Jun 24, 2016 - In binary propositional logic modus ponens is a tautology known as the law of detachment. It also has importance as a rule of inference in logic ...
Why Modus Ponens is an inference rule but not a tautology By Graeme Heald 24th June 2016 Abstract Are modus ponens and its counterpart, modus tollens, universal laws of logic? A deconstruction of modus ponens in binary propositional logic finds that the propositional formula of binary logic has exceeded the definition of modus ponens. In a number of logic systems, including U4, modus ponens is not a theorem. An examination of the truth table for modus ponens in U4 will show that modus ponens as an inference rule can be preserved. Discussion In binary propositional logic modus ponens is a tautology known as the law of detachment. It also has importance as a rule of inference in logic systems. But the question can be raