PHIL 120: Jan 8

1. PHIL 120: Jan 8 • Basic notions of logic • More implications of the recursive definition of SL • Kinds of sentences • Main connectives and sentential components • Symbolizing sentences with more than one connective

2. Argument • An argument is a set of 2 or more sentences, one of which is the conclusion and the others premises. • Standardized form Premise Another premise (if there is one) And another (if there is one) … ---------------------------------------------------- Conclusion

3. Arguments • Premise indicator terms: ‘because’, ‘since’, ‘for the reason that’… • Conclusion indicator terms: ‘therefore’, ‘thus’, ‘it follows that’, ‘so’ … • One compound sentence can contain an argument: • ‘Sarah will do well in logic because she studies and anyone who studies does well in logic’

4. Arguments ‘Sarah will do well in logicbecause [she] Sarah studies andanyone who studies does well in logic’ Sarah studies. Anyone who studies does well in logic. --------------------------------------------------- Sarah will do well in logic.

5. Arguments • Famous arguments: • Descartes’ “cogito” “I think, therefore I am” “Cogito, ergo sum” • John Locke (and Samuel Johnson) Men (humans) are free by nature. As they are born free, no other man has the right to claim complete and total power over another man. --------------------------------------------------------------- Governments can only legitimately rule just as long as men, who are free by nature, consent to it.

6. Arguments • Arguments and philosophy • 2008 UW Philosophy Club tee shirt: • On the front: • Sketches of Plato, Socrates, Aristotle, Newton, Descartes, etc. • “Great minds don’t just think” • On the back: • “They argue”

7. Basic notions of logic: validity • An argument is deductively valid if and only if it is not possible for the premises to be true and the conclusion false. • An argument is deductively invalid if and only if it is possible for the premises to be true and the conclusion false. • Every argument is either deductively valid or deductively invalid. • Only arguments are valid or invalid.

8. All men are mortal. Socrates is a man. ------------------------- Socrates is mortal. If it is true that all men are mortal and that Socrates is a man, then it must be true that Socrates is a mortal. All men are green. Socrates is a man. ------------------------- Socrates is green. If it true that all men are green… Validity: A matter of form rather than content

9. Venn Diagram of the two categories and of their potential overlap

10. The first premiseThere are no men who aren’t mortal, hence the part of ‘men’ that doesn’t overlap with ‘mortals’ is empty

11. The second premise using X for SocratesAs he is a man, he must be in that portion of the category “men” that is not empty – and that overlaps with “mortal”. So, once we diagram the two premises, the conclusion is already diagrammed.

12. Basic notions of logic that apply to sentences • Logical truth: a sentence is logically true if and only if it is not possible for the sentence to be false. • 2 + 2 = 4 • A rose is a rose • If it rains, then it rains • Logical falsity: a sentence is logically false if and only if it is not possible for the sentence to be true. • If it rains, then it doesn’t rain. • A rose is not a rose.

13. Basic notions of logic that apply to sentences • Logical indeterminacy: a sentence is logically indeterminate if and only if it is neither logically true nor logically false. • Most sentences are logically indeterminate though all sentences are either true or false. • The truth or falsity of logically indeterminate sentences is determined by the way the world is. • The earth is a sphere • Copper conducts electricity • All men are mortal

14. Basic notions of logic that apply to sentences • Logical equivalence: the members of a pair of sentences are logically equivalent iff it is not possible for one of the members of the pair to be true while the other is false. • “Neither Mary nor John went to the store” • “Both Mary didn’t go to the store and John didn’t go to the store” • It follows that: • The sentences of any pair of logical truths (logically true sentences) are logically equivalent • The sentences of any pair of logical falsehoods (logically false sentences)

15. Basic notions of logic that apply to sentences • Strange sentences: “This sentence is false.” (The liar’s paradox) Strange pairs of sentences: On the front of a tee shirt: The sentence on the back of this shirt is false. On its back: The sentence on the front of this shirt is true.

16. Basic notions of logic that apply to sets of sentences • Logical consistency: A set of sentences is logically consistent iff it is possible for all the members of that set to be true. • A set of sentences is logically inconsistent iff it is not possible for all the members of that set to be true. • Put another way: logically consistent sets of sentences do not contain any contradictions; logically inconsistent sets of sentences do.

17. Basic notions of logic that apply to sets of sentences • Questions: • If a set of sentences includes only logically indeterminate sentences could it be logically consistent? • If a set of sentences includes one logically false sentence could it be logically consistent? • Could a set of sentences each one of which is logically true be logically inconsistent? • Could a set of sentences, none of which includes the ~, be logically inconsistent?

18. Implications of the definition of validity An argument is deductively valid iff it is not possible for the premises to be true and the conclusion false. • From which it follows that valid arguments can have: • True premises and a true conclusion • False premises and a false conclusion • False premises and a true conclusion • Only arguments with true premises and a false conclusion cannot be valid.

19. Implications of the definition of validity A valid argument with a false premise and false conclusion: All men are green. Socrates is a man. ------------------------- Socrates is green. A valid argument with false premises and a true conclusion: If the sun didn’t rise 01/06/2009, PHIL 120 met at 11:30 that day as scheduled. The sun didn’t rise 01/06/2009. ------------------------------------------------ Phil 120 met at 11:30 that day as scheduled.

20. Implications of the definition of validity “Special” cases of validity: • All arguments containing one logically-false premise are deductively valid. Why? • All arguments whose premises form a logically inconsistent set are deductively valid. Why? • All arguments with a logically true conclusion are deductively valid. Why?

21. Logical soundess • An argument is deductively sound if and only if it is deductively valid and its premises are true. • So valid arguments with one or more false premises are not sound arguments. • And valid arguments with one or more logically false premises are not sound arguments. • And valid arguments whose premises form an inconsistent set are not sound arguments.

22. Part 2 Further implications of the recursive definition of SL

23. Implications of the recursive definition • All sentences of SL are one of the following: • An atomic or simple sentence (A, B… Z) • A negation (a sentence of the form ~P) • A conjunction (a sentence of the form (P & Q) • A disjunction (a sentence of the form (P v Q) • A material conditional (a sentence of the form (P  Q) • A biconditional (a sentence of the form (P  Q) • This means that no matter how many connectives a sentence may contain, it has one (and only one) main connective.

24. Implications of the recursive definition Note: in practice, we drop the outside premises of a whole sentence. Take the sentence: (A v B) & ~C Our question: which kind of sentence is it? Put another way, what is the main connective of this entire sentence? One clue: the & is the only binary connective that is not inside parentheses So the entire sentence is a conjunction whose conjuncts are: A v B and ~C

25. Implications of the recursive definition The sentence (A v B) & ~C is a conjunction whose conjuncts are: A v B and ~C To determine whether the entire sentence is true or false, we need to determine if the left conjunct (A v B) is true or false and whether the right conjunct (~C) is true or false. The parentheses around ‘A v B’ indicate that it is a disjunction that forms one conjunct of the entire sentence.

26. Implications of the recursive definition Compare the sentence (A v B) & ~C To the sentence A v (B & ~C) In the second sentence ‘v’ is the only binary connective outside of parentheses indicating that the entire sentence is a disjunction whose disjuncts are ‘A’ and ‘B & ~C’ The left disjunct is the atomic sentence A; the right disjunct is the conjunction B & ~C, the left conjunct is B and the right is ~C

27. Implications of the recursive definition Compare the sentence (A v B) & ~C To the sentence (A v B)  ~C Here the only binary connective outside of parentheses is the  indicating that the sentence is a material conditional whose antecedent is ‘A v B’ and whose consequent is ~C. A v B is a disjunction. And ~C is a negation.

28. Implications of the recursive definition Compare the sentence (A v B) & ~C To the sentence ~[(A v B)  C] Here the only connective outside of parentheses is the ~ indicating that the sentence is a negation, a sentence of the form ~P, where P is the conditional (A v B)  C The antecedent of this conditional is the disjunction ‘A v B’ and the consequent is the atomic sentence ‘C’

29. So every sentence of SL that is not atomic has a main connective (and this determines what kind of sentence it is). For a sentence of the form ~P, P is its immediate component For sentences of the forms P & Q, P v Q, P  Q, or P  Q, P and Q are the immediate components If an immediate component itself has a connective, it too has a component or 2 components. And the atomic components of a sentence are all the components that are atomic. Finally, the sentential components of a sentence include the sentence itself.

30. Main connectives and sentential components Take the sentence: (A v B)  ~(~A & ~B) Its sentential components include it and its immediate components: (A v B) and ~(~A & ~B) That in turn have components, respectively: A, B and ~A & ~B A and B are atomic (so atomic components of the whole sentence) ~A & ~B has the components ~A and ~B That, respectively, have the components A and B

31. Part 3 Symbolizing sentences with more than 1 connective

32. Symbolizing • Sentences with more than one connective • Sentences with one binary connective and a tilde • Sentences with more than one binary connective (with or without one or more tildes) • Compare the sentences: Bob jogs regularly but Carol doesn’t. Although Carol doesn’t jog regularly, Bob does. B & ~C ~C & B They are logically equivalent.

33. Symbolizing Underline simple declarative sentences and choose sentence letters for them Identify the main connective (if there is one), and any other connectives Begin to build the sentence ‘Bob jogs regularly if Carol doesn’t B C ‘Bob jogs regularlyif it is not the case that Carol jogs regularly ~C  B

34. Symbolizing Alice and Bob jog regularly if and only if Carol or Dan do not [jog regularly] A: Alice jogs regularly B: Bob jogs regularly C: Carol jogs regularly D: Dan jogs regularly Main connective:  Immediate sentential components: A & B and ~C v ~D (A & B)  (~C v ~D) OR (A & B)  ~(C & D)

35. Symbolizing Either Alice, Bob or Carol jogs regularly. A: Alice jogs regularly B: Bob jogs regularly C: Carol jogs regularly Main connective: v Either: (A v B) v C OR A v (B v C) NOT: A v B v C

36. Symbolizing If Carol jogs regularly, then if Bob does too, so does Alice. C: Carol jogs regularly B: Bob jogs regularly A: Alice jogs regularly Main connective:  But there are 2 sentences of the form P  Q. Which contains the main connective? C  (B  A)