PHIL 115: Philosophical Anthropology. I. Propositional Forms (in Stoic Logic) Lecture #4: Stoic Logic


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1 HIL 115: hilosophical Anthropology Lecture #4: Stoic Logic Arguments from the Euthyphro: Meletus Argument (according to Socrates) [3ab] Argument: Socrates is a maker of gods; so, Socrates corrupts the youth. Two ways to state the connection: Stoic Logic: If Socrates is a maker of gods, then Socrates corrupts the youth Aristotelian Logic: The difference Anyone who is a maker of gods corrupts the youth Stoic logic is based on the logical connections if then either or not both and Aristotelian logic is based on the logical connections All Some None 2 Overview 3 4 Classification of ropositions Four Basic Modes (Forms) of Argument Modus onens Modus Tollens Disjunctive Conjunctive Two More Complex Argument Forms Chain Arguments Dilemmas I. ropositional Forms (in Stoic Logic) Kinds of roposition Simple (atomic) Socrates is a maker of gods. Any son who prosecutes his father is impious. Compound (in which two simple propositions are connected) Euthyphro is at court and Socrates is at court. If Socrates is a maker of gods, then Socrates corrupts the youth. Either Socrates is prosecuting a case or he is defending in one. Socrates is not both a maker of gods and a disbeliever in gods. 5 Some are trivial Compound ropositions I e.g., Euthyphro is at court and Socrates is at court. not expressing any logical connection between the two propositions not the basis for syllogistic inference simplifiable: each conjunct is true and, symbolized (! ) Some are clearly logical connectives e.g., If Socrates is a maker of gods, then Socrates corrupts the youth. expressing a logical connection between two propositions the basis for syllogistic inference knowing that one is true (or false) sometimes allows one to infer the truth (or falsity) of the other if symbolized: ( " ) 6
2 Compound ropositions II What about or and notboth? Symbolized or ( # ) not both ( $ ) or ~(! ) They are like and The order does not matter These two are the same: Either Socrates is prosecuting a case or he is defending in one Either Socrates is defending in a case or he is prosecuting one. They are like if From the truth of the whole and the truth (or falsity) of the right one of the parts, one can infer the truth (or falsity) of the other e.g., Either Socrates is prosecuting a case or he is defending in one.» He is not prosecuting a case.» So, he must be defending in one. Socrates is not both a maker of gods and a disbeliever in gods.» He is a maker of gods.» So, he is not a disbeliever in the gods. 7 Three Basic Logical Relations Name Form Symbolization Names of arts Hypothetical (Conditional) if then ( ) or ( ) Antecedent & Consequent Disjunction or ( ) Disjuncts Denial of a Conjunction not both and ( ) or ~( ) Conjuncts 8 Logical Symbols for Stoic Logic 9 10 not p ~p if p then q p " q p or q p #q II. The Basic Stoic Forms p and q not both p and q p! q p $ q or ~ (p! q) therefore % Stoic s 11 Four Basic Stoic Modes of Inference 12 A discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so Aristotle Stoic s begin with [i.e., have as premises] two propositions a compound proposition stating a logical relation between two propositions (if, or, notboth) & the assertion or denial of one of the component propositions end with [i.e., have as a conclusion] the assertion or denial of the other of the component propositions Mode Names Modus ponens Modus tollens Disjunctive Conjunctive remise If p then q If p then q Either p or q Not both p!and q remise p not q not p p Conclusion q not p q not q
3 s of the Four Basic Stoic Modes of Inference 13 Hypothetical s: Recognizing Modus onens 14 Mode Modus onens Modus Tollens Disjunctive Conjunctive If Socrates is a maker of gods, then Socrates corrupts the youth. Socrates is a maker of gods. So, Socrates corrupts the youth. If Socrates were corrupting the youth of Athens, then their relatives would be in court to testify against him. But their relatives are not here. So, Socrates is not corrupting the youth of Athens. Either Socrates is prosecuting a case or he is defending in one. He is not prosecuting a case. So, he must be defending in one. Socrates is not both a maker of gods and a disbeliever in gods. He is a maker of gods. So, he is not a disbeliever in the gods. The key to M. ponens is matching the antecedent of the conditional and detaching the consequent, not the absence of the negation sign These are all Modus ponens ( ) ( ) ( ) ( ) These are not Modus ponens (& neither is valid) ( ) ( ) Hypothetical s: Recognizing Modus Tollens 15 Disjunctive s: Recognizing Disjunctive s 16 The key to M. ponens is mismatching the