work? What is it? Are all these criteria equally 'valid'? Why? b. Can we develop a coherent view of the nature of political science? 1) Is it characterized by an agreement as to substance or methods, or both? 2) Is there a pretense, interest in, science as an inherent part of the discipline's roots? c. What is the basis for the empirical / normative split in the discipline? Has it changed over time? d. Are we more able to understand the current frontiers of the discipline better by seeing its history? 12. a bit of methods (review of Giere Chap. 2) A. statements, real world (correspondence theory) and truth B. knowledge, beliefs [do a little prelude to the next material on logic lightly, perhaps through not, and mention validity from the outline section (on validity). 13. Lead Articles in A. apsr a. B. ajps C. jop intro notes: 700 Page 11 GVPT 700 - Intro Weds. August 29, 2001 1. My def of the course: considering if there is a definable way (or set of ways) to do research wh A. lets you make knowledge claims B. lets them be corrected C. lets us increase the understanding of what we know about politics 2. Thus, at the center must be an agreement as to what is knowledge, and knowledge claims A. Justified True Belief B. But certainly we don’t know what is true if and hence at most these are knowledge claims 3. Why I am interested in this: The problem of justification A. Too often is by authority B. Fascination w the question of justification of knowledge claims C. My start in physics and questioning of the relationship between knowledge in sciences, economics & politics a. Why couldn’t they be studied the same way? b. No good answers. c. discovery of a branch of inquiry which political sci (e.g. Lewis, Lowi, Rossiter, Einaudi, Hacker) didn't know at all. d. only Lowi really knew any stats, or any philosophy of inquiry D. perhaps that's why I do the sorts of poli sci that I do. 4. Why might you be interested in this - or is it only for credit - (handout studinfo and then) I call out names - adding if there are others (give them time to fill it out) - collect it. 5. my office hrs: W after class till 4; Th 1:45-4& by appt. 6. Handout and collect POPQUIZ -- comments? How many didn't do the readings? Serious about them. 7. Handout syl - and give time for them to read it. A. Note changes: office hours; no presentations (in grading, bottom of p. 3) B. answer questions 8. Grading: (0-10 scale): A > 8.5; B > 7 9. Participation (75), papers (160), quizes (as they come up, <80) 10. The course will be somewhat descriptive: that's the scope side. 11. The course will be primarily prescriptive. How one makes judgements on research questions, and the methods which underlie the answers. A. Methods will be defined broadly so as to focus on epistemological as well data handling sorts of questions. It will cover both the empirical and normative sides of the discipline. B. How can we evaluate methods? a. a way of getting to the objective. Does it work? intro notes: 700 Page 12 b. implies an objective C. Thus the course will certainly not be value free. 12. But can we agree on a set of values? Something we will have to return to a number of times. Perhaps, perhaps not. TODAY’S WORK. 1. Is there a common thread to the readings - esp Ion and Szasz? A. Let's perhaps start w Ionesco and move from there. a. What's the subj or argument in the Ionesco play? 1) The problem of the (WHAT IS THIS:) relationship betw evidence and knowledge claims? - doorbell (p. 22-7) , child (p. 15-19, 20) 2) (WHAT IS THIS:) consistency - bobby watson, died, married, kids? 3) (WHAT IS THIS:) explanation - 1st par. 4) (WHAT IS THIS:) the relationship betw logic and the world and our statements about it (p. 20-1) re the heart, ageless, no, in betw. . . the issue of language, usage and communiction. b. Why would this all be important to us - bec it deals w the issues of how to deal with facts and their relationship to the larger claims of truth, of theories (falsification, corroboration, etc.) and to the relationship between language and reality c. The importance of the issue of truth, knowledge in the evaluation of situations and policy B. It isn't frightening in Ionesco - e.g. the door ringing scene. C. What about Szasz - is he giving evidence regarding the same problems? a. What's Szasz say about the right to stand trial (don’t go into this unless raised by the classs). (Instead, go directly to “b” or §1.C.b. 1) the problem of those called mentally ill are faced w the following questionable assumptions: (p. 34) A) mental illness deprives a patient of their fitness to stand trial B) only those fit should stand trial C) the hospitalization, even wo treatment, willrestore a patient to capacity to stand trial. 2) should note (p. 35) A) competence varies across people B) the relation betw mental illness and competence is uncertain and irrelevant to the aims of criminal law C) the right to a trial should not be contingent upon mental illness D) the prosecutor should not be able to make the decision that the defendant must be subject to pretrial psychiatric exam is wrong. .