Highlights, Section 1.2: Propositional Logic

Propositional wff: represent some sort of argument, to be tested, or proven, by propositional logic.
valid arguments, e.g.

P₁ and P₂ and ... and P_n -> Q
have hypotheses (we suppose that the P_i are true), and a conclusion (Q). To be valid, this argument must be a tautology (always true). To be an argument, Q must not be identically true (i.e. a fact, in which case the hypotheses would be irrelevant!).
Proof Sequence: a sequence of wffs in which every wff is a hypothesis or the result of applying the formal system's derivation rules (truth-preserving rules) in sequence.
Objective: to reach the conclusion Q from the hypotheses P₁, P₂, ..., P_n.
Types of derivation rules:
- Equivalence rules (see Table 1.12, p. 23):
  we can substitute equivalent wffs in a proof sequence. One way of showing that two wffs are equivalent is via their truth tables.
  - commutative
  - associative
  - De Morgan's laws
  - implication
  - double negation
  Implication seems somewhat unusual, but it is suggested by Exercise 6a, section 1.1. You're asked to prove it in Practice 9, p. 23. That is, prove that
  (P -> Q) <-> (P' or Q) is a tautology. How would you do it?
- Inference rules: from given hypotheses, we can deduce certain conclusions (see Table 1.13, p. 24)
  - modus ponens:
    If Q follows from P, and P is true, then so is Q.
  - modus tollens:
    If Q follows from P, and Q is false, then so is P.
  - conjunction:
    If Q is true, and P is true, then they're both true together.
  - simplification:
    If both Q and P are true, then they're each true separately.
  - addition:
    If P is true, then either P or Q is true.
  Practice 10, p. 24. Also give step 4!
  For a more elaborate example, let's look at #27, p. 32, which shows that one can prove anything if one introduces a contradiction (e.g. the mensa quiz). Also called an inconsistency.
- The difference between equivalence rules and inference rules is that equivalence rules are bi-directional (work both ways), whereas some inference rules are uni-directional (work in only one direction - this is what inference is all about: from this we can infer that, but we cannot necessarily infer this from that!).
  Notice that in the table 1.14 (p. 31) some rules appear twice: two uni-directionals can make a bi-directional!
  Note for your homework: you are not allowed to invoke the rule that you are trying to prove! Notice that the entries in this table are followed by exercise numbers: it is in those exercises that the results are obtained!
- Deduction method: if we seek to prove an implication, we can simply add the hypothesis of this conclusion implication to the hypothesis of the argument, and prove the conclusion of the remaining implication:
  
  P₁ and P₂ and ... and P_n -> (R-> S)
  can be replaced by
  
  P₁ and P₂ and ... and P_n and R -> S
  If you're interested in seeing why this rule works, you might try exercise 45, p. 33, but think of it this way: we're interested in assuming that all the P_i are true, and see if we can deduce the implication R-> S. If R is false, then the implication is true. The only potentially problematic case is where R is true, and S is false. Then what we want to know is: given that
  P₁ and P₂ and ... and P_n and R are true, is S true?
  Exercise #32, p. 32
- Hypothetical syllogism: if P -> Q and Q -> R, then P -> R. (and see a whole long list of rules in Table 1.14). This rule might be referred to as transitivity.
A new rule is created each time we prove an argument; but we don't want to create so many rules that we keel over under their weight! Keep just a few rules in view, and learn how to use them well....
Our goal may well be to turn a "real argument" into a symbolic one. This allows us to test whether the argument is sound (that is, that the conclusion follows from the hypotheses).
Exercise #39, p. 32.
The propositional logic system is complete and correct:
- complete: every valid argument is provable.
- correct: only a valid argument is provable.
The derivation rules are truth-preserving, so correctness is pretty clear; completeness is not! How can we tell if we can prove every valid argument?!

longa@nku.edu