2: Logic and Quantifiers
If at first you don’t succeed, try again. Then quit. There’s no use being a damn fool about it.
–W. C. Fields
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- 2.1: Predicates and Logical Connectives
- In every branch of Mathematics there are special, atomic, notions that defy precise definition. In Geometry, for example, the atomic notions are points, lines and their incidence. The atomic concepts in Set Theory are “set”, “element” and “membership”. The atomic concepts in Logic are “true”, “false”, “sentence” and “statement”. Regarding true and false, we hope there is no uncertainty as to their meanings.
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- 2.2: Implication
- Suppose a mother makes the following statement to her child: “If you finish your peas, you’ll get dessert.” This is a compound sentence made up of the two simpler sentences P = “You finish your peas” and D = “You’ll get dessert.” It is an example of a type of compound sentence called a conditional. Conditionals are if-then type statements. In ordinary language the word “then” is often elided (as is the case with our example above).
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- 2.3: Logical Equivalences
- Some logical statements are “the same.” For example, we discussed the fact that a conditional and its contrapositive have the same logical content. However, the equals sign (=) has already got a job; it is used to indicate that two numerical quantities are the same. The formal definition of logical equivalence is two compound sentences are logically equivalent if in a truth table, the truth values of the two sentences are equal in every row. Thus, we use the symbol (≅) instead.
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- 2.4: Two-Column Proofs
- It may be an impossible goal to get “the average Joe” to perform algebraic manipulations with clarity, but those of us who aspire to become mathematicians must certainly hold ourselves to a higher standard. Two-column proofs are usually what is meant by a “higher standard” when we are talking about relatively mechanical manipulations – like doing algebra, or more to the point, proving logical equivalences.
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- 2.5: Quantified Statements
- All of the statements discussed in the previous sections were of the “completely unambiguous” sort; that is, they didn’t have any unknowns in them. Admittedly, we’ve used variables to refer to sentences (or sentence fragments) themselves, but we’ve said that sentences that had variables in them were ambiguous and didn’t even deserve to be called logical statements. The notion of quantification allows us to use the power of variables within a sentence without introducing ambiguity.
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- 2.6: Deductive Reasoning and Argument Forms
- Deduction is the process by which we determine new truths from old. It is sometimes claimed that nothing truly new can come from deduction, the truth of a statement that is arrived at by deductive processes was lying (perhaps hidden somewhat) within the hypotheses. This claim is something of a canard, as any Sherlock Holmes aficionado can tell you, the statements that can sometimes be deduced from others can be remarkably surprising.
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- 2.7: Validity of Arguments and Common Errors
- An argument is said to be valid or to have a valid form if each deduction in it can be justified with one of the rules of inference listed in the previous section. The form of an argument might be valid, but still the conclusion may be false if some of the premises are false. So to show that an argument is good we have to be able to do two things: show that the argument is valid (i.e. that every step can be justified) and that the argument is sound which means that all the premises are true.