Information about Critical Thinking 04 Soundness

Slides on conditionals, disjunction, validity, soundness, modus ponens, modus tollens, chain argument, disjunctive syllogism, and dilemma

Review Logical negation, often expressed in English by ‘not’, is true when the component claim is false, false when the component claim is true. It is symbolized by ‘~’ and has the logical form ~P. Logical conjunctions, often expressed in English by ‘and’, is true when the component claims it joins are true, otherwise it is false. It is symbolized by ‘&’. It’s logical form is P & Q.

Review Contradiction, a special form of conjunction in which a claim and its negation are joined— they are always false. The logical form of a contradiction is P & ~P. The Principle of Noncontradiction, states that no thing can, at the same time and in the same manner, both have and not have the same property.

Review The Standard of Consistency—accept only those beliefs which are consistent with each other and any accessible evidence. Reductio ad ridiculum, appealing to ridicule (making fun of an opposing view) rather than providing reasons against it—it is a fallacy. Equivocation, to use a term ambiguously or vaguely in an argument—it is a fallacy.

Review double negation—any even number of negations cancel each other out. to prove a conjunction false prove that one of the component claims is false. to evaluate by contradiction—isolate the subject and predicate, generate lists of things that fall under each, stopping when you determine that they are not identical. proof by counterexample—Choose an item that is not on both lists, explain how the definition says it should be, then explain why it is not, indicate the inconsistency, and reject or revise the definition.

Proof by Counterexample A Method for Reasoning with Contradictions Line of Reasoning An explanation showing that the definition should be true of a specific example (thing or event). Reject the original definition Original definition. Another Line of Reasoning Another explanation showing that the definition is not true of the same example.

Reductio ad absurdam Reductio ad absurdam Indirect Proof, Proof by Counterexample Line of Reasoning 1. Claim 2. reasons 3. conclusion 6. P & ~P Another Line of 4. other reasons Reasoning 5. other conclusion 7. Rejection

The Logical Form of a Reductio Reductio ad absurdam, Indirect Proof, Proof by Counterexample 1.claim 2.reasons 3.conclusion 4.other reasons 5.other conclusion 6.contradiction 7.rejection

Review to avoid equivocating— define key terms by giving them one (to disambiguate) clear (to avoid vagueness) meaning. to avoid equivocating—use the Principle of Charity to settle on the best interpretation, whether normative or descriptive. to avoid reductio ad ridiculums—use the Principle of Sufficient Reason and attempt to provide reasons for each claim.

The Conditional Logical Complexity

A Logically Simple Truth Two logical possibilities Sarah attends Stanford. 1 True 2 False Given that we’ve filled in the indices, made the ceteris paribus explicit, and defined key terms.

Combining Logically Simple Truths Four states of affairs (states) or possible worlds Sarah attends Stanford. Sarah goes into debt. 1 True True 2 True False 3 False True 4 False False

Logical Conjunction Sarah attends Stanford AND Sarah goes into debt 1 True True True 2 True False False 3 False False True 4 False False False

Logical Conditional IF Sarah attends Stanford THEN Sarah goes into debt 1 True ? True 2 True ? False 3 False ? True 4 False ? False

Logical Conditional IF Sarah attends Stanford THEN Sarah goes into debt 1 True True True 2 True ? False 3 False ? True 4 False ? False

Logical Conditional IF Sarah attends Stanford THEN Sarah goes into debt 1 True True True 2 True False False 3 False ? True 4 False ? False

Logical Conditional IF Sarah attends Stanford THEN Sarah goes into debt 1 True True True 2 True False False 3 False True True 4 False True False

Logical Conditional IF Sarah attends Stanford THEN Sarah goes into debt 1 True True True 2 True False False 3 False True True 4 False True False

Logical Conditional Taking ‘if’ seriously If Sarah goes to Stanford then she will incur debt. ≠ If Sarah will incur debt then she goes to Stanford. Either Sarah will incur debt or she goes the Stanford. Both Sarah will incur debt and she goes the Stanford. = = Either Sarah goes the Stanford or she will incur debt. Both Sarah goes the Stanford and she will incur debt.

Logical Conditional Order Matters Antecedent Consequent If Sarah goes to Stanford then she will incur debt.

The Case of Iffy Advice Sarah Scatterleigh weighed her options. She could transfer to Stanford, which had a stronger program for her major and a better track record of placing graduates into the job market. But Stanford cost quite a bit more than the school she was presently attending, Jefferson University. She sought advice from her friend, Johnny Nogginhead, musing that If I go to Stanford then I’ll go into debt. But, replied Johnny, You don’t go to Stanford. I know, said Sarah, I said If I go to Stanford…. But you don’t, retorted Johnny, you go to Jefferson! I never said I didn’t, said an exasperated Sarah, I know I don’t go to Stanford, my point is that going to Stanford might mean going into debt. Why didn’t you just say that, said Johnny.

Logical Interpretations of ‘if’ In a certain sense, ‘if’ means the antecedent isn’t true. If Sarah goes to Stanford then she will incur debt. Either Sarah doesn’t go to Stanford OR she does AND will incur debt. It is NOT that Sarah could go to Stanford and not incur debt.

Logical Interpretations of ‘if’ IF THEN Sarah goes into debt 1 True True True 2 True False False 3 Truth Values match line for line (across all possible worlds) Sarah attends Stanford False True True 4 False True False Sarah doesn’t attend Stanford OR (she does AND goes into debt) 1 False True True True True 2 False False True False False 3 True True False False True 4 True True False False False Either It’s NOT that (Sarah attends Stanford AND doesn't go into debt) 1 True True False False 2 False True True True 3 True False False False 4 True False False True

Logical conditional, often expressed in English by ‘if…then….’, is true when the antecedent is true and the consequent is false, otherwise it is true. It is symbolized by ‘⊃’. It’s logical form is P ⊃ Q.

Logical Form of Conditionals IF P THEN Q P⊃Q P→Q P ⊃ Q True True True 2 True Fals e False 3 False True True 4 False True False 1

to prove a conditional false Prove that the antecedent is true while the consequent is false.

to interpret conditionals logically translate it as ‘Either not p or (p and q)’ or ‘It not that (p and not q)

indicators for conditionals a. if p then q b. q if p c. p only if q d. not p unless q e. supposing p, q f. imagine p ... q g. assuming p, q h. all p are q i. whenever p, q j. when p, q

Validity A Standard of Critical Thinking

Argument, a set of claims in which some claims (premises) are offered to show the truth (or falsehood) of another claim (the conclusion). A line of reasoning.

Argument s Lines of reasoning If it is a mammal then it gives live birth. It lays eggs. If it is red then it has color, So it’s not a mammal. if it has color then it emits or reflects a wavelength of light, thus if it is red then it emits or reflects a wavelength of light. If anything is a dog then it is a mammal. If anything is a mammal then it is an animal. which proves that if anything is a dog then it is an animal. When water is heated to 212° it boils. It’s not boiling, which demonstrates it hasn’t been heated to 212°. If living pigeons didn’t all come from rock pigeons then they must have come from other kinds of pigeons. There are no other kinds of pigeons. This established they all come from rock pigeons. All dogs are mammals. All mammals are animals. Hence all dogs are animals. When a government abuses rights it ought to be removed. The king abuses rights and so he ought to be removed.

indicators for premises a. as b. as shown by c. because d. deduce from e. derive from f. finally, the last reason g. first, second, third,… next h. follows from i. for j. inasmuch as k. indicated by l. is the reason that m.it is the case that n. may be deduced from o. may be derived from p. may be inferred from q. one reason being… r. since s. the fact that t. the reason

indicators for conclusions a. as a result b. consequently c. demonstrates d. entails e. establishes f. hence g. I conclude that h. implies i. in conclusion j. infer k. it follows that l. justifies m.means n. proves o. shows p. so q. then r. therefore s. thus

Sets of Claims Order Are Without Any Particular Sarah’s beliefs The alarm did not go off. Today is either Tuesday or Thursday. She will recognize her teacher. The class meets in the same She has chemistry today. room. Today is Monday. She will recognize her classmates. She went to the right room. She’s not dreaming. She is late.

Argument s therefore) Have Order (‘∴’ means 1.If it is a mammal then it gives live birth. 2.It lays eggs. 1.If it is red then it has color, 3.∴ It’s not a mammal. 2.if it has color then it emits or reflects a wavelength of light, 3.∴ If it is red then it emits or reflects a wavelength of light. 1. If anything is a dog then it is a mammal. 2. If anything is a mammal then it is an animal. 3. ∴ If anything is a dog then it is an animal. 1.When water is heated to 212° it boils. 2.It’s not boiling, 3.∴ It hasn’t been heated to 212°. 1.If living pigeons didn’t all come from rock pigeons then they must have come from other kinds of pigeons. 2.There are no other kinds of pigeons. 3.∴ They all come from rock pigeons. 1. All dogs are mammals. 2. All mammals are animals. 3. ∴ All dogs are animals. 1.When a government abuses rights it ought to be removed 2.The king abuses rights and 3.∴ He ought to be removed.

Validity, if the premises are true then the conclusion is true.

Validity, either the premises are false, or they are true and so it the conclusion. …it is not possible that the premises are true while the conclusion is false.

Validity Another Emergent Property Wetness emerges as a property of water when hydrogen and oxygen are properly combined—though neither are wet themselves. In a similar manner, validity emerges when claims are properly structured into an argument.

Validity Versus Truth Validity Truth Applies to whole arguments Applies to claims, both simple and complex Does not apply to claims Does not apply to arguments As Technical Terms

Valid Arguments Premises 1 If anything is a dog then it is mammal. 2 If anything is a mammal then it is an animal. ∴3 If anything is a dog then it is an animal. Conclusion

Valid Arguments � animals � Here, by premise 1, no dog can be at the bottom of the blue (it is outside of mammals). By premise 2 no mammal can be at the bottom of the green (it is outside of the animals). So there is no place left for a dog to be. � mammals � � dogs � � � � ✔ 1 If anything is a dog then it is mammal. � ✔ 2 If anything is a mammal then it is an animal. ✔ ∴3 If anything is a dog then it is an animal.

Valid Arguments ✔ 1 If anything is a dog then it is mammal. ✔ 2 If anything is a mammal then it is an animal. ✔ ∴3 If anything is a dog then it is an animal. So this argument is valid

Valid Arguments Premises If: hectagon means 1,000,000 sided, and; chiliogon means 1,000 sided, and; megagon means 100 sided; then the conclusion would have to be true. 1 If anything is a hectagon then it has more sides than a chiliogon. 2 If anything is a chiliogon then it has more sides than a megagon. ∴3 If anything is a hectagon then it has more sides than a megagon. Conclusion

Valid Arguments 1 If anything is a hectagon then it has more sides than a chiliogon. 2 If anything is a chiliogon then it has more sides than a megagon. ∴3 If anything is a hectagon then it has more sides than a megagon. So this argument is valid

Valid Arguments But: megagon means 1,000,000 sided, and hectagon means 100 sided (chiliogon does mean 1,000 sided) so the premises are in fact false. ✘ ✘ 1 If anything is a hectagon then it has more sides than a chiliogon. 2 If anything is a chiliogon then it has more sides than a megagon. ∴3 If anything is a hectagon then it has more sides than a megagon.

Valid Arguments ✘ If anything is a hectagon then it has more sides than a chiliogon. 2 If anything is a chiliogon then it has more sides than a megagon. ∴3 ✘ 1 If anything is a hectagon then it has more sides than a megagon. But this argument is valid, because if the premises were true then the conclusion would be true too.

Valid Arguments ✘ If anything is a hectagon then it has more sides than a chiliogon. 2 If anything is a chiliogon then it has more sides than a megagon. ∴3 ✘ 1 If anything is a hectagon then it has more sides than a megagon. Test it by replacing ‘hectagon’, ‘chiliogon’ and ‘megagon’ with ‘triangle’, rectangle’, and ‘octogon’.

Valid Arguments Both Arguments have the Form: 1. P ⊃ Q 2. Q ⊃ R 3. ∴ P ⊃ R

Invalid Arguments ✔ 1 If anything is a camel then it has four legs. ✔ 2 If anything is a pig then it has four legs. ✘ ∴3 If anything is a pig then it is a camel. This argument is invalid, because even if the premises are true then the conclusion is not.

Invalid Arguments This Argument Has the Form: P⊃R Q⊃R ∴P⊃Q It is not a valid form.

to determine validity check to see if the form of the argument fits one of the valid patterns.

Validity: The Test Yes If it is… …then it is invalid Is it possible for the premises to be true while the conclusion is false? No If it is not… …then it is valid

Validity: A Quick Check Yes If it does… …then it is valid Does the argument have a valid form? No If it does not… …then it is invalid

Soundness A Complex Standard of Critical Thinking

Soundness, valid arguments with true premises.

Arguments arguments valid arguments sound arguments A Taxonomy 1. If anything is a camel then it is a has humps. 2. Thor has no humps. 3. ∴ Thor is not a camel. 1. If anything is a camel then it has four legs. 2. If anything is a pig then it has four legs. 3. ∴ If anything is a pig then it is a camel. 1. If anything is a dog then it is mammal. 2. If anything is a mammal then it is an animal. 3. ∴ If anything is a dog then it is an animal. 1. If anything is a hectagon then it has more sides than a chiliogon. 2. If anything is a chiliogon then it has more sides than a megagon. 3. ∴ If anything is a hectagon then it has more sides than a megagon. 1. If anything is a pig then it is a quadruped. 2. Trakr is a quadruped. 3. ∴ Trakr is a pig. 1. If anything is wild then it is free. 2. Peter is not wild. 3. ∴ Peter is not free. 1. If anything is a dog then it is has four legs. 2. If anything is a cat then it is has four legs. 3. ∴ If anything is a dog then it is a cat.

Soundness Yes Are the premises true? Yes it is sound Is the argument valid? No No it is unsound

to determine soundness check to see if the form of the argument fits one of the valid patterns, then check to see if the premises are true.

Validity Some Common Forms Chain Arguments 1. If anything is a dog then it is mammal. 2. If anything is a mammal then it is an animal. 3. ∴ If anything is a dog then it is an animal. Modus Ponens 1. When a government abuses rights it ought to be removed. 2. The king abuses rights . 3. ∴ He ought to be removed. Modus Tollens 1. If it is a mammal then it gives live birth. 2. It lays eggs. 3. ∴ It’s not a mammal.

Chain Argument A Common Form Chain Arguments 1.If anything is a dog then it is mammal. 2.If anything is a mammal then it is an animal. 3.∴ If anything is a dog then it is an animal.

Chain Argument The Parts of a Chain Argument If anything is a dog then it is mammal. 1.Conditional Premise 2.Conditional Premise 3.Conditional Conclusion If anything is a mammal then it is an animal. ∴ If anything is a dog then it is an animal.

Chain Argument The antecedents and consequents of the premises link up as in a chain. The Structure of a Chain Argument 1.If anything is a dog then it is a mammal. 2.If anything is a mammal then it is an animal. 3.∴ If anything is a dog then it is an animal. The conclusion has the same antecedent as the first premise… The conclusion has the same consequent as the last premise…

Chain Arguments Also called ‘hypothetical syllogisms’ 1. P ⊃ Q 2. Q ⊃ R 3. ∴ P ⊃ R

Chain Arguments Can have indefinitely many premises 1. 2. 3. 4. 5. P⊃Q Q⊃R R⊃S S⊃T ∴P⊃T

to calculate the number of possible worlds raise two to the power of the number of claims being evaluated, here there are three: P, Q, & R 3 2 =2•2•2=8

Proving Chain Arguments Valid P Q Q R P Q) (Q R) (P 1 True True True True True True 2 True True True False True False 3 True False False True True True 4 True False False False True False 5 False True True True False True 6 False True True False False False 7 False False False True False True 8 False False False False False False (P ⊃ ⊃ R ⊃ R) Step One: Assign values to the simplest atomic claims, P, Q, & R

Proving Chain Arguments Valid (P ⊃ Q) (Q ⊃ R) (P ⊃ R) 1 True True True True True True True True True 2 True True True True False False True False False 3 True False False False True True True True True 4 True False False False True False True False False 5 False True True True True True False True True 6 False True True True False False False True False 7 False True False False True True False True True 8 False True False False True False False True False Step Two: Determine the values of the next simplest or molecular claims.

Proving Chain Arguments Valid (P ⊃ Q) 1 True True True (Q ✔ ⊃ R) True True True (P ✔ ⊃ R) True True True 2 True True True True False False True False False 3 True False False False True True True True True 4 True False False False True False True False False 5 False True True 6 False True True ✔ True True True True False False ✔ False False ✔ False ✔ False True True False True False ✔ False False ✔ False 7 False True False True True True True 8 False True True True False Step Three: Determine if there is a possible world where the premises are both true while the conclusion is false.

Proving Chain Arguments Valid There is no possible world where the premises are true while the conclusion is false. (P ⊃ Q) 1 True True True (Q ✔ ⊃ R) True True True (P ✔ ⊃ R) True True True 2 True True True True False False True False False 3 True False False False True True True True True 4 True False False False True False True False False 5 False True True 6 False True True ✔ True True True True False False ✔ False False ✔ False ✔ False True True False True False ✔ False False ✔ False 7 False True False True True True True 8 False True True True False So Chain Arguments are valid.

An Unnamed Fallacy There is a possible world where the premises are true while the conclusion is false. (P ⊃ R) 1 True True True (Q ✔ 2 True False False 3 True True True 6 False True False R) True True True ✔ False True True False True False ✔ True True True True False False ✔ False False ✔ False ⊃ (P ✔ True False False 4 True False False 5 False True True ⊃ Q) True True True True True True ✘ True False False True False False ✔ False True True False True True ✔ False False ✔ False 7 False True True True True True False 8 False True True True False So arguments of this form are invalid.

Modus Ponens A Common Form modus ponens 1.When a government abuses rights it ought to be removed. 2.The king abuses rights . 3.∴ He ought to be removed.

Modus Ponens The Parts of a Modus Ponens Argument 1.Conditional Premise 2.Premise Affirming the Antecedent of the Conditional 3.Concluding the Consequent of the Conditional When a government abuses rights it ought to be removed. The king abuses rights . ∴ He ought to be removed.

Modus Ponens The Structure of a Modus Ponens Argument A conditional premise. 1. When a government abuses rights it ought to be removed. 2. The king abuses rights . 3. ∴ He ought to be removed. A premise which affirms the antecedent of the conditional premise. The conclusion is the consequent of the conditional premise.

Modus Ponens Also called ‘Affirming the Antecedent’ and ‘Conditional Elimination’ 1. P ⊃ Q 2. Q 3. ∴ P

Modus Ponens Can be extended by Chain Argument 1. 2. 3. 4. 5. P P⊃Q Q⊃R R⊃S ∴S

to calculate the number of possible worlds raise two to the power of the number of claims being evaluated, here there are two: P & Q 2 2 =2•2=4

Proving Modus Ponens Valid P P Q Q (P (P Q) Q 1 True True True True 2 True True False False 3 False False True True 4 False False False False ⊃ Step One: Assign values to the simplest atomic claims, P & Q

Proving Modus Ponens Valid (P (P ⊃ Q) Q 1 True True True True True 2 True True False False False 3 False False True True True 4 False False True False False Step Two: Determine the values of the next simplest or molecular claims.

Proving Modus Ponens Valid (P (P ✔ ⊃ Q) True True True Q ✔ 1 True True 2 True True False False False 3 False False True True True 4 False False True False False Step Three: Determine if there is a possible world where the premises are both true while the conclusion is false.

Proving Modus Ponens Valid There is no possible world where the premises are true while the conclusion is false. (P (P ✔ ⊃ Q) True True True Q ✔ 1 True 2 True True False False False 3 False False True True True 4 False False True False False So Modus Ponens Arguments are valid. True

An Attendant Fallacy: Affirming the Consequent There is a possible world where the premises are true while the conclusion is false. (P 1 True 2 False 3 True 4 False ✔ ✔ ⊃ Q) True True True True (Q False False False True True False True False P ✔ True True ✘ False False So arguments which affirm the consequent are invalid—and so such arguments are fallacies

Modus Tollens A Common Form modus tollens 1.If it is a mammal then it gives live birth. 2.It lays eggs. 3.∴ It’s not a mammal.

Modus Tollens The Parts of a Modus Tollens Argument 1.Conditional Premise 2.Premise Denying the Consequent of the Conditional 3.Concluding the Denial of the Antecedent of the Conditional If it is a mammal then it gives live birth. It lays eggs. ∴ It’s not a mammal.

Modus Ponens The Structure of a Modus Tollens Argument A conditional premise. 1. If it is a mammal then it gives live birth. 2. It lays eggs. 3. ∴ It’s not a mammal. A premise which denies the consequent of the conditional premise. The conclusion is the denial of the antecedent of the conditional premise.

Modus Tollens Also called ‘Denying the Consequent’ 1. P ⊃ Q 2. ~Q 3. ∴ ~P

Modus Tollens Can be extended by Chain Argument 1. 2. 3. 4. 5. P⊃Q Q⊃R R⊃S ~S ∴ ~P

to calculate the number of possible worlds raise two to the power of the number of claims being evaluated, here there are two: P & Q 2 2 =2•2=4

Proving Modus Tollens Valid ~Q P Q ~P (~Q (P Q) ~P 1 False True True False 2 True True False False 3 False False True True 4 True False False True ⊃ Step One: Assign values to the simplest atomic claims, P & Q, keeping track of negation.

Proving Modus Tollens Valid (~Q (P ⊃ Q) ~P 1 False True True True False 2 True True False False False 3 False False True True True 4 True False True False True Step Two: Determine the values of the next simplest or molecular claims.

Proving Modus Tollens Valid (~Q (P ⊃ Q) ~P 1 False True True True False 2 True True False False False 3 False False True True True 4 True False True False ✔ ✔ True Step Three: Determine if there is a possible world where the premises are both true while the conclusion is false.

Proving Modus Tollens Valid There is no possible world where the premises are true while the conclusion is false. (~Q (P ⊃ Q) ~P 1 False True True True False 2 True True False False False 3 False False True True True 4 True False True False ✔ ✔ So Modus Tollens Arguments are valid. True

An Attendant Fallacy: Denying the Antecedent There is a possible world where the premises are true while the conclusion is false. (~P (P ⊃ Q) ~Q 1 False True True True False 2 False True False False True 3 True False True True 4 True False True False ✔ ✘ False True So arguments which affirm the consequent are invalid—and so such arguments are fallacies

Disjunction Logical Complexity

A Logically Simple Truth Two logical possibilities The coast is foggy. 1 True 2 False Given that we’ve filled in the indices, made the ceteris paribus explicit, and defined key terms.

Combining Logically Simple Truths Four states of affairs (states) or possible worlds The coast is foggy. The coast is sunny. 1 True True 2 True False 3 False True 4 False False

Logical Conjunction The coast is foggy. AND The coast is sunny. 1 True True True 2 True False False 3 False False True 4 False False False

Logical Conditional IF The coast is foggy. THEN The coast is sunny. 1 True True True 2 True False False 3 False True True 4 False True False

Logical Disjunction The coast is foggy. OR The coast is sunny. 1 True ? True 2 True ? False 3 False ? True 4 False ? False

Logical Disjunction The coast is foggy. OR The coast is sunny. 1 True ? True 2 True ? False 3 False ? True 4 False False False

Logical Disjunction The coast is foggy. OR The coast is sunny. 1 True ? True 2 True True False 3 False True True 4 False False False

Logical Disjunction The coast is foggy. OR The coast is sunny. 1 True True/False True 2 True True False 3 False True True 4 False False False

Logical Disjunction The Ambiguity of ‘or’ Exclusive ‘or’ Inclusive ‘or’ Either the Giants win the division Either the Giants make the or the A’s do (but not both) playoffs or the A’s do (or both) Either heads or tails (but not both) Either by plane or by car (or both) Latin: aut Latin: vel

Logical Disjunction Logic settles on an inclusive way The coast is foggy. OR The coast is sunny. 1 True True True 2 True True False 3 False True True 4 False False False

Logical disjunction, often expressed in English by ‘Either…or….’, is false when the both components are false, otherwise it is true. It is symbolized by ‘V’. It’s logical form is P V Q.

Logical Form of Disjunctions Either P OR Q PVQ P V Q 1 True True True 2 True True False 3 False True True False Fals e False 4

to prove a disjunction true Prove that one of the component claims is true.

to interpret a disjunction specify if you are using it inclusively or exclusively.

Tautolog y Putting Negation and Disjunction Together Which claim is not a disjunction? Hockey is better than basketball but it is not better than basketball.* Jupiter is bigger than Mars or it is not Drinking milk is healthy or unhealthy.* bigger than Mars. New York either is or isn’t the largest city in the US.* Same-sex schools are optimal unless same-se Eleven is a prime number or schools are less than optimal. eleven is not a prime number. Romeo and Juliette is a tragedy or it is not a tragedy.* The child looks at the jellyfish or looks away from it*. The jellyfish has tentacles—or not! The music is loud or the Either Jacqui thinks black is more alluring than pink music is quiet.* or she doesn’t. The Constitution of the United States was adopted on either September 17, 1787 or July 4, 1776.*

Tautologie s Tautology: P V ~P The Logical Form of a The square is white V The square is not white 1 True ? False 2 False ? True Given that disjunctions are false when all component claims are false, what is the truth value of this disjunction?

Tautologie s Tautology: P V ~P The Logical Form of a The square is white V The square is not white 1 True True False 2 False True True Tautologies are true in all possible worlds.

Tautolog y Tautology: P V ~P The Logical Form of a P V ~P 1 True True False 2 False True True Tautologies are true in all possible worlds.

Tautology, a special form of disjunction in which a claim and its negation are joined—they are always true. The logical form of a tautology is P V ~P.

Tautolog yPrinciple of Sufficient The Logical Form of the Reason For every claim, give a reason why it is true or not true. T V ~T 1 True True False 2 False True True The Principle of Sufficient Reason covers all possible worlds.

Controlling the Question Is drinking milk healthy for humans? What are the healthiest drinks for humans? What Constitutional rights should we keep? Has the Constitutional right to bear arms outlived its usefulness? Are single-sex schools better for education? What is the best method of education?

Controlling the Question Open Questions What are the healthiest drinks for humans? What Constitutional rights should we keep? What is the best method of education? Yes-or-no Questions Is drinking milk healthy for humans? Has the Constitutional right to bear arms outlived its usefulness? Are single-sex schools better for education?

Open Questions Are Topic or Theme Questions What are the healthiest drinks for humans? What Constitutional rights should we keep? What is the best method of education?

A Topic Question: an open question. Such questions require disjunctive reasoning to treat the alternates.

Disjunctive Reasoning Reasons supporting 1st alternate: this alternate or refuting the others. the question Reasons supporting 2nd alternate: this alternate or refuting the others. . . . Final alternate: . . . Reasons supporting this alternate or refuting the others. Which alternate has the better reasons?

Disjunctive Reasoning Water: What are the healthiest drinks for humans? Reasons supporting water or refuting the others. Milk: Reasons supporting milk or refuting the others. . . . . . . Reasons supporting Electrolyte electrolyte solutions or Solutions refuting the others. Which alternate has the better reasons?

Validity Some Common Forms Involving Disjunctions Disjunctive Argument 1. Either Kierkegaard can be a Christian or a philosopher. 2. He cannot be a philosopher. 3. ∴ So he must be a Christian. Simple Dilemma 1. If Johnny’s friendship is for pleasure then he is not a true friend. 2. If Johnny’s friendship is for utility then he is not a true friend. 3. Either Johnny’s friendship is for pleasure or utility. 4. ∴ Johnny’s friendship is not a true friendship. Dilemma 1. If existence precedes essence then humanity is free. 2. If there is no God then we we alone can justify ourselves, without excuse. 3. Either existence precedes essence or there is not God. 4. ∴ Either humanity is free or is without any justifications or excuses but those they provide.

Disjunctive Argument A Common Form Disjunctive Argument 1.Either Kierkegaard can be a Christian or a philosopher. 2.He cannot be a philosopher. 3.∴ So he must be a Christian.

Disjunctive Argument The Parts of a Disjunctive Argument 1.Disjunctive Premise. 2.Premise Denying one of the disjuncts of the Disjunction. 3.Concluding the remaining Disjunct. Either Kierkegaard can be a Christian or a philosopher. He cannot be a philosopher. ∴ So he must be a Christian.

Disjunctive Argument The Structure of a Disjunctive Argument A disjunctive premise. 1. Either Kierkegaard can be a Christian or a philosopher. 2. He cannot be a philosopher. 3. ∴ So he must be a Christian. A premise which denies one of the component claims of the disjunctive premise. The conclusion is the remaining component claim of the disjunctive premise.

Disjunctive Argument More commonly called ‘Disjunctive Syllogism’ and also called ‘modus tollendo ponens' 1. P V Q 2. ~Q 3. ∴ P 1. P V Q 2. ~P 3. ∴ Q Can be run either way

Disjunctive Syllogism Can be extended indefinitely 1. 2. 3. 4. 5. PVQVRVS ~Q ~R ~S ∴P

to calculate the number of possible worlds raise two to the power of the number of claims being evaluated, here there are two: P & Q 2 2 =2•2=4

Proving Disjunctive Argument Valid ~P P Q Q ~P (P Q) Q 1 False True True True 2 False True False False 3 True False True True 4 True False False False V Step One: Assign values to the simplest atomic claims, P & Q, minding the negations.

