A Deduction Problem #1

If Barcelona wins the championship, fans riot. If Barcelona doesn’t win the championship, fans will not go to work. The fans either go to work or cry out of depression. The fans do not riot.
Are the fans crying out of depression?
1. If Barcelona wins the championship, fans riot.

> Fans do not riot.
> Therefore, Barcelona do not win the championship.

2. If Barcelona does not win the championship, fans do not go to work.

> Barcelona does not win the championship.
> Therefore, fans do not go to work.

3. Fans either go to work or cry out of depression.

> Fans do not go to work.
> Therefore, fans are crying out of depression.

Solution using Symbolic Logic
Barcelona wins = w
Fans riot = r
Fans go to work = g
Fans cry out of depression = c
1. w > r
2. ~ w > ~ g
3. g ∨ c
4. ~ r
5. ~ w (1,4 MT)
6. ~ g (2,5 HS)
7. c (3,6 DS) (Conclusion)
Therefore, fans will cry out of depression.

Source of the problem statement
A fantastic Youtube video playlist called “Logic 101 MOOC” by William Spaniel.
Link> Logic 101 MOOC


Rules of Inference and Replacement قوانين الإستدلال والإستبدال

Below are the 18 rules of Inference and Replacements that Logicians use for deductions and proofs.
But first we will introduce the symbols of logical operators:
∧ AND (also &)
∨ OR
~ NOT (also ¬)
→ IF..THEN (also >)
⇔ IFF (also equivalence sign ≡)

I. The Rules of Inference قوانين الإستدلال

1. Modus Ponens (MP) قياس استثنائي وضعي
p > q
Therefore q
2. Modus Tollens (MT) قياس استثنائي رفعي
p > q
∴ ~p

3. Disjunctive Syllogism (DS) قياس منفصل حملي
p V q
∴ q

4. Hypothetical Syllogism (HS) قياس شرطي
p > q
q > r
∴ p > r

5. Constructive Dilemma (CD) 
∴ q∨s

6. Conjunction (Conj) إتصال
∴ p&q

7. Simplification (Simp) تبسيط
∴ p

8. Addition (Add) إضافة
∴ p∨q

II. The Rules of Replacement قوانين الإستبدال

9. Demorgan’s Law (DM) قانون ديمورغن
~(p&q)  (~p∨~q)
~(p∨q)  (~p&~q)

10. Commutivity (Com) تبادل
q∨p  p∨q
q&p  p&q

11. Associotivity (Assoc) تجميع
p∨(q∨r)  (p∨q)∨r
p&(q&r)  (p&q)&r

12. Distribution (Dist) توزيع
p∨(q&r)  (p∨q)&(p∨r)
p&(q∨r)  (p&q)∨(p&r)

13. Double Negation (DN) نفي مزدوج
~~q  q

14. Transportation (Trans) نقل
p>q  ~q>~p

15. Material Implication (Impl)
p>q  ~p∨q

16. Material Equivalence (Equiv)
p≡q  (p>q)&(q>p)

17. Exportation (Exp) تصدير
p>(q>r)  p>(q&r)

18. Tautology (Taut)
p∨p  p
p&p  p

A brilliant YouTube video series called “100 Days of Logic” by Carneades.org
Link > 100 Days of Logic
In addition to my friend λόγος, who is a great Thinker and Logician.