June 2023 – MScMathematics

Reference: Introduction to Logic by Patrick Suppes

I) Tautological Implications:

i) Law of Detachment: $P \& (P \rightarrow Q) \rightarrow Q$

ii) Modus tollendo tollens : $\neg{Q} \& (P \rightarrow Q) \rightarrow \neg{P}$

iii) Modus tollendo ponens : $\neg {P} \& (P \vee Q) \rightarrow Q$

iv) Law of Simplification : $P \& Q \rightarrow P$

v) Law of Adjunction : $P \& Q \rightarrow P \rightarrow Q$

vi) Law of Hypothetical Syllogism : $(P \rightarrow Q) \& (Q \rightarrow R) \rightarrow (P \rightarrow R)$

vii) Law of Exportation : $((P \& Q) \rightarrow R) \rightarrow (P \rightarrow (Q \rightarrow R))$

viii) Law of Importation: $(P \rightarrow(Q \rightarrow R) ) \rightarrow (P \& Q \rightarrow R)$

ix) Law of Absurdity : $(P \rightarrow Q \& \neg {Q}) \rightarrow \neg {P}$

x) Law of Addition : $P \rightarrow P \vee Q$

II) Tautological Equivalences

a) Law of Double Negation: $P \Longleftrightarrow \neg \neg P$

b) Law of Contraposition : $(P \rightarrow Q) \Longleftrightarrow (\neg {Q} \rightarrow {\neg {P}})$

c) De Morgan’s Laws: $\neg (P \& Q) \Longleftrightarrow \neg{P} \vee \neg{Q}$ and $\neg(P \vee Q) \Longleftrightarrow \neg{P} \& \neg{Q}$

d)Commutative Laws: $P \& Q \Longleftrightarrow Q \& P$ and $P \vee Q \Longleftrightarrow Q \vee P$

e) Law of Equivalence for Implication and Disjunction: $(P \rightarrow Q) \Longleftrightarrow \neg{P} \vee Q$

f) Law of Negation for Implication: $\neg(P \rightarrow Q) \Longleftrightarrow P \& \neg{Q}$

g) A Law for Biconditional Sentences : $(P \Longleftrightarrow Q) \Longleftrightarrow (P \rightarrow Q) \& (Q \rightarrow P)$

h) Another Law for Biconditional Sentences : $(P \Longleftrightarrow Q) \Longleftrightarrow (P \& Q) \vee (\neg{P } \& \neg{Q})$

III) Two Further Tautologies:

i) Law of Excluded Middle : $P \vee \neg {P}$

ii) Law of Contradiction: $\neg{P \& \neg{P}}$

Cheers, cheers, cheers,

Nalin Pithwa

MScMathematics