Highlights, Section 1.2: Propositional Logic

Propositional wff: represent some sort of argument, to be tested, or proved 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).
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 P.
Types of derivation rules:
- Equivalence rules (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 what we did in 6a, section 1.1. We can prove it using Practice 9, p. 23. That is, prove that (P -> Q) <-> (P' or Q) is a tautology.
- Inference rules: from given hypotheses, we can deduce certain conclusions (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 #18/27, p. 31/32, 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 (found in the exercises) some rules appear twice: two uni-directionals can make a bi-directional!
- 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 36/45, p. 32/33, 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 #23/32, p. 31/32
- 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 #30/39, 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