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ᕕ(ᐛ)ᕗ ID: 285b99 March 15, 2020, 8:56 p.m. No.8433054   🗄️.is 🔗kun

Probability calculus

Modus ponens represents an instance of the Law of total probability which for a binary variable is expressed as:

 

{\displaystyle \Pr(Q)=\Pr(Q\mid P)\Pr(P)+\Pr(Q\mid \lnot P)\Pr(\lnot P)\,}{\displaystyle \Pr(Q)=\Pr(Q\mid P)\Pr(P)+\Pr(Q\mid \lnot P)\Pr(\lnot P)\,},

 

where e.g. {\displaystyle \Pr(Q)}{\displaystyle \Pr(Q)} denotes the probability of {\displaystyle Q}Q and the conditional probability {\displaystyle \Pr(Q\mid P)}{\displaystyle \Pr(Q\mid P)} generalizes the logical implication {\displaystyle P\to Q}P\to Q. Assume that {\displaystyle \Pr(Q)=1}{\displaystyle \Pr(Q)=1} is equivalent to {\displaystyle Q}Q being TRUE, and that {\displaystyle \Pr(Q)=0}{\displaystyle \Pr(Q)=0} is equivalent to {\displaystyle Q}Q being FALSE. It is then easy to see that {\displaystyle \Pr(Q)=1}{\displaystyle \Pr(Q)=1} when {\displaystyle \Pr(Q\mid P)=1}{\displaystyle \Pr(Q\mid P)=1} and {\displaystyle \Pr(P)=1}{\displaystyle \Pr(P)=1}. Hence, the law of total probability represents a generalization of modus ponens [11].

 

Subjective logic

Modus ponens represents an instance of the binomial deduction operator in subjective logic expressed as:

 

{\displaystyle \omega {Q\|P}^{A}=(\omega {Q|P}^{A},\omega {Q|\lnot P}^{A})\circledcirc \omega {P}^{A}\,}{\displaystyle \omega {Q\|P}^{A}=(\omega {Q|P}^{A},\omega {Q|\lnot P}^{A})\circledcirc \omega {P}^{A}\,},

 

where {\displaystyle \omega {P}^{A}}{\displaystyle \omega {P}^{A}} denotes the subjective opinion about {\displaystyle P}P as expressed by source {\displaystyle A}A, and the conditional opinion {\displaystyle \omega {Q|P}^{A}}{\displaystyle \omega {Q|P}^{A}} generalizes the logical implication {\displaystyle P\to Q}P\to Q. The deduced marginal opinion about {\displaystyle Q}Q is denoted by {\displaystyle \omega {Q\|P}^{A}}{\displaystyle \omega {Q\|P}^{A}}. The case where {\displaystyle \omega {P}^{A}}{\displaystyle \omega {P}^{A}} is an absolute TRUE opinion about {\displaystyle P}P is equivalent to source {\displaystyle A}A saying that {\displaystyle P}P is TRUE, and the case where {\displaystyle \omega {P}^{A}}{\displaystyle \omega {P}^{A}} is an absolute FALSE opinion about {\displaystyle P}P is equivalent to source {\displaystyle A}A saying that {\displaystyle P}P is FALSE. The deduction operator {\displaystyle \circledcirc }{\displaystyle \circledcirc } of subjective logic produces an absolute TRUE deduced opinion {\displaystyle \omega {Q\|P}^{A}}{\displaystyle \omega {Q\|P}^{A}} when the conditional opinion {\displaystyle \omega {Q|P}^{A}}{\displaystyle \omega {Q|P}^{A}} is absolute TRUE and the antecedent opinion {\displaystyle \omega {P}^{A}}{\displaystyle \omega {P}^{A}} is absolute TRUE. Hence, subjective logic deduction represents a generalization of both modus ponens and the Law of total probability [12].

ᕕ(ᐛ)ᕗ ID: 285b99 March 15, 2020, 8:59 p.m. No.8433087   🗄️.is 🔗kun

Modus ponendo tollens (MPT;[1] Latin: "mode that denies by affirming")[2] is a valid rule of inference for propositional logic. It is closely related to modus ponens and modus tollendo ponens.