~ 3 ~ -LOGIC WITH UNIVERSAL GENERALIZATIONS- Validity

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1 ~ 3 ~ -LOGIC WITH UNIVERSAL GENERALIZATIONS- i. DEDUCTION, VALIDITY, AND LOGIC Validity Deductive arguments are those that are supposed to be valid. In a valid deductive argument, the premises support the conclusion in a special way: they absolutely guarantee it. This means that if the premises are true, then it is absolutely impossible literally unthinkable, or unimaginable that the conclusion could be false. Any deductive argument that is not valid is called invalid. 5 A valid argument can have false premises. However, if it does, the truth of the conclusion is no longer guaranteed: the conclusion could be true or false. Valid deductive arguments are truth-preserving : if we put truth into the argument (in its premises), we get truth out (in its conclusion). This means that if we know that its conclusion is false, we can know that it has at least one false premise. Logic Logic is the study of good argument patterns, patterns of inference that reliably lead to a true conclusion when we start from true premises. Logic usually means deductive logic, the study of valid argument patterns. Here are two examples: Argument 3.1: All A s are B s. And is an A. Therefore is a B. Argument 3.2: If P, then Q. And P is true. Therefore Q is true. 5 Vocabulary Alert: In ordinary language, the word deduce often means simply infer, not necessarily with a logically deductive argument. And in ordinary language, valid often means true, relevant, legitimate, justified, or good. [ 35 ]

2 In Argument 3.1, the letters stand for things; in Argument 3.2, the letters stand for statements. It does not matter what statements we fill in for P and Q, or what things we fill in for A, B, and. Any argument that follows one of these patterns (or others that we will learn) is valid. These kinds of short deductive arguments are called syllogisms. Valid syllogisms re-organize or re-combine information from their premises for the conclusion. ii. UNIVERSAL GENERALIZATIONS The ancient Greek philosopher Aristotle ( BCE) invented the first kind of logic. We will learn a simplified version of this logic, which constructs valid syllogisms using universal generalizations. A universal generalization relates one type of thing to another type of thing. There may be lots, or just one, or none at all, of each type of thing. Universal means that the statement gives a rule that allows not even one exception (case where it is false). We ll look at universal generalizations constructed with the quantifiers all, only, or no. Every universal generalization can be written in this form: [Quantifier] A s are B s. A and B are plural nouns (things we can count). Statement 3.3: All disasters are earthquakes. Statement 3.4: Only professional athletes are celebrities. Statement 3.5: No films are comedies. Not every universal generalization is written in this form, but every one can be written in this form. Writing generalizations in this form will be necessary for doing logic with them. To write them this way correctly, we need to pay close attention to the structure of sentences. [ 36 ]

3 Subject and Predicate Every sentence has two parts: subject and predicate. The subject is what the sentence is about. The predicate is what the sentence says about the subject; it begins with the verb. Subject Predicate Sentence 3.6: Bob runs. Subject Predicate Sentence 3.7: Bob and Abby run away from the zombies and don t look back. In a generalization, the subject of the sentence gives the quantifier and the first type of thing; the predicate gives the second type of thing. Subject Predicate Sentence 3.8: Only birds fly. Fly is a verb, not a plural noun. But we can easily turn it into a plural noun. Statement 3.8: Only birds fly. = Only birds are flying things. Flying things is a plural noun; we can count flying things. The method for writing a generalization in this form is: 1) find the predicate, 2) turn the predicate into a plural noun. Predicate Statement 3.9: All smartphones have an operating system (OS). All smartphones are things with an OS. Predicate Statement 3.10: Only planets with water support life. Only planets with water are places that support life. [ 37 ]

4 All A s are B s = Only B s are A s Once we ve identified the two types related in the statement, we can draw a simple diagram of the generalization. Each type is shown with a circle. Statement 3.8 Statement 3.9 SP TWOS These generalizations each show one type of thing entirely contained within another type of thing. The diagrams say what the statements say: all flying things () are birds, and all smartphones (SP) are things with an OS (TWOS). We can make these same statements using the word only instead of all. To convert a generalization from one form to the other, we need to switch the order of the two things in the sentence. Statement 3.8: Only birds are. All are birds. Statement 3.9: All SP are TWOS. Only TWOS are SP. Every all/only universal generalization works this way. All A s are B s. = Only B s are A s. A s B s [ 38 ]

5 All and only A s are B s It is also possible to combine an all generalization with an only generalization: All and only A s are B s. To diagram this, we combine the diagrams for All A s are B s and Only A s are B s. The A s circle and the B s circle overlap. All A s are B s. All B s are A s. All and only A s are B s. Only B s are A s. Only A s are B s. + = A s A s B s B s A s B s No A s are B s We can also re-write and diagram generalization that use the quantifier no. Predicate Statement 3.11: No cities have more than 15 million people. No cities are places with more than 15m people (PWMT15MP). No PWMT15MP are cities. Cities PWMT15MP Every no universal generalization works this way. No A s are B s. = No B s are A s. A s B s [ 39 ]

6 iii. COUNTER-EAMPLES Universal generalizations state a rule that allows not even one exception. An exception to a generalization is called a counter-example. Since a generalization relates two things, a counter-example has a two-part description. Generalization Counter-example () All A s are B s. = Only B s are A s. = A that is not a B. No A s are B s. = No B s are A s. = A that is also a B. Statement 3.9 is true because there are no counter-examples. There is no smartphone that not a thing with an OS. Statement 3.8 is false because there is a counter-example: flying fish. These are flying things that are not birds. For any counter-example, we can name it and describe it. Counter-example to Only birds are. = All are birds. Name Flying fish Description that is not a bird Flying fish show that some flying things are not birds. The circle is not contained within the circle it must extend outside of it to include the counter-examples. We can draw a corrected diagram with s to show the counter-examples. Generalization Corrected Diagram Only birds are. (Some are not birds.) All are birds. Flying fish [ 40 ]

