Deductive Reasoning Patterns PDF

MODULE 3 PT.1: COMMON PATTERNS OF DEDUCTIVE REASONING THINKIN – A65C DISCUSSION Module 2 Recap 1. Arguments and non-arguments OUTLINE 2. Reviewing of individual exercise results Module 3 pt. 1: Common Patterns of Deductive Reasoning...

MODULE 3 PT.1: COMMON PATTERNS OF DEDUCTIVE REASONING THINKIN – A65C DISCUSSION Module 2 Recap 1. Arguments and non-arguments OUTLINE 2. Reviewing of individual exercise results Module 3 pt. 1: Common Patterns of Deductive Reasoning 3. Five common patterns 4. Categorical Logic Reminder for our Asynchronous Session this Week RECAP OF MODULE 2 What are arguments? What are the common examples of non-arguments? MODULE III This module exhibits the common patterns that display PT. 1: the use of deductive and inductive reasoning. COMMON Achieving familiarity with these common patterns also PATTERNS allows familiarity with the variety of ways a necessary OF inferential relation (for deductive reasoning) and a DEDUCTIVE probabilistic inferential relation (for inductive reasoning) REASONING can possibly happen. COMMON This section is divided according to the common patterns in which deductive reasoning is PATTERNS expressed: OF A. Hypothetical syllogism B. Categorical syllogism DEDUCTIVE C. Argument by elimination REASONING D. Argument based on mathematics E. Argument from definition  A syllogism is a two-premise deductive argument. In a A. hypothetical syllogism, one of the premises is a conditional HYPOTHETICA statement. Examples: L SYLLOGISM If it rains, the streets get wet. It rained. So, the streets are wet. If I want to keep my financial aid, I’d better study hard. I do want to keep my financial aid. Therefore, I’d better study hard. Both arguments display the same logical form: If A, then B. A. Therefore B. This pattern is called modus ponens. Arguments with this pattern consist of (1) one conditional premise, (2) a second premise that asserts as true the antecedent (the “if” part) of the conditional, and (3) a conclusion that asserts as true the consequent (the “then” part) of the conditional. OTHER COMMON VARIETIES OF HYPOTHETICAL SYLLOGISMS: Chain Argument Modus Tollens (Denying The Consequent) Denying The Antecedent Affirming The Consequent Chain arguments consist of three conditional statements that link together in the following way: If we don’t stop for gas soon, then we’ll run out of gas. If we run out of gas, then we’ll be late for the wedding. Therefore, if we don’t stop for gas soon, we’ll be late for the wedding. Modus tollens arguments are also called “denying the consequent” because they consist of (1) one conditional premise, (2) a second premise that denies (i.e., asserts to be false) the consequent of the conditional, and (3) a conclusion that denies the antecedent of the conditional. Example: If we run out of gas, then we’ll be late for the wedding. We were not late for the wedding. Therefore, we did not run out of gas. TWO PATTERNS OF HYPOTHETICAL SYLLOGISMS THAT ARE LOGICALLY UNRELIABLE 1. Denying the antecedent Example: If Shakespeare wrote War and Peace, then he’s a great writer. Shakespeare didn’t write War and Peace. Therefore, Shakespeare is not a great writer. 2. Affirming the consequent Example: If we’re on Neptune, then we’re in the solar system. We are in the solar system. Therefore, we’re on Neptune. In the above examples of Denying the antecedent and Affirming the consequent, the premises are true, but the conclusion is false. Therefore, they are not reliable patterns of reasoning. B. CATEGORICAL SYLLOGISM For present purposes, a categorical syllogism may be defined as a three-line argument in which each statement begins with the word all, some, or no. Examples: All oaks are trees. All trees are plants. So, all oaks are plants. Some Democrats are elected officials. All elected officials are politicians. Therefore, some Democrats are politicians. Because categorical reasoning like this is such a familiar form of rigorous logical reasoning, such arguments should nearly always be treated as deductive. C. ARGUMENT BY ELIMINATION This argument seeks to logically rule out various possibilities until only a single possibility remains. Examples: Either Joe walked to the library, or he drove. But Joe didn’t drive to the library. Therefore, Joe walked to the library. Either Dutch committed the murder, or Jack committed the murder, or Celia committed the murder. If Dutch or Jack committed the murder, then the weapon was a rope. The weapon was not a rope. So, neither Dutch nor Jack committed the murder. Therefore, Celia committed the murder. Because the aim of such arguments is to logically exclude every possible outcome except one, such arguments are always deductive. D. ARGUMENT BASED ON MATHEMATICS In an argument based on mathematics, the conclusion is claimed to depend largely or entirely on some mathematical calculation or measurement (perhaps in conjunction with one or more nonmathematical premises). Examples: Light travels at a rate of 186,000 miles per second. The sun is more than 93 million miles away from the Earth. Therefore, it takes more than eight minutes for the sun’s light to reach the earth. My blind uncle told me that there were eight men, six women, and twelve kids at the party. By simple addition, therefore, it follows that there were twenty-six people at the party. Arguments based on mathematics can also be inductive, just like when the argument makes use of statistics or generalizes from a given sample. E. ARGUMENT FROM DEFINITION In an Argument from Definition, the conclusion is presented as being “true by definition,” that is, as following simply from the meaning of some keyword or phrase used in the argument. Example: Bachelors are unmarried men. Jose is an unmarried man. So, Jose is a bachelor. CATEGORICAL Categorical logic is logic based on the relations of inclusion and exclusion among classes (or “categories”) as stated in categorical LOGIC claims. Categorical Claims and Standard-Form Categorical Claims A categorical claim says something about classes (or “categories”) of things. A standard-form categorical claim is a claim that results from putting names or descriptions of classes into the blanks of the following structures: A: All _________ are _________. (All Lasallians are Career-oriented persons.) E: No _________ are _________. (No Muslims are Christians.) I: Some _______ are _________. (Some Christians are Filipinos.) O: Some ______ are not _____. (Some critical thinkers are not philosophers.)  