consequent of the conditional and then (as a conclusion) contradicting the antecedent, not the presence of the negation sign. These are all Modus tollens ( ) These are not Modus tollens ( ) ( ) ( ) ( ) ( ) (valid, but not M. tollens) The key to Disjunctive is contradicting either disjunct, not the presence of the negation sign. Also, drawing the correct conclusion. These are all Disjunctive s ( ) ( ) ( ) ( ) These are not Disjunctive s (& neither is valid) ( ) ( ) Conjunctive s: Recognizing Conjunctive s The key to Conjunctive is affirming either conjunct, not the absence of the negation sign. Also, drawing the correct conclusion. These are all Disjunctive s ( ) ( ) ( ) ( ~ ) These are not Disjunctive s (& neither is valid) ( ) ( ) 17 Distinguishing Conditional from Arguments An argument connects a premise & a conclusion: Socrates is a maker of gods. So, Socrates corrupts the youth. Analysis remise: Socrates is a maker of gods Conclusion: Socrates corrupts the youth. Here both the premise & the conclusion are asserted. A conditional (or hypothetical) statement connects an antecedent & a consequent: If Socrates is a maker of gods, then Socrates corrupts the youth. Analysis Antecedent: Socrates is a maker of gods Consequent: Socrates corrupts the youth. Here neither the antecedent nor the consequent is asserted. It only asserts a logical connection between the two. Or, what else would be true if the antecedent were. Conditionals are useful because we often want to state a consequence of a proposition before we know whether the proposition is true. E.g., in scientific work we may not know whether a theory is true The best way to find out is to draw some predictions (consequences) from it and see whether the predictions correspond with experimental or observational results. 18
4 Disjunctive s: Two Meanings of or Is this valid? 1. Either Socrates is prosecuting a case or he is defending in one. 2. Socrates is defending in a case. 3. So, Socrates is not prosecuting a case. That depends how we understand the word or. It has two senses:  An inclusive sense: at least one, one or both E.g., (Speaking about someone who got into a very selective college) Either he had very high test scores or he had very good letters of recommendation from his teachers. Such a person, the speaker says, must have had at least one, but there s no reason to think that a student could not have had both.  An exclusive sense: one, but not both E.g., Either resident Obama will win November s election or Governor Romney will. They cannot both win.  Some arguments that are invalid on the inclusive sense are valid on the exclusive sense. 19 Symbolization of Or The Symbols Inclusive or! ( ) Exclusive or! ( )" or" (( ) ()) The Choice First, note that both senses of or include the connection ( ). So, any time one has two statements connected by an or one should write at least that. Second, one should only add ( ) if there is a particular reason to do so The most common particular reasons would be An explicit statement of the speaker to the effect that not both Background knowledge that justifies this» as in the election example above Charity» If the validity of an argument requires that the speaker mean not both, one might note that rather than just say that the argument is invalid. 20 Necessary & Sufficient Conditions Conditional statements can express two kinds of condition: Necessary conditions E.g., It is a necessary condition of being resident of the United States that one be at least 35 years of age. For N is a necessary condition of, write ( N). Sufficient conditions E.g., It is a sufficient condition of being elected resident of the United States that one receive a majority of votes in the Electoral College. Sometimes conditionals state only relative necessity: If this paper is combustible, then it will catch fire when I put this match to it (provided there is oxygen in the room, &c.). For S is the sufficient condition of, write (S ). III. The Complex Stoic Forms Chain Arguments 23 Chain Arguments 24 If guns are outlawed, only outlaws will have guns. If only outlaws have guns, our country will be a more dangerous place to live. So, if guns are outlawed, our country will be a more dangerous place to live. Form G O O D G D limitation This form of argument only gets us from one conditional to another. If, Meletus, you agree that Euthyphro is wise in these matters, then you should consider me, too, to have the right beliefs. If you consider me to have the right beliefs, they you should not bring me to trial. So, if you agree that Euthyphro is wise in these matters, then you should not bring me to trial. Form E I I ~B E ~B limitation This form of argument only gets us from one conditional to another.