7 What about a penguin? A penguin is 1) a bird, but 2) not a flying thing. This is not a counterexample; it fits into the original, uncorrected diagram. No correction necessary. (A penguin would be a counter-example to a different generalization: All birds are.) Statement 3.11 is also false. Again, there are counter-examples. Shanghai, Delhi, and Lagos are 1) cities, and 2) also places that have more than 15 million people. Again we can correct the diagram of the generalization. Counter-examples to No cities are PWMT15MP. = No PWMT15MP are cities. Name Shanghai Delhi Lagos Description City that is also a PWMT15MP Generalization No cities are PWMT15MP. Corrected Diagram (Some cities are also PWMT15P.) No PWMT15MP are cities. Cities PWMT15MP Cities PWMT15MP Shanghai Delhi Lagos [ 41 ]

8 All and only combines two generalizations, so two sorts of counter-examples apply to it. Predicate Statement 3.12: All and only bacteria cause disease in normally healthy humans. All and only bacteria are things that cause disease in normally healthy humans (TTCDINHH). Statement 3.12 is doubly false. The all generalization is false, and so is the only generalization. Bacteria in our gut and on our skin, as well as bacteria in soil, do not cause disease in normally healthy humans. And some viruses (e.g. influenza virus, which causes flu), genetic abnormalities (e.g. a mutated form of the gene for CR protein, which causes cystic fibrosis), and many of the causes of cancer (e.g. ultraviolet radiation, which causes skin cancer) cause disease in normally healthy humans, but are not bacteria. Counter-examples to All and only bacteria are TTCDINHH. Name Gut bacteria (GB) Skin bacteria (SkB) Soil bacteria (SB) Influenza virus (IV) CR gene mutation (CGM) Ultraviolet radiation (UVR) Description Bacteria that is not a TTCDINHH TTCDINHH that is not bacteria Generalization All and only bacteria are TTCDINHH. Corrected Diagram (Some bacteria are not TTCDINHH.) (Some TTCDINHH are not bacteria.) Bacteria GB IV SkB CGM TTCDINHH SB UVR Bacteria TTCDINHH [ 42 ]

9 Rejecting a Counter-example Suppose that Bob believes Statement 3.8, Only birds fly, and Abby proposes (suggests) the counter-example of an owl. Bob should not be convinced that some flying things are not birds. An owl is a flying thing. However, an owl is also a bird. (How did Abby not know that?) No correction necessary. Suppose Abby claims that a cloud is a counter-example. Again, Bob should not be convinced. Clouds are not birds. However, a cloud is really not a, either, even though it is in the sky. It is just floating, not flying. No correction necessary. Suppose Abby suggests a dragon, a giant, flying, fire-breathing lizard. A dragon is a counterexample because it is a that is not a bird. However, again Bob should not be convinced. The problem with Abby s suggestion, of course, is that there are no dragons. They are mythical creatures that do not really exist. No correction necessary. [ 43 ]

10 Revising a Generalization Bob should not be convinced that Statement 3.8 is false by owls, clouds, or dragons. However, as we ve seen, he should be convinced by the flying fish counter-example. The generalization diagram must be corrected to include flying fish. Corrected Diagram (Some are not birds.) Flying fish Bob may revise the generalization in to avoid the flying fish counter-example. He can do that in two ways: restrict (make smaller) the type or expand (make larger) the type. Restrict to Animals that Fly by Flapping their Wings (AFFW). Flying fish jump from the water and glide a short distance. They do not push themselves through the air by flapping their wings; they are not AFFW. Expand birds to animals. Flying fish are and also animals. Revised Generalization ( Restricted) Only birds are AFFW. All AFFW are birds. Revised Generalization ( Expanded) Only animals are. All are animals. AFFW Animals [ 44 ]

11 These revised generalizations avoid the flying fish counter-example. However, this does not mean that either new statement is true! There are other counter-examples that apply to each. Bats and flying insects both fly by flapping their wings, as did pterosaurs, which went extinct many millions of years ago. But these are not birds. Airplanes and remote controlled drones (RCD) both fly. But they are not animals. New Corrected Diagram (Some AFFW are not birds.) New Corrected Diagram (Some are not animals.) Bat Flying insect Pterosaur Airplane RCD Flying fish AFFW Flying fish Animals We could do something similar with Statement 3.11 to avoid the counter-examples we found to it, except in this case, it will do no good to expand either type. The only options are to restrict one or the other. For example, we could restrict Cities to North American cities (NAC). Or we could restrict PWMT15MP to places with more than 30 million people (PWMT30MP). [ 45 ]

12 Corrected Diagram (Some cities are also PWMT15P.) Cities PWMT15MP Shanghai Delhi Lagos Revised Generalization (Cities Restricted) No NAC are PWMT15MP. Revised Generalization (PWMT15MP Restricted) No cities are PWMT30MP. NAC PWMT30MP Cities PWMT15MP Cities PWMT15MP These revised generalizations avoid the counter-examples of Shanghai, Delhi, and Lagos. Once again, this does not mean that either new statement is true. There could be other counter-examples (although there are not any). We ve now seen what counter-examples are, as well as the different ways that we can respond to a proposed counter-example. How to respond to a proposed counter-example? Reject the Generalization The statement is false. Reject the Counter-example That thing That thing has does not not been correctly exist. described. Revise the Generalization Modify (restrict Modify (restrict or or expand) one expand) the other type. type. [ 46 ]