Modus ponens and modus tollens are valid deductive inference rules in logic, often represented with arrows (
→right arrow
→ for “if…then” and
∴∴
∴ for “therefore”). Modus ponens affirms the antecedent (
P→Q,P,∴Qcap P right arrow cap Q comma cap P comma ∴ cap Q
𝑃→𝑄,𝑃,∴𝑄), while modus tollens denies the consequent (
P→Q,¬Q,∴¬Pcap P right arrow cap Q comma logical not cap Q comma ∴ logical not cap P
𝑃→𝑄,¬𝑄,∴¬𝑃).
Modus Ponens (Method of Affirming)
Structure:
If
Pcap P
𝑃, then
Qcap Q
𝑄 (
P→Qcap P right arrow cap Q
𝑃→𝑄)
Pcap P
𝑃
∴Q∴ cap Q
∴𝑄
Example: If it rains (
Pcap P
𝑃), then the ground is wet (
Qcap Q
𝑄). It is raining (
Pcap P
𝑃). Therefore, the ground is wet (
Qcap Q
𝑄).
Modus Tollens (Method of Denying)
Structure:
If
Pcap P
𝑃, then
Qcap Q
𝑄 (
P→Qcap P right arrow cap Q
𝑃→𝑄)
Not
Qcap Q
𝑄 (
¬Qlogical not cap Q
¬𝑄)
∴∴
∴ Not
Pcap P
𝑃 (
¬Plogical not cap P
¬𝑃)
Example: If it rains (
Pcap P
𝑃), then the ground is wet (
Qcap Q
𝑄). The ground is not wet (
¬Qlogical not cap Q
¬𝑄). Therefore, it is not raining (
¬Plogical not cap P
¬𝑃).
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