This week we will talk about sections 1.3-1.5, and we will use everything we have learned last week. Here is a summary:
Proofs (or arguments) are about statements, composed of statements with logical connectives. We hope to make arguments/write down proofs that are valid, so when all premises are true, we can only say the conclusion must be true. Truth tables are nice ways of organizing information, and you can use truth tables to list all logical possibilities.
Statements: sentences that can only be either true or false. You might need more information to know the truth value of it. We use capital letters to represent statements.
To connect these statements, we use logical connectives. We have three so far: , , , and one more will be introduced in section 1.5. These connect the statements you make.
In an argument, you might see words like: therefore/then/hence/thus/we conclude, etc. Statements before these words are called premises, and statements after these words are called conclusions. An argument is invalid, if all premises are true, but the conclusion can be false. So if you want to judge the validity of an argument, try to see if you can come up with something that satisfies all the premises but doesn’t satisfy the conclusion. These are call counterexamples.
Remind you that logical forms (statements with logical connectives) are not unique, as you can see in the equivalent formulas on page 21 and 23.
Finally, truth table is a great tool for organizing information. Above equivalent formulas can be discovered via such tables.