1. Copyright © Peter Cappello Logical Inferences Goals for propositional logic 1.Introduce notion of a valid argument & rules of inference. 2.Use inference rules to build correct arguments.

2. Copyright © Peter Cappello What is a rule of inference? A rule of inference allows us to specify which conclusions may be inferred from assertions known, assumed, or previously established. A tautology is a propositional function that is true for all values of the propositional variables (e.g., p  ~p).

3. Copyright © Peter Cappello Modus ponens A rule of inference is a tautological implication. Modus ponens: ( p  (p  q) )  q

4. Copyright © Peter Cappello Modus ponens: An example Suppose the following 2 statements are true: If it is 11am in Miami then it is 8am in Santa Barbara. It is 11am in Miami. By modus ponens, we infer that it is 8am in Santa Barbara.

5. Copyright © Peter Cappello Other rules of inference Other tautological implications include: (Is there a finite number of rules of inference?) p  (p  q) (p  q)  p [~q  (p  q)]  ~p [(p  q)  ~p]  q [(p  q)  (q  r)]  (p  r) hypothetical syllogism [(p  q)  (r  s)  (p  r) ]  (q  s) [(p  q)  (r  s)  (~q  ~s) ]  (~p  ~r) [ (p  q)  (~p  r) ]  (q  r ) resolution

6. Copyright © Peter Cappello Common fallacies 3 fallacies are common: Affirming the converse: [(p  q)  q]  p If Socrates is a man then Socrates is mortal. Socrates is mortal. Therefore, Socrates is a man.

7. Copyright © Peter Cappello Common fallacies... Assuming the antecedent: [(p  q)  ~p]  ~q If Socrates is a man then Socrates is mortal. Socrates is not a man. Therefore, Socrates is not mortal.

8. Copyright © Peter Cappello Common fallacies... Non sequitur: p  q Socrates is a man. Therefore, Socrates is mortal. The following is valid: If Socrates is a man then Socrates is mortal. Socrates is a man. Therefore, Socrates is mortal. The argument’s form is what matters.

9. Copyright © Peter Cappello Examples of arguments Given an argument whose form isn’t obvious: Decompose the argument into premise assertions Connect the premises according to the argument Check to see that the inference is valid. Example argument: If a baby is hungry, it cries. If a baby is not mad, it doesn’t cry. If a baby is mad, it has a red face. Therefore, if a baby is hungry, it has a red face.

10. Copyright © Peter Cappello ( (h  c)  (~m  ~c)  (m  r) )  (h  r) r m c h

11. Copyright © Peter Cappello Examples of arguments... Argument: McCain will be elected if and only if California votes for him. If California keeps its air base, McCain will be elected. Therefore, McCain will be elected. Assertions: m: McCain will be elected c: California votes for McCain b: California keeps its air base Argument: [(m  c)  (b  m)]  m (valid?)