# Recall. Validity: If the premises are true the conclusion must be true. Soundness. Valid; and. Premises are true

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1 Recall Validity: If the premises are true the conclusion must be true Soundness Valid; and Premises are true

2 Validity In order to determine if an argument is valid, we must evaluate all of the sets of truth values (interpretations) that an argument can have. This can be quite difficult. Logicians have devised many tools for establishing validity using a variety of shortcuts. In contrast, establishing invalidity is quite simple: all you need to do is to provide one counter example (the Method of Counter Example).

3 Valid forms Modus Ponens If A then B A Therefore B Modus Tollens If A then B ~B Therefore ~A

4 Conditional truth table Truth table for A->B i.e. If A then B We can verify both MP and MT. Note: A->B = ~A or B.

5 Examples of if then Equivalent to S->B: If you re smart then you re beautiful. If you re smart, you re beautiful. You re beautiful if you re smart. You re smart only if you re beautiful.

6 Valid forms (cont.) Disjunctive syllogism A or B not A Therefore B MP Equivalence A->B (i.e. ~A v B) A (i.e., not not A) Therefore B

7 Valid forms (cont.) Hypothetical syllogism A -> B B -> C Therefore A->C Two MP applications if A then B if B then C A (hypothesis, i.e., 'if') B C Therefore, if A then C

8 Quantifiers Quantifiers are all or some in a sentence MP applies for all but not some Example: Some dogs are black Some black things are bigger than earth Therefore some dogs are bigger than earth (invalid) Instead, use Venn diagrams

9 Venn diagrams A B A B All A are B Some A are B Etc (No A are B; i is B; No C is A and B)

10 Valid forms (cont.) Universal syllogism All A are B All B are C So All A are C A B C Two MP applications All A are B = (x)ax->bx All B are C Ax (hypothesis, i.e., 'if') Bx Cx Therefore, All A are C All A are B All B are C So, All A are C

11 Valid forms (cont.) Universal-particular syllogisms All A are B Individual i is A Therefore, i is B All A are B Individual i is non-b Therefore, i is non-a B A *i *i B A Modus Ponens (x)ax->bx Ai Therefore Bi Modus Tollens (x)ax->bx ~Bi Therefore ~Ai (Note: non is often not the same as not e.g. non-friendly versus not friendly).

12 Summary There is no point in trying to memorize all of the possible valid argument forms. There are an infinite variety. Nevertheless, the ones above are by far the most common. The power of formal logic lies in its ability to systematically organize valid argument forms (e.g., seeing how they are all versions of MP).

13 Invalid arguments To show an argument is invalid, you can: describe a situation in which the premises are true and the conclusion is false; or show that the argument fits a typical form of invalid argument For instance

14 Invalid forms Affirming the consequent If A then B B Therefore A Example of affirming the consequent If I punch you in the nose it bleeds. Your nose is bleeding. Therefore I punched you in the nose. (Invalid because of other possible nasal issues)

15 Invalid forms Denying the antecedent If A then B not A Therefore not B Example of denying the antecedent If I push the book then it will move. I didn't push the book. Therefore it didn't move. (Invalid because of other possible causes)

16 Sailing the three C s Need to read arguments: carefully critically charitably In brief this means: identifying each and all assertions determining if the argument is valid and/or sound trying to make the arguments as plausible as possible

17 Being charitable Often possible to paraphrase the explicit statements in an argument Essential that the premise as paraphrased says exactly same thing as the original. The economy will be in big trouble, because it cannot tolerate the continuation of low interest rates, but such rates will continue. Better re-phrasing: "If low interest rates continue, then the economy will be in big trouble." (plus Such rates will continue )

18 Determining validity We can summarize the steps needed to determine the validity of an argument as follows: A. Identify the conclusion. (careful) B. Identify the premises. (careful) C. Identify reasonable and stated premises. (charitable) D. Determine whether the argument is valid (if not considered conservative modifications that would make it valid). (critical)

19 Islamic-world argument If America and the world's other governments in their completely justified, very necessary, and much overdue "war against terrorism" will also commit part of this long and difficult struggle to the creation of a good, right, and just society, if they do so in a way that follows the accepted standards of justice, and if they harm no innocent citizens of any nation in the process, then it should be expected they will receive the full and continued support of virtually every Muslim in the World. Regarding the issue of exploitation, particularly economic exploitation, we view as highly significant a comment made in an address to the people of America several years ago by then President of the United States, Bill Clinton, who said, "America with about five percent of the world's population utilizes about forty percent of the world's resources; and, we are going to have to work ever harder to keep it that way." When you are an African parent sitting homeless in the dust, hungry and thirsty, while your children die slowly before your eyes it is easy to see the blatant unfairness of this economic disparity. Is it then surprising that there will be anger toward America? Allah has told us that wealth must be equitably distributed among the world's people and nations.

20 Islamic-world argument Main argument P1) If America commits part of their struggle to the creation of a just society and follow accepted standards of justice and harm no innocent citizens then they will receive the support of all Muslims. P2) America does not follow the accepted standards of justice. C1) Therefore they won't receive support of all Muslims. (invalid) Argument for P2 P3) If you exploit other nations you do not follow the accepted standards of justice. P4) Bill Clinton made a statement showing that the Americans will continue to attempt to exploit other nations. Therefore C2/P2. (valid)

21 Argument structure Draw the argument structure: P3 P4 P2/C2 P1 Simple repair? C1 If they want to get support then follow justice Or, If and only if i.e. A->B & B->A

22 Criticizing valid arguments If the argument is definitely valid, then, as a critic you have at least three remaining possibilities: 1. Show that one of the premises is false (the argument is unsound) 2. Suggest that one of the premises is unjustified (that is, demand more reasons for thinking that premise is true) 3. Show that the argument is unclear or vague, or of poor form

23 Classifying arguments Good (sound) Sound (and valid) Exception: Tautology Invalid Bad (not sound) Valid not sound contradiction

24 Classifying arguments invalid contra etc. sound (good) bad valid

25 Vagueness 2000 years ago, Sophists exploited vagueness (still do). What are sources and pitfalls of vagueness? Vagueness and Context Example: Ann went to the store. She bought some muffins. & & & ('She' refers to the individual 'Ann'.) Example: Ann went near the store. She got muffins while she was there. ('There' refers to 'a vicinity' near the store, which could be any number of places.)

26 Vagueness Ambiguity Lexical ambiguity Example: He kicked his can. Syntactic ambiguity Example: Defining your terms clearly strengthens your argument. Contextual ambiguity Example: John's father wanted the key with him.

27 Vagueness Equivocation (scope fallacies) & & & & & P1) Hot dogs are better than nothing. P2) Nothing is better than steak. C) Therefore, hot dogs are better than steak. The existence of a law means that there must be a law maker. But we know that the law of gravity and other scientific laws have not been made by any human law maker. So it follows that there must be a non-human law maker, God. Everybody loves somebody. Therefore, everybody loves one of us.

28 Vagueness Connotation (i.e. using emotion to affect reasoning) Example: He sure is <<debonaire, easy-going, devil-may-care, reckless>> Euphemism Example: neutralize, collateral damage, friendly fire, Instead of content Example: "Have a coke and a smile" "Pepsi, taste of a new generation" "Toyota Corolla. It just wants to have fun." Jargon/Bafflegab (why? it works) Example: Digging a hole: The author of this report utilized a manually operated soil displacement implement to facilitate the synergistic excavation of a negative volume of mineral aggregate relative to a datum plane defined at ground level.

29 Connotation (cont.) Loaded Verbs Vagueness Example: Surely you realize that using jargon just confuses the audience. Versus: Surely you believe that using jargon just confuses the audience. Insinuation Example: Today, you were on time Example: You should not speak ill of your friends (move emphasis).

30 Question and Example Question: Example: Char Margolis Cold Reading Example: Florsheim Magnetic Shoes (others include BIOFlex insoles/shoes).

31 Bonus: Is it valid? The pen in my hand is red and five hundred pounds. It is not red. Therefore it is five hundred pounds. 1. A & B 2. ~A 3. A (from 1) 4. ~A v B (from 2; i.e. A=>B) 5. B (conclusion; from 4 and 3 or directly from 1) Is it a valid deduction? Technica"y it is (i.e. given the agreed upon definition of what to do with contradiction). What s wrong, then? It cannot possibly be sound, given the law of non-contradiction.

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