L05A: Lecture on Types of Arguments

MODR 1760 C 2024-25 L05A: Lecture on Types of arguments In Lecture 2 we introduced the ARG conditions as a way to help us assess an argument. The idea is that a good argument has all of these three features, Acceptability, Relevance and Good Grounds. This is discussed in Govier’s text, chapter 4 section “The ARG conditions” p. 87-8. But how these features work, and how we assess an argument, can vary according to the type of argument we are looking at. So we review some types of argument. Remember, A = Acceptability R = Relevance G = Good Grounds Types of Argument This discussion is based on Govier, chapter 4 section “More on the R and G conditions,”p. 88-93. A. Deductive Validity A deductively valid argument is the strongest form of argument. Defn: An argument is deductively valid the truth of the premises guarantees the truth of the conclusion. (If the premises are true, then the conclusion must be true.) Example: (with true premises) 1. All mammals are warm-blooded. 2. Whales are mammals. Therefore, 3. All whales are warm-blooded. In a deductively valid argument, it is impossible for the premises to be true and the conclusion false. In order to test this impossibility, we have to be able to distinguish the difference between whether some statement is in fact true, from the hypothetical scenario, “what if the statement was true.” I warn you now that this is a point that my teaching experience tells me many students will find difficult. But it is important, not just to understanding deductive validity, but for understanding many aspects of the course. In the above example both of the premises, All mammals are warm-blooded, and Whales are mammals are both in fact true. Given that they are true, the conclusion, All whales are warmblooded, must be true. We can see what this means better, if we look at a very similar argument in which not both of the premises are true. 1 Example: (with a false premise) 1. All mammals are warm-blooded. 4. Frogs are mammals. Therefore, 5. Frogs are warm-blooded. In this case, the second premise is not true. Frogs are not mammals in fact. But we can still see that if they were mammals, then the conclusion would be true. It is this feature, (if all premises are true, then the conclusion would be true) that constitutes deductive validity. Deductive validity does not tell us anything about the truth of the premises on its own. Deductive validity when it holds, guarantees the relevance of the premises to the conclusion, and it establishes the strongest possible connection between the premises and the conclusion, so strong that it automatically provides good grounds for the conclusion if the premises are true. In this unit we are beginning Unit II, we turn to propositional logic to examine examples of deductively valid arguments. Those that we study here in Unit II are arguments that are valid because of the meanings of the logical words in them. The study of propositional logic will allow us to see that relevance and acceptability can be completely separate things. Many deductively valid arguments are ones in which the premises are linked together in their support for the conclusion. B. Conductive Argument Defn: an argument is conductive when there is more than one premise that independently supports the conclusion. This type of support was initially discussed in Lecture 3. Govier’s example: Conductive arguments occur in legal reasoning. Consider an example of the prosecution of someone for a crime of violence. In order to establish the guilt of someone, several distinct lines of argument will want to be taken in to account. Each one alone would raise suspicions, as each gives some reason to suppose that the accused person committed the crime. 1. The accused has no alibi for the time of the crime. 2. The accused has a motive for committing the crime. 3. The accused has been identified as being near the scene of the crime by a reliable witness. 4. The accused has lied in describing his relationship to the victim of the crime. Therefore, 5. The accused committed the crime. Govier has a diagram of this as an example of a conductive support on p. 90. Each premise is not logically related to the other premises. That is, having an alibi and having a motive are not logically related to one another. One could be true while the other is false and vice versa. (There is the hypothetical being used again.) This is why they are independent of each other. 