5 Logical Dilemma: An The Logic of Dilemmas The text: If, Meletus, you agree that Euthyphro is wise in these matters; [then you should] consider me, too, to have the right beliefs, and do not bring me to trial. If he prosecutes his father, he will anger his family If you do not think so, If he does not prosecute, he will anger the gods then [you should] prosecute that teacher of mine, and not me. In the colloquial sense, being in a dilemma means being in a situation each one of which is bad (i.e., each one of which has bad consequences). Euthyphro might feel that he is in such a dilemma Extraction of the logical form if you agree with Euthyphro, In the logical sense, a dilemma is an argument with three premises: two conditionals, & a disjunction asserting that then you should not prosecute me (his student). one of the antecedents is true, or if you disagree with Euthyphro, one of the consequents is false then you should not prosecute me (his student) [rather, you should first prosecute him (the teacher). a conclusion asserting that the corresponding consequent is true, or the corresponding antecedent is false Either you agree with Euthyphro or you disagree with him So, you should not prosecute me E.g., the above thoughts from Euthyphro, plus Either he prosecutes or he does not So, he will anger his family or he will anger the gods Simplified Dilemmas Kinds of Dilemma The dilemma as just stated If Euthyphro prosecutes his father, he will anger his family If he does not prosecute, he will anger the gods Conditionals > Disjunction Either he prosecutes or he does not So, he will anger his family or he will anger the gods There are four kinds of dilemma: Simplified If Euthyphro prosecutes his father, he will anger his family Not Convergent ( ) (R S) Convergent ( ) (R ) ( 䏕 R) Complex Constructive Concl.: ( 䏕 S) Simple Constructive Concl.: () ( 䏕 ~S) Complex Destructive Concl.: ( 䏕 ~R) If he does not prosecute, he will anger the gods Convergent ( ) ( S) Either he prosecutes or he does not So, he will anger someone Simple Destructive Concl.: 29 of a Complex Constructive Dilemma George Will on Evel Knievel s attempt to make a 1500foot jump across Snake River Canyon (from the Washington ost): If the jump is not as risky as it is advertised to be it is a fraud. If, as seems probable, it does involve a serious risk of Knievel s life, it is obscene. Symbolization: Explicit (~R F) (R O) Implicit ( R 䏕 ~R) (F 䏕 O) 30 of a Simple Constructive Dilemma Caliph Omar (according to legend) after the conquest of Cairo on whether the books at the library of Alexandria should be preserved If what is written in them agrees with the Koran, they are not required; if it disagrees, they are not desired. Destroy them [X] therefore. Symbolization: Explicit (A ~ R) (~A ~ D) X Implicit (~ R X) (~ D X) ( A 䏕 ~A)
6 : lato on poets (Republic iii, 391c e) If Homer speaks the truth then the heroes are the sons of the gods and if he speaks the truth then they did many wicked things. But either they are not the sons of gods or they did not to many wicked things. of a Simple Destructive Dilemma Form: (T S) (T W) (~S ~W) 31 The historical situation St. Thomas More s Dilemma In 1534, Henry VIII had arliament declare him Supreme Head of the Church in England and demanded of some of his subjects that they state whether they accepted this law. In May, 1535, Thomas Cromwell visited More in prison to demand his opinion of the new statutes. Thomas More s Dilemma (in both logical & colloquial senses) If I say that I disagree with the law, then I will be beheaded (for treason). If I say that I agree with the law, then I will be sent to Hell (for perjury). When I m asked, either I will say that I agree with the law or say that I disagree with it. So, either I will be beheaded or I will be sent to Hell. 32 So, Homer does not speak the truth. ~T Two Ways of Responding to a Dilemma 33 assing through the horns of the dilemma : More s attempt to avoid his dilemma by refusal to say anything (i.e., neither speaking the truth nor telling a lie) The soundness of a dilemma depends on the disjunction being exhaustive. assing through is a challenge to the truth (exhaustiveness) of the disjunction; i.e., (( ) R) rather than ( ). Taking the dilemma by the horns : John Stuart Mill s response to arguments like Omar s: however true [a given opinion] may be, if it is not fully, frequently, & fearlessly discussed, it will be held as a dead dogma, not as a living truth, &c. (On Liberty) The soundness of a dilemma depends on the hypotheticals being true. Taking by the horns is a challenge to the truth of one of the hypotheticals. Rebutting the dilemma (see next slide)
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