Proving Disjunctive Argument Valid ~P (P V Q) Q 1 False True True True True 2 False True True False False 3 True False True True True 4 True False False False False Step Two: Determine the values of the next simplest or molecular claims.

Proving Disjunctive Argument Valid ~P (P V Q) Q 1 False True True True True 2 False True True False False 3 True False True True 4 True False False False ✔ ✔ True False Step Three: Determine if there is a possible world where the premises are both true while the conclusion is false.

Proving Disjunctive Argument Valid There is no possible world where the premises are true while the conclusion is false. ~P (P V Q) Q 1 False True True True True 2 False True True False False 3 True False True True 4 True False False False ✔ ✔ So Disjunctive Arguments are valid. True False

Simple Dilemma A Common Form Simple Dilemma 1.If Johnny’s friendship is for pleasure then he is not a true friend. 2.If Johnny’s friendship is for utility then he is not a true friend. 3.Either Johnny’s friendship is for pleasure or utility. 4.∴ Johnny is not a true friend.

Simple Dilemma The Parts of a Simple Dilemma Either Johnny’s friendship is for pleasure or utility. 1. Disjunctive Premise. 2. A Conditional Premise whose antecedent is one of the disjuncts of the Disjunctive Premise and whose consequent is the same as the other Conditional Premise. 3. Another Conditional Premise whose antecedent is the other disjunct of the Disjunctive Premise and whose consequent is the same as the other Conditional Premise. 4. Concluding the Consequent of the Conditional Premises. If Johnny’s friendship is for pleasure then he is not a true friend. If Johnny’s friendship is for utility then he is not a true friend. ∴ Johnny is not a true friend.

Simple Dilemma The Structure of a Simple Dilemma One antecedent is a component of the disjunction. The other antecedent is the other component of the disjunction. 1. If Johnny’s friendship is for pleasure then he is not a true friend. 2. If Johnny’s friendship is for utility then he is not a true friend. 3. Either Johnny’s friendship is for pleasure or utility. 4. ∴ Johnny is not a true friend. A disjunctive premise. The conclusion is the consequent of the conditional premises. Both Conditional Premises share a consequent.

Simple Dilemma Also called ‘Disjunctive Elimination’ 1. 2. 3. 4. PVQ P⊃R Q⊃R ∴R

Simple Dilemma Can be extended indefinitely 1. 2. 3. 4. 5. PVQVR P⊃S Q⊃S R⊃S ∴S

Destructive Dilemma Simple Dilemma combined with Modus Tollens 1. 2. 3. 4. ~R P⊃R Q⊃R ∴ ~P V Q

to calculate the number of possible worlds raise two to the power of the number of claims being evaluated, here there are three: P, Q, & R 3 2 =2•2•2=8

Proving Simple Dilemma Valid P Q P R Q Q) (P R) (Q 1 True True True True 2 True True True 3 True False 4 True (P V ⊃ R ⊃ R R) R True True True False True False False True True False True True False True False False False False 5 False True False True True True True 6 False True False False True False False 7 False False False True False True True 8 False False False False False False False Step One: Assign values to the simplest atomic claims, P, Q, & R

Proving Simple Dilemma Valid (P V Q) (P ⊃ R) (Q ⊃ R) R 1 True True True True True True True True True True 2 True True True True False False True False False False 3 True True False True True True False True True True 4 True True False True False False False True False False 5 False True True False True True True True True True 6 False True True False True False True False False False 7 False False False False True True False True True True False True False False True False 8 False False False Step Two: Determine the values of the next simplest or molecular claims. False

Proving Simple Dilemma Valid (P V Q) 1 True True True 2 True True True (P ✔ ⊃ R) True True True 5 False True True ✔ True False False 3 True True False ✔ True True True 4 True True False (Q True True R) True True True R ✔ True False False ✔ False True False False ✔ False ⊃ True True False ✔ False True False ✔ True True True True True False ✔ True 6 False True True False True False True False False False 7 False False False False True True False True True True 8 False False False False True False False True False False Step Three: Determine if there is a possible world where the premises are both true while the conclusion is false.

Proving Simple Dilemma Valid There is no possible world where the premises are true while the conclusion is false. (P V Q) 1 True True True 2 True True True (P ✔ ⊃ R) True True True 4 True True False 5 False True True ✔ True False False 3 True True False ✔ True True True True True R) True True True R ✔ True False False ✔ False True False False ✔ False ⊃ (Q True True False ✔ False True False ✔ True True True True True False ✔ True 6 False True True False True False True False False False 7 False False False False True True False True True True 8 False False False False True False False True False False So Simple Dilemmas are valid.

Dilemma A Common Form Dilemma 1.If existence precedes essence then humanity is free. 2.If there is no God then we we alone can justify ourselves, without excuse. 3.Either existence precedes essence or there is no God. 4.∴ Either humanity is free or is without any justifications or excuses but those they provide.

