1. Today’s Topics Argument forms and rules (review)
Rules of inference and equivalence
Applying rules of inference
Justifying steps in a proof
Beginning to construct proofs

2. Argument Form An argument form set of statement forms
Every substitution instance of an argument form is an argument
Argument forms are either valid or non-valid (truth tables verify the validity of valid argument forms)

3. Valid Argument Forms Justify Two Types of RULES
Rules of Inference
Operate on whole lines only
Generate new lines with unique truth values
Rules of Equivalence
Operate on whole lines or parts of lines
Generate lines equivalent to the original lines

4. Rules of Inference
Generate new lines whose truth value follows from, but is not identical to, the truth of the source lines.
Operate on lines whose statement forms match the statement forms of the lines in the argument form of the rule.
Can be applied ONLY to entire lines, not parts of lines.

5. Eight Basic Inference Rules
Modus Ponens (MP)
From a conditional and a line identical to its antecedent, you may derive a line identical to its consequent
Modus Tollens (MT)
From a conditional and the negation of its consequent, you may derive the negation of its antecedent
Disjunctive Syllogism (DS)
From a disjunction and the negation of one disjunct, you may derive the other disjunct
Hypothetical Syllogism (HS)
From 2 conditionals, if the consequent of the first is identical to the antecedent of the second, you may derive a new conditonal whose antecedent is identical to the antecedent of the first and whose consequent is identical to the consequent of the second.

6. Modus Ponens Modus Tollens
p  q p  q
p ~q
q ~p
Disjunctive Syllogism Hypothetical Syllogism
p ▼ q p  q
~p q  r
q p  r

7. Simplification Constructive Dilemma Conjunction Addition
From a conjunction you may derive either conjunct.
Conjunction
From any 2 lines you may derive a conjunction which has those lines as conjuncts
Addition
From any line you may derive a disjunction with that line as a disjunct
Constructive Dilemma
From a disjunction and 2 conditionals, if the antecedents match the disjuncts, you may derive a disjunction of the consequents

8. Simplification
p  q q
Conjunction p q
p  q
Addition p
p ▼ q
Constructive Dilemma
p ▼ q
p  r
q  s
r ▼ s

9. Rules of Equivalence
Equivalent expressions are true and false under exactly the same circumstances. So, one expression, or part of an expression, can be replaced with an equivalent expression (or part) without any change in meaning.
Rules of Equivalence allow replacement of “equals for equals”
Rules of equivalence are bi-directional

10. Use rules of equivalence to manipulate the shape of a line to make it fit the strict pattern of an inference rule.

11. Rules of Equivalence Double Negation (DN) p :: ~ ~ p ~ ~ p :: p
DeMorgan (DM)
~(p ▼ q) :: (~p  ~q)
~(p  q) :: (~p ▼ ~q)
Association (Assn)
(p ▼ q) ▼ r :: p ▼ (q ▼ r)
(p  q)  r :: p  (q  r)
Commutation (Comm)
p ▼ q :: q ▼ p
p  q :: q  p

12. Distribution (Dist) Material Implication (MI) Exportation (Exp)
[p  (q ▼ r)] :: [(p  q) ▼ (p  r)]
[p ▼ (q  r)] :: [(p ▼ q)  (p ▼ r)]
Material Implication (MI)
(p  q) :: (~p ▼ q)
Exportation (Exp)
((p q)  r) :: (p  (q  r))

13. Contraposition (Cont) (p  q) :: (~q  ~p) Redundancy (Re)
p :: (p  p)
p :: (p ▼ p)
Material Equivalence (ME)
(p  q) :: [(p  q) ▼ (~p  ~q)]
(p  q) :: [(p  q)  (q  p)]

14. One Additional Rule of Equivalence:
Absorption
From a conditional you may derive a new conditional whose antecedent is that of the original and whose consequent is a conjunction of the original antecedent and the original consequent
p  q : : p  (p ● q)

15. That’s It! We now have all 19 of the rules of inference and equivalence
MP, MT, DS, HS, Conj, Simp, Add, CD, Abs
NOTE: there are no inference rules for dealing with biconditionals
DN, DeM, Assoc, Comm, Dist, Imp, Exp, Cont, Taut, Equiv

16. Applying Rules of Equivalence to Formulas
Download the Handout on Equivalence Rules and apply the rules indicated to the formulas given. Discuss your results on the bulletin board.

17. Remember
A proof is a finite set of formulae, beginning with the premises of an argument and ending with its conclusion, in which each formula following the premises is derived from the preceding formulae according to established rules of inference and equivalence.

18. Justifying Steps In a Proof
Each line in a proof must be justified.
Premises justify themselves, we assume them to be true.
Derived Lines (those lines after the premises) must be justified according to valid rules of inference or equivalence as following from previous lines.

19. Here’s how it works: Consider the argument A  B, A B
First, list the premises and give them line numbers.
1. A  B Premise
2.A Premise
Now, justify lines leading to the conclusion;
3.B ,2 Modus Ponens (MP)
That is the proof.

20. Try a few
Download the Handout on Justification of Proofs and try to justify all the derived lines (those that follow the premises).
Now try the next homework set on justification of proofs.

21. Thinking About Proofs
Proofs in logic work just like proofs in geometry
The 18 rules we have allow us to manipulate a basic set of assumptions (the premises) so as to show that the conclusion is a logical consequence of them.
A proof is a set of instructions on how to get from the premises to the conclusion.

22. Constructing a proof is like giving instructions
Constructing a proof is like giving instructions. The question is “How do I get there (the conclusion) from here (the premises)?”
The rules are the allowable moves or turns you can take.
Proceed stepwise. Suppose you want to get to D from A, B, and C. Well, if from A and B you can get to E, and from E and C you can get to D, you have your instructions.
That is all there is to constructing proofs

23. Key Ideas Argument form Rules of inference Rules of equivalence Proofs
Justifying steps in a proof