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 ≡)
∀ ALL
∃ SOME

I. The Rules of Inference قوانين الإستدلال

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

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

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

5. Constructive Dilemma (CD) 
(p>q)&(r>s)
p∨r
∴ q∨s

6. Conjunction (Conj) إتصال
p
q
∴ p&q

7. Simplification (Simp) تبسيط
p&q
∴ p

8. Addition (Add) إضافة
p
∴ 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

Source:
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.

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