Dilemma The Parts of a Dilemma Either existence precedes essence or there is no God. 1. Disjunctive Premise. 2. A Conditional Premise whose antecedent is one of the disjuncts of the Disjunctive Premise. 3. Another Conditional Premise whose antecedent is the other disjunct of the Disjunctive Premise. 4. Concluding a Disjunction of the Consequents of the Conditional Premises. If existence precedes essence then humanity is free. If there is no God then we we alone can justify ourselves, without excuse. ∴ Either humanity is free or is without any justifications or excuses but those they provide.

Dilemma The Structure of a Dilemma One antecedent is a component of the disjunction. 1. 2. 3. 4. The other antecedent is the other component of the disjunction. If existence precedes essence then humanity is free. If there is no God then we we alone can justify ourselves, without excuse. Either existence precedes essence or there is no God. ∴ Either humanity is free or is without any justifications or excuses but those they provide. A disjunctive premise. The conclusion is a disjunction of the consequents of the conditional premises.

Dilemma Also called ‘Constructive Dilemma’ 1. 2. 3. 4. PVQ P⊃R Q⊃S ∴RVS

Simple Dilemma Can be extended indefinitely 1. 2. 3. 4. 5. PVQVR P⊃S Q⊃T R⊃U ∴SVTVU

Destructive Dilemma Dilemma combined with Modus Tollens 1. 2. 3. 4. ~R V ~S P⊃R Q⊃S ∴ ~P V ~Q

Destructive Dilemma Dilemma combined with Modus Tollens and a Tautology 1. 2. 3. 4. ~R V ~R P⊃R Q⊃R ∴ ~P V ~Q

to calculate the number of possible worlds raise two to the power of the number of claims being evaluated, here there are three: P, Q, & R 4 2 = 2 • 2 • 2 • 2 = 16

Proving Dilemma Valid P Q (P V P Q) (P R ⊃ Q R) (Q S ⊃ R S) R S V S 1 True True True True True True True True 2 True True True True True False True False 3 True True True False True True False True 4 True True True False True False False False 5 True False True True False True True True 6 True False True True False False True False 7 True False True False False True False True 8 True False True False False False False False 9 False True False True True True True True 10 False True False True True False True False 11 False True False False True True False True 12 False True False False True False False False 13 False False False True False True True True 14 False False False True False False True False 15 False False False False False True False True 16 False False False False False False False False Step One: Assign values to the simplest atomic claims, P, Q, R,

Proving Dilemma Valid (P V Q) (P ⊃ R) (Q ⊃ S) R V S 1 True True True True True True True True True True True True 2 True True True True True True True False False True True False 3 True True True True False False True True True False True True 4 True True True True False False True False False False False False 5 True True False True True True False True True True True True 6 True True False True True True False True False True True False 7 True True False True False False False True True False True True 8 True True False True False False False True False False False False 9 False True True False True True True True True True True True 10 False True True False True True True False False True True False 11 False True True False True False True True True False True True 12 False True True False True False True False False False False False 13 False False False False True True False True True True True True 14 False False False False True True False True False True True False 15 False False False False True False False True True False True True 16 False False False False True False False True False False False False Step Two: Determine the values of the next simplest or molecular claims.

Proving Dilemma Valid (P ⊃ R) (Q ⊃ S) True True True True True True True True True True True False True True True False False True True True True True False False 5 True True False ✔ True True True 6 True True False ✔ True True True 7 True True False True False 8 True True False True 9 False True True 10 False True True 11 False True True 12 False True 13 False 14 (P V Q) R V S 1 True True True True True True 2 True True False True True False 3 True True True False True True 4 True False False False False False ✔ False True True ✔ True True True ✔ False True False ✔ True True False False False True True False True True False False False True False False False False False True True True True True True True True False True True True False False True True False False True False True True True False True True True False True False True False False False False False False False False True True False True True True True True False False False False True True False True False True True False 15 False False False False True False False True True False True True 16 False False False False True False False True False False False False ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ Step Three: Determine if there is a possible world where the premises are both true while the conclusion is false.

Proving Dilemma Valid There is no possible world where the premises are true while the conclusion is false. (P ⊃ R) (Q ⊃ S) True True True True True True True True True True True False True True True False False True True True True True False False 5 True True False ✔ True True True 6 True True False ✔ True True True 7 True True False True False 8 True True False True 9 False True True 10 False True True 11 False True True 12 False True 13 False 14 (P V Q) R V S 1 True True True True True True 2 True True False True True False 3 True True True False True True 4 True False False False False False ✔ False True True ✔ True True True ✔ False True False ✔ True True False False False True True False True True False False False True False False False False False True True True True True True True True False True True True False False True True False False True False True True True False True True True False True False True False False False False False False False False True True False True True True True True False False False False True True False True False True True False 15 False False False False True False False True True False True True 16 False False False False True False False True False False False False ✔ ✔ ✔ ✔ ✔ ✔ So Dilemmas are valid. ✔ ✔ ✔

Fallacies A Relevant Fallacy

False Dilemma, to provide a non exhaustive disjunction as a premise—it is a fallacy.

to avoid false dilemma Provide an exhaustive list of the possible answers to the topic question, listing explicitly those you may not wish to treat.

Assignment What does it mean to be ethical? Ethics How do you come to an ethical decision? What is hypocrisy?

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