Symbolic Logic: Conditional/Indirect Proofs and Proving Theorems

Hey guys! So, this time, we’re going look at other methods we can use to construct proofs when just deriving from the premises isn’t enough.

Conditional Proof (CP)

The setup:

Basically, you use this method when the conclusion or a part of the conclusion you want is a conditional. This makes it so you assume the predicate in order to derive the consequent. Here’s an example:

Indirect Proof (IP)

The setup:

For this method, you use this primarily when the conclusion is a negated statement. You assume the un-negated form of the conclusion and attempt to find a contradiction so that the assumption is false, thus ending at the negated form. It also works the other way around where the conclusion isn’t negated so you make the assumption negated instead and then use the DN rule at the end. It’s also super useful when proving theorems where you have a limited plan of action. An example:

Theorems

Theorems are formulas that can be proven true without premises so the proofs for theorems have the additional challenge of not being able to build off of premises. Therefore, the above two methods are essential to be able to do proofs of theorems. Here’s an example:

RULES

  1. All assumptions must be discharged(closed).
  2. Lines between different assumptions must not cross.
  3. Once discharged, steps within the subproof cannot be used anymore.

On to the next page for a few practice problems!

Symbolic Logic: 10 Replacement Rules

This is part two of our little intro to Symbolic Logic. We’re going to expand our repertoire of rules we can employ in our proofs. These rules are all about putting logic statements into an alternative form. A lot of these rules will be familiar as they’re used in mathematics. One of the differences between these rules and the basic 8 is that these are reversible, hence the :: symbol to denote a two-way operation. The format of this post is going to be similar to the last one; the rules will be listed first, then some simple examples and then a couple of practice problems on the following pages.

1. Double Negation (D.N)

p :: ~~p

2. Commutation (Comm.)

p V q :: q V p

p • q :: q • p

3. Association (Assoc.)

[(p V (q V r)] :: [(p V q) V r)]

*applies to AND operators the same way*

4. Duplication (Dup.)

p :: p V p

5. DeMorgan’s Law (DeM.)

~(p V q) :: ~p • ~q

This law describes how a negation gets distributed into a parenthesised statement. It negates the two variables and switches the operator from an AND to an OR or vice versa. It only works on AND and OR operators though so if you have a (BI)CONDITIONAL operator inside the parenthesis, you’ll need to use one of the later replacement rules to make them into one.

6. Biconditional Exchange (B.E.)

(p ≡ q) :: [(p ⊃ q) • (q ⊃ p)]

7. Contraposition (Contra.)

(p ⊃ q) :: (~q ⊃ ~p)

8. Conditional Exchange (C.E.)

p ⊃ q :: (~p V q)

9. Exportation (Exp.)

[(p • q) ⊃ r] :: [(p ⊃ (q ⊃ r)]

10. Distribution (Dist.)

[p • (q V r)] :: (p • q) V (p • r)

[p V (q • r)] :: (p V q) • (p V r)


Examples




Alright y’all, on to the practice problems.

Symbolic Logic: 8 Basic Inference Rules

Hi y’all! So, if you’re computer science majors/philosophy major/etc., you probably have to take this class in college. I love this stuff because it’s very procedural and the proofs they give for you to solve are like puzzles and puzzles are super fun. Today, we’re gonna look at the 8 basic rules and then we’ll look at the replacement rules and more. I’m going to assume that y’all know the basic structure of sentential logic including operators and truth tables. Let’s get started.

Structure

A proof is a procedure which is supposed to derive the desired conclusion from a set of premises. To do this, the proof has to be set up in a certain way. First, all lines of a proof must be numbered. The premises make up the first lines of the proof along with the desired conclusion. Then, all subsequent derivations from the premises are listed below with the justification for each step listed along the right side, noting which rule was used and what lines of the proof were referenced. Here’s an example:

  1. Premise Pr. (for premise)
  2. Premise Pr. /:. Conclusion
  3. Derivation (Name of Rule) 1, 2 (lines 1 & 2 referenced)

For our purposes on this page, the visualisations for each of the rules below will not be written in this vertical fashion as they are cumbersome to format in the WordPress editor so it’ll be horizontal.

Let’s get into the rules and then work on some examples which will be on page 2.

1. Simplification (Simp.)

p • q /:. p

OR

p • q /:. q

If there is a conjunction, then both conjuncts can be individually represented as being true by themselves.

2. Conjunction (Conj.)

p; q /:. p • q

If two variables are true, then they can be joined in a conjunction.

3. Addition (Add.)

p /:. p V q

This rule is incredibly powerful as it allows you to introduce new elements into a disjunction as long as we have one of its disjuncts as true.

4. Disjunctive Syllogism (D.S.)

p V q; ~q /:.p

OR

p V q; ~p /:. q

If one of the disjuncts is stated to be false, then the remaining disjunct is true.

5. Modus Tollens (M.T)

p q; ~q /:. ~p

If the consequent of a conditional is false, then its antecedent is also false.

6. Modus Ponens (M.P.)

p q; p /:. q

If the antecedent of a conditional is true, then the consequent is also true.

7. Hypothetical Syllogism (H.S.)

p q; q r /:. p r

If the antecedent of a conditional leads to an antecedent of another conditional, then you can infer that the first antecedent leads to the consequent of the second conditional.

8. Rule of Dilemma/ Constructive Dilemma (C.D)

p V q; p s; q r /:. s V r

If either of the antecedents of two conditionals is true, then either of their consequents must also be true.


Alright, off to examples! We’re gonna start off easy and build onto harder and longer proofs.





Alright, now on to real problems; I have included two. I’ll give you guys the premises and the desired conclusion and I’ll post answers on how to derive the conclusion on the page after that. On to the next page!