The phrases that go in the blanks are terms; the one that goes into the first blank is the subject term of the claim, and the one that goes into the second blank is the predicate term. Only nouns and noun phrases will work as terms. An adjective alone, such as “red,” won’t do. “All fire engines are red” does not produce a standard-form categorical claim, because “red” is not a noun or noun phrase. The proper translation into a standard-form categorical claim can be: “All fire engines are red vehicles.”  Translation into Standard Form is significant to enable the use of the mechanism of Categorical Logic Two claims are equivalent claims if, and only if, they would be true in all and exactly the same circumstances—that is, under no circumstances could one of them be true and the other false. The word “only,” used by itself, introduces the predicate term of an A-claim. The phrase “the only” introduces the subject term of an A-claim. “Only sophomores are eligible candidates.” = All eligible candidates are sophomores. “The only people admitted are people over twenty-one.” = All people admitted are people over twenty-one. Claims about single individuals should be treated as A-claims or E- claims. TESTING SYLLOGISMS FOR VALIDITY USING VENN DIAGRAMS To test a syllogism for validity, we Venn diagram the premises and inspect the result. If the diagram of the premises excludes the possibility of the conclusion being false, then the syllogism is valid. In other words, if the Venn diagram of the premises includes a representation of the conclusion, then the syllogism is valid. Otherwise, it is invalid. Since any syllogism has exactly three terms, the Venn diagram of a syllogism will have three circles. Although it doesn't really matter which circle represents which terms, standardly the left circle represents the minor term, the right circle represents the major term, and the bottom circle the middle term. Diagram this argument: All fish are swimmers. All bass are fish. All bass are swimmers. “All fish are swimmers" Now we add the diagram for "all bass is diagrammed as are fish." follows: Now we ask: Does the Venn diagram contain a diagram of the conclusion? We see that it does since the area in B but not in S is completely shaded. Thus, the argument is valid. CONSIDER OUR DOG SYLLOGISM: SOME DOGS ARE HAPPY SOME DOGS ARE NOT BROWN. NO BROWN THINGS ARE HAPPY. For this syllogism, we have two particular In our dog premises. syllogism, we We use "x"s to mark areas where something rewrite it as: exists in the class represented by the subject and predicate circles. When we diagram syllogisms, we are using Some D are H. three cirles, and so the area where we would Some D are not place our "x" is segmented into two regions. B. To handle this potental ambiguity, we place the No B are H. "x" on the border of the two regions, to indicate that the "x" could be in either of the two regions.  Here our "x" is in the area of D outside of B. But it Note that the "x" is in the area that is both is on the line of the border of H. Do we have a D and H. But it is on the border of the area diagram of the conclusion, "No B are H"? Clearly that is B. It might be in B or it might not. not. So the argument is invalid. The first premise doesn't make that clear. So we express the lack of information by In cases where one of the premises is universal placing the "x" on the line. and the other premise is particular, it is a rule that we diagram the universal premise first, and only after do we diagram the particular premise. If we diagramed the particular premise first before diagramming the universal premise, the diagram would give us an invalid reading of what is actually a valid syllogism. THE RULES METHOD OF These rules are based on two ideas, the first of which TESTING FOR has been mentioned already: affirmative and negative categorical claims. (Remember, the A- and I- VALIDITY claims are affirmative; the E- and O-claims are negative.) The other idea is that of distribution. Terms that occur in categorical claims are either distributed or undistributed: Either the claim says something about every member of the class the term names, or it does not. The A-claim distributes its subject term, the O-claim distributes its predicate term, the E-claim distributes both, and the I-claim distributes neither of its terms. We can now state the three rules of the syllogism. A syllogism is valid if, and only if, all of these conditions are met: 1. The number of negative claims in the premises must be the same as the number of negative claims in the conclusion. (Because the conclusion is always one claim, this implies that no valid syllogism has two negative premises.) 2. At least one premise must distribute the middle term. 3. Any term that is distributed in the conclusion of the syllogism must be distributed in its premises. PROPOSITIONAL LOGIC (TO BE CONTINUED) REFERENCES Bassham, Gregory, Irwin, William, et. al. 2023. Critical Thinking: A Student’s Introduction, 7th ed. McGraw Hill LLC, New York. Moore, Brooke, Parker, Richard. 2021. Critical Thinking, 13th ed. McGraw Hill, New York. ASYNCHRONOUS SESSION THIS WEEK Group exercise 2: Task per individual member for the group discussion: Think of songs you listen to (with English or Filipino lyrics). Pick out 5 lines from however many songs you like. The lines you selected could serve as either a premise or a conclusion. Out of those lines, formulate 5 valid deductive arguments: 1 hypothetical syllogism, 1 categorical syllogism, 1 argument by elimination, 1 argument based on mathematics, and 1 argument from definition. Discuss your work with the rest of the group so that the group can deliberate on which arguments to be presented in the work. Task of the leader: Compile the peer review forms submitted by the members, and keep the record in the meantime. Written output submission (as a group): Write down the date and time of your group meeting, as well as the platform used (if online, otherwise indicate “in- person”) List the names of all participating members Synthesise the discussion (a paragraph will do) and decide as a group to select 5 arguments from the members (could be from just one member, or could be a mix). Each properly categorised argument is equivalent to 1 point, for a total of 5 points for the group activity.