Rules of Inference | Readstall

Rules of Inference

12-02-19 Himaja G 0 comment

Rules of inference: We can describe the process of derivation by which demonstrates that a particular formula is a valid consequence of a given set of premises. Before we do this, we give two rules of inference which are called Rule P and Rule T.

Rule P: A premises may be introduced at any point in the derivation.

Rule T: A formula S may be introduced in a derivation if S is tautologically implied by any one or more of the preceding formulas in the derivation.

We list some important implications and equivalences, before we proceeding of the actual process. That will be referred to frequently.

Some of the Implications are:

I1: P∧Q⇒P                     I6: Q⇒PQ                            I11: P,PQ⇒Q

I2: P∧Q⇒Q                   I7: ~(PQ) ⇒P                     I12: ~Q,PQ⇒~P

I3: P⇒P∨Q                   I8: ~(PQ) ⇒~Q                  I13:PQ,QR⇒PR

I4: Q⇒P∨Q                    I9: P,Q⇒P∧Q                        I14: P∨Q,PR,QR⇒R

I5:~P⇒PQ                 I10: ~P,P∨Q⇒Q

Some of equivalences are:

E1:~(~P) P                                              E12:R∨ (P∧~P) R

E2:P∧QQ∧P                                           E13:R∧ (P∨~P) R

E3:P∨QQ∨P                                           E14:R∨ (P∨~P) T

E4:(P∧Q) ∧RP∧ (Q∧R)                           E15:R∧ (P∧~P) F

E5:(P∨Q) ∨RP∨ (Q∨R)                           E16: PQ~P∨Q

E6:P∧ (QR) (P∧Q)(P∧R)                        E17:~(P∨Q) P∧~Q

E7:P∨ (Q∧R) (P∨Q) ∧ (P∨R)                 E18:PQ~Q~P

E8:~(P∧Q) ~P∨~Q                                 E19:P(QR) (P∧Q)R

E9:~(P∨Q) ~P∧~Q                                  E20:~(PQ) (P~Q)

E10:P∨PP                                               E21: (PQ) (PQ) ∧ (QP)

E11:P∧PP                                                E22: (PQ) (P∧Q)(~P∧~Q)

Ex: 1. Demonstrate that R is valid inference from the premises

           P→Q, Q→R, and P

Ans:  PQ                    Rule P

P                           Rule P

Q                          Rule T

QR                    Rule P

R                           Rule T

  1. ~P is valid inference from the premises


Ans: PQ                   Rule P

~P~Q                Rule T

~Q                        Rule P

~P                         Rule T

  1. R∧ (P∨Q) is a valid inference from the premises

    P∨Q, Q→R, P→M and ~M

Ans:   PM                            Rule P

~M                                 Rule P

~P                                  Rule T

P∨Q                               Rule P

Q                                    Rule T

QR                             Rule P

R                                    Rule T

R∧ (P∨Q)                       Rule T

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