Elements of Science (cont.); Conditional Statements Phil 12: Logic and Decision Making Fall 2010 UC San Diego 9/29/2010 1
Why cover statements and arguments Decision making (whether in science or elsewhere) involves reasoning based on evidence Question: when does some piece of information count as (good) evidence for or against a conclusion? - When does some piece of information (evidence) serve to support (confirm) or count against (disconfirm) a hypothesis or theory? To answer these questions we need to discuss arguments, and the statements of which they re composed 2
Clicker question 1 What type of statement is the following sentence: It is sunny, but it is not sunny. A. tautology B. contingent statement C. contradiction 3
Review - 1 Statements are sentences that have a truth value are either true or false - Contingent statements: could be true or could be false - Contradictions: always false - Tautologies: always true 4
Trying to define ordinary terms Necessary and sufficient conditions can generally only be provided for technical terms (e.g., in mathematics or in legal contracts) Most ordinary terms defy such definition: For any attempted definition, a counterexample can be found A counterexample is either: 1. An example that fits the definition but we would not count as an instance of a term; or 2. An example that does not fit the definition but we would count as an instance of a term 5
Clicker question 2 Which of the following is an adequate definition of bird? A. an animal that can fly B. an animal with feathers C. an animal with a beak D. an animal with feathers and a beak that can fly 6
Define bird Flying: is every animal that flies a bird? - Not all birds fly, and most insects do Feathers: is every animal with feathers a bird? - Caudipteryx, Microraptor seem to have had feathers but not be birds 7
Doing without definitions Start with typical cases Robin Blue jay Sparrow Extend to unusual cases 8
Arguments and justification If someone asserts something which you do not believe,you frequently ask them to justify what they say An argument is a set of statements, some of which are offered as support for other statements in the set - An argument provides reasons to believe something - An argument need not involve another person: you can construct an argument to demonstrate that something is true without showing it to anyone 9
Elements of arguments Premises: statements offered in support - Often indicated by words such as because, since, given that, on account of, etc. Conclusion: the statement that is supported - Often indicated by words such as thus, therefore, this establishes that, etc. Example: Premises Conclusion The car has a large dent in it. Dents don t just appear in cars. You had an accident. 10
Good and bad arguments We don t just care whether the conclusion is true We also want to know whether the reasons stated in the premises give us good logical grounds for thinking that the conclusion is true The goal is not actual persuasion (people can be persuaded for bad reasons), but establishing the truth Two factors relevant to the evaluation of arguments: 1. Are the premises true? 2. Is the connection between the premises and the conclusion such that: if the premises were true, would they establish that the conclusion is true? 11
Valid arguments An argument is valid iff it meets the following condition: if the premises were true, the conclusion must also be true - A valid argument cannot have true premises and a false conclusion This relationship is modal: it tells us what would be the case were certain conditions to be met - These conditions might not be satisfied in reality - The definition tells you nothing about what happens when they are not satisfied A way to test for validity: if you can imagine a situation in which the premises are true and the conclusion false, then the argument is not valid 12
Sound arguments An argument is sound iff: 1. the premises are true 2. the argument is valid This definition of a sound argument does not say anything about the truth of the conclusion - But the conclusion of a sound argument must be true A sound argument meets both of the desiderata of a good argument: - True premises - Valid 13
Clicker question 3 A valid argument cannot have a false conclusion. A. True B. False 14
Clicker question 4 A sound argument cannot have a false conclusion. A. True B. False 15
Clicker question 5 An argument with a true conclusion is sound. A. True B. False 16
Conditional statements Conditional statements consist of two component statements linked by the logical connective IF, THEN - If it rains today, then there will be no game If and then are not marking premises and conclusion of an argument antecedent consequent - If it rains today, then there will be no game is not an argument! It simply asserts a conditional relationship between two statements - Compare: On account of the fact that it is raining today, there will be no picnic. 17
Truth values of conditional statements IF, THEN is a truth functional connective: the truth of a compound (whole) statement depends only on the truth values of the component statements If A, then B is false when the antecedent is true and the consequent is false. Otherwise, it is true. You trespass You are arrested If you trespass, then you will be arrested TRUE FALSE FALSE TRUE TRUE TRUE FALSE FALSE TRUE FALSE TRUE TRUE 18
Clicker question 6 The statement The alarm will not sound if the window is opened is false when: A. The alarm sounds and the window is opened B. The alarm sounds and the window is not opened C. The alarm does not sound and the window is opened D. The alarm does not sound and the window is not opened 19
Reversing antecedent and consequent IF A, THEN B is not equivalent to IF B, THEN A - - IF A, THEN B is false when A is true and B is false IF B, THEN A is false when B is true and A is false IF A, THEN B is equivalent to IF NOT B, THEN NOT A. If you trespass, then you will be arrested is equivalent to If you are NOT arrested, then you did NOT trespass 20