2 C. Inductive Argument An inductive argument, commonly used in science, may be a strong form of argument. But in an inductive argument, even a very good one, the premises do not guarantee the conclusion as in a deductive argument, but instead they make the conclusion more likely. In some contexts, the strength of an inductive argument can be measured using statistical reasoning. But whether we use statistical methods or we do not, the central idea is that we have some direct experience stated in one (or more) of our premises, and we use that data to draw a conclusion about something. We are engaged in some form of generalization. Usually, the premises of the argument are working together to show it is likely that the conclusion is true. Defn: An inductive argument is one in which the truth of the premises makes the truth of the conclusion likely. That is, the premises’ truth supports the conclusion being true. So here is an example. Inductive Example: Samir has always taken the GO bus when travelling to York University from home. Samir travelled from home to York today. Samir took the GO bus to York today. Is this argument valid? It is not valid because there are circumstances that would make the premises true and the conclusion false. Consider, perhaps Samir is carpooling with some people and he came by car with them. This is a counterexample to the argument that concludes he came by GO by us. A counterexample shows that the argument is not deductively valid. But nevertheless knowing that Samir as travelled to York on the GO bus in the past, gives us some basis to think he came that way today. Govier’s example (e) p. 91: 1. These 80 students who graduated from school X all have good marks in mathematics. 2. The best and most plausible explanation of this is that mathematics is well taught at school X. Therefore, probably 3. Mathematics is well-taught at school X. She gives a diagram of this argument on p. 91 as well. This example involves an explanatory hypothesis, 2, which is said to the “best explanation” of the data in premise 1. This is joined together with the arguer’s observation to give the conclusion. Like deductive arguments we see the premises of inductive arguments working together to the conclusion. We will return to talk more about inductive arguments in the winter term of the course. 3 D. Analogical Argument Defn: An analogical argument is when the conclusion is based on an analogy, giving premises that state that some type of thing, A behaves a certain way, and then extending that statement to things that are similar to A in some respects. Govier’s example is based on a medical or biological example (f) p. 92: 1. When given a particular shot or injection, 50 out of 200 rats developed breast cancer in our experiment. So, it seems likely that, 2. The shot had the effect of causing breast cancer in about 25% of the rats. So it seems likely that, 3. The shot may cause breast cancer in humans. This is based on the similarity of the biology of rats and humans, which biological and medical science have studied. Govier calls this a linear pattern of argument in her diagram on p. 92. Do exercises from p. 93-94 of Govier to practice using these distinctions. Most of these questions are posted in the file L05A Questions. 4 Lecture 5B L05B: Introduction to Propositional (often called Sentence) Logic Our study of sentence logic will give us a deeper understanding of deductive validity, and as well it will give us many examples of deductively valid arguments. A. Logical Words: “and,” “or,” “not” and similar words This Photo by Unknown Author is licensed under CC BY-SA Consider the following argument from Govier: P1. If global warming continues, then the polar ice caps will melt. P2. Global warming will continue. Therefore, Conc. The polar ice caps will melt. This argument is deductively valid. If the premises are true, the conclusion must be true. The first premise (likely originally written in 1988) does not assert either that global warming will continue, or that the polar ice caps will melt. It asserts a relationship between them. To make a conditional statement like this, is to say that if the first statement – the antecedent—is true, then the second statement is true. 5 Consider the following example: P1. If the legal drinking age in Ontario is 19, then a person who is 19 years old or more can legally enter a bar in Toronto. P2. The legal drinking age in Ontario is 19 years old. Therefore, (Conc). A person who is 19 years old or more can legally enter a bar in Ontario. This argument has the same logical form as the first argument did. It is about a different subject completely, but the structure is the same. We notice the logical form when we replace the same statement with a letter and different statements with a different letter. Modus Ponens then looks like this when it is expressed as a logical form. P1. If P, then Q. P2. P Therefore, Conc: Q. This deductively valid pattern of argument is called Modus Ponens. It was known to medieval logicians. B. The Connectives In this section of the course, we will use symbols to represent the following four logical ideas: English Words Replaced: English idea English replaced Name of common symbol Tilde Govier Textbook Symbol - Name of Symbol “Not “ Most common version of this symbol ~ Negation Conjunction “And” & Ampersand • Dot Disjunction “Or”  Wedge  Wedge Conditional “If … then” -> Arrow  Horseshoe Dash These connectives are defined in terms as truth-functions. This means: a. The meaning of the connective sentence is determined only by the truth-value of its component sentences. 6 b. They are given a standard meaning here that is included in (but is not exactly the same as) their English meaning. Translations show logical words prominently. Suppose we have the key, A: Alberta has mountains. B: British Columbia has mountains. Negation: ~A ( or -A) read “not-A” ~A means, "It is not true that Alberta has mountains." or “Alberta does not have mountains.” The whole sentence is a negation, its main connective is -, and it applies to A. A True False The negative of a sentence reverses its truth-value. ~A False True Conjunctions: A&B (A●B) read “A and B” “Alberta has mountains and BC has mountains.” The whole sentence is a conjunction, its main connective is &. The component sentences are A, B, they are called conjuncts. A True True False False B True False True False A&B T F F F Conjunction is true when both conjuncts are true, false otherwise. Disjunction: AB read “A or B” “Either Alberta has mountains or BC has mountains.” The whole sentence is a disjunction, its main connective is . Each component sentence, A, B is called a disjunct. A T T F F B T F T F AB T T T F 7 A disjunction is false n the one case in which both disjuncts are false. In the case where A and B are both true, the disjunction is true. We call our disjunction definition “inclusive” because it makes the disjunction true when both disjuncts are true. Conditionals: A-> B ( or A B) read “if A then B” The whole sentence is a conditional sentence, main connective ->. The first part (before the arrow) is called the antecedent of the conditional. The second part (after the arrow) is called the consequent of the conditional. A T T F F B T F T F A -> B T F T T A-> B is false in only the case in which A is true and B is false. C. Brackets If a complex sentence has more than one connective, we use brackets to make sure it is not ambiguous. A&BC A & (BC) or (A&B)C. A&(BC) has & as the main connective and one component is a disjunction. If we have A&B, it is clear the whole sentence is a conjunction: A, B are parts. *Here is an important rule about negations: A negation sign always applies to the shortest full sentence that follows it. a. So when a negation symbol is in front of a single letter, it applies just to the single letter. In ~A & B the negation applies just to A. The conjunction applies to ~A and to B. The letter A, by itself, is the “shortest full sentence following the ~.” b. When a negation is in front of a bracket, it applies to exactly what is in those brackets. And that is the “shortest full sentence following the ~.” So in ~(A&B) the negation applies to the whole sentence (A&B). 8 Suppose we have A & ~(A -> B) The whole thing is a conjunction. A is the first conjunct (or the first part), and ~(A -> B) is the second conjunct. D. Translations using more than one connective Suppose we have the key, A: Alberta has mountains. B: British Columbia has mountains. Double Negatives Suppose we have: ~ ~ A, It means,” it is not true that Alberta does not have mountains.” ~ ~ A is logically equivalent to A. That is, it is not true that Alberta doesn’t have mountains is the same as Alberta has mountains. The two negatives cancel each other out. We do not need any brackets to tell us the main connective. It is the first ~. Look at this: ~(~(A)) The smallest full sentence after the first ~ sign is (~(A)), but that is the same as ~(A). So we don’t need the first set. And ~(A) is the same as ~A, because ~(A) is obviously the same as ~A. So ~(~(A)) is equal to ~~A. Disjunction AB Which says “Either Alberta or BC have mountains.” Or “Alberta or BC have mountains” Negation of a Disjunction ~(AB) Which denies the whole disjunction, so it means “It is not true that Alberta or BC have mountains.” ~(AB) is logically equivalent to saying ~A & ~B. Neither Samantha nor Simone is coming to the party. A: Samantha is coming B: Simone is coming. ~A&~B or I could say, ~ (AB). 9 Conditional (if, then) If Alberta has mountains, then BC has mountains. A -> B If it is true that Alberta has mts, then BC has mountains. Which is different from saying, B -> A If BC has mountains, then Alberta does. Maybe this is brought out better by: C: Jasmeet owns a car D: Jasmeet has a driver’s license. C -> D may be true, but it is not the same as D ->C. We could have complex sentences as part of a conditional as well so (A&B) -> A Is a sentence, saying “If Alberta and BC have mountains, then Alberta has mountains. or (A & ~C) -> ~D If Alberta has mountains and Jasmeet does not own a car, then Jasmeet does not have a driver’s license. Practice some translations exercises on your own based on this lecture L05B. The file is called L05B Questions. This lecture is based on Govier’s text chapter 8, p. 216-22. 10