SSH Validity, Part 2 Patterns of Argument - Week 7 (Lecture notes)

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October 21st, 2023 Validity part 2 - Patterns of Argument Week 7 A2. Patterns of Argument - Pattern of argument displays the underlying form or logical structure of an argument. Argument Pattern 1 (Always valid) 1. x is an A 2. All As are Bs 3. x is a B OR Pluto is a planet All planets are more than 2 feet in diameter Pluto is more than two feet in diameter - Letter x standing for names like “Boris” or “Pluto” and letters A and B standing for the predicate (descriptive phrases) like “is a student”, “is a planet” and so on. This represents predicate logic. Argument Pattern 2 (Always valid) 1. Either P or Q 2. -p (p isn’t true) 3. Q. OR 1. Either my son is on the phone or my wife is on the phone 2. It is not the case that my son is on the phone 3. My wife is on the phone - We use letters standing for whole sentences, but nothing standing for the parts, such as “my wife” or “on the phone”. The latter kind of argument illustrates what is known as sentential logic, or propositional logic. - We use the capital letters P, Q, R to stand for complete sentences,. Compound sentences are formed by combining two or more simpler sentences, and we abbreviate compound sentences by using sentences as “P and Q” or “P or Q” or “If P then Q” - When displaying the pattern of arguments in predicate logic, we use lowercase letters such as x anf y to abbreviate names of individuals and uppercase letters such as A and B to abbreviate words and phrases expressing properties or characteristics of individuals.

EXAMPLE: “If x is an A, then x is a B” This can abbreviate: “If Jones is an astronaut, then Jones is brave.” - When we are discussing arguments containing premises such as “All astronauts are brave” we might use the abbreviation “All As are Bs.” Finally, we can also make use of the ~ to abbreviate a sentence such as ‘Jones is not an astronaut’. We might write this as “x is ~A.” - The key thing to look for is whether the whole conclusion occurs inside one of the premises. If it does, then the argument is almost definitely best seen as an argument from sentential logic. - If the whole conclusion does not appear in a premise, but the parts of the conclusion appear in different premises, then the argument is most likely best taken to be one from predicate logic. A3. Some Patterns of Argument in Sentential Logic Conjunction: Sentences formed by combining two sentences with the world “and” Disjunction: “Or” sentences Conditionals: “if-then” sentences. Argument by Elimination: - The first premise says that one or another of two propositions is true. Its second premise “eliminates” or says one of the possibilities are not true, then the arguments conclusion is that the remaining possibility is true. 1. P or Q 2. -P 3. Q

Conjunction - There are two simple patterns of argument that make use of conjunctions. If each of two statements istrue, then so is conjunction. 1. P. 2. Q. 3. P and Q. Simplification: - And if a conjunction is true, then each conjunct is true 1. P and Q 2. P - In a conditional, the first part- the “if” clause, or antecedent is claimed to be sufficient for the truth of the “then” clause, or consequent. Affirming the Antecedent (ALWAYS valid) 1. If P then Q 2. P. 3. Q. Example 1. If the president is in the White House, then the president is in Washington, D.C. 2. The president is in the White House. 3. The president is in Washington, D.C. Denying the Consequent (always valid) 1. If P then Q 2. -Q 3. -P Example 1. If the president is in the White House, then the president is in Washington, D.C. 2. The president is not in Washington. 3. The president is not in the White House.

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Hypothetical Syllogism - Arguments string together a few conditionals 1. If P then Q 2. If Q then R 3. If P then R Example 1. If the president is in the White House, then the president is in Washington, D.C. 2. If the president is in Washington, D.C., then the president is not on vacation. 3. If the president is in the White House, then the president in not on vacation. Denying the Antecedent (not valid) - Consider next a case in which one premise is a conditional and the other premise denies its antecedent 1. If P then Q 2. -P 3. -Q Example 1. If the president is in the White House, then the president is in Washington, D.C. 2. The president is not in the White House. 3. The president is not in Washington, D.C.

A5. A Modification of the Definition of Validity Two ways in which the premises of an argument can be said to guarantee the truth of the arguments conclusion - One concerns the form of arguments alone - Second takes the meanings of key terms of the argument into account D1b: An argument is valid if and only the argument follows a pattern such that it is impossible for any argument following that pattern to have true premises and a false conclusion. Video Lecture Notes: - If P then Q - P is the antecedent - Q is the consequent ( part that comes after the then) Example Modus Tollens (Denying the consequent) so denying Q 1. If P, then Q. 2. Not Q. 3. Therefore, not P. If Bob is in Toronto, then he is in Ontario Bob is not in Ontario Therefore Bob is not in Toronto. - In this case, were are denying that “he is in ontario” since that is the consequent - Then since we deny that part of the conditional we also deny the first part which is the antecedent. Modus Ponens (affirming the antecedent) - Affirming whatever comes BEFORE the Q 1. If P, then Q. 2. P. 3. Therefore Q. Example 1. If Bob is in Toronto, then he is in Ontario. 2. Bob is in Toronto 3. Therefore, Bob is in Ontario We are affirming P so affirming Bob is in Toronto If thats the case the second part of the conditional is true. Invalid: Affirming the consequent 1. If P, then Q. 2. Q. 3. Therefore, P.

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Here we are affirming Q which is the SECOND part of the conditional, if we affirm that first then conclude P must be true since we affirm Q then it is invalid. The first part must be affirmed (Modus Ponens) or you must deny the second part (consequent) for it to be true. You cannot affirm the consequent. Denying the antecedent If P, then Q. Not P. Therefore not Q. Tips for finding validity and invalidity - Find examples to establish invalidity (“counterexamples”) (come up with a case where all the premises are true and the conclusion is false.) - Look for arguments that follow the same pattern, where validity or invalidity might be easier to identify. - Check the pattern TWO KINDS OF VALIDITY 1. Donatello is a turtle 2. Therefore, Dontaello is an animal Semantic: Valid because it follows a valid argument form Modal: Valid because of the meaning of the terms Modally valid because Donatello is a turtle. Can turn modally valid arguments into semantic valid arguments. 1. Donatello is a turtle 2. If Donatello is a turtle, Donatello is an animal 3. Therefore, Dontatello is an animal. What has been covered - 6 valid argument forms - 3 tips for determining (in)validity. - Two kinds of validity -