For Wednesday, read chapter 2, sections 3 and 4. As nongraded homework, do the problems at the end each section. Also try exercises 7.1, C, D, and E on the Power of Logic web tutor: It uses a dot in place of the ampersand. Also, look at the ‘help’ link to see how to enter the symbols.

Soundness If possible, we would like our arguments to be valid and to have all true premises. An argument with both characteristics (valid, with all true premises) is sound. Any argument that is not sound is unsound. An unsound argument is either invalid or has at least one false premise (or both).

All whales are fish. No fish live in trees. Therefore, no whales live in trees. Valid? Sound? Answer: Valid, but unsound; one false premise is enough to make the argument unsound.

Some fruits are green. Some fruits are apples. Therefore, some fruits are green apples. Answer: Invalid, and unsound—even though all three statements in the argument are true. A note about ‘some’: For our purposes, ‘some’ means ‘at least one’.

These arguments involve category relations: all humans are said to be in the category of reptiles. But for the next few weeks, we’ll be studying sentential logic; we will treat entire simple sentences as single units: If I win the lottery, then I’ll be rich. I just won the lottery. Therefore, I’m rich

Sentential Logic The system contains three kinds of symbol, each playing a different role: 1. The five sentential connectives: ~ (tilde), & (ampersand), v (vee), → (arrow), and ↔ (double- arrow). These connect smaller formulae together (or preface them, in the case of the tilde). 2. Capital letters A-Z abbreviate simple (or atomic) statements. These are grammatically simple sentences that have no words corresponding to sentential connectives in them.

‘~’ abbreviates ‘not’ and equivalent phrases, such as ‘it is not the case that’ Joanne is not tall. (J: Joanne is tall) ~ J ~ J It is not the case that Tom is a lawyer. (T: Tom is a lawyer) ~ T

‘&’ abbreviates ‘and’ and equivalent terms José is a doctor and Martin is a lawyer. (J: José is a doctor; M: Martin is a lawyer) J & M Theresa is poor, even though she is a doctor. (P: Theresa is poor; D: Theresa is a doctor) P & D

‘v’ abbreviates the inclusive ‘either...or...’ and equivalent phrases (inclusive ‘or’ is ‘and/or’) Either Jane is over four years old or over forty pounds. (F: Jane is over four years old; P: Jane is over forty pounds) F v P We often use ‘or’ to mean ‘either...or...but not both’; this is the exclusive ‘or’. We can symbolize it if need be, but the formula is somewhat complex.

3. Parentheses are used for grouping. Either Tom is a doctor or Jill is a doctor, but not both. (T: Tom is a doctor; J: Jill is a doctor) (T v J) & ~ (T & J) This expresses the exclusive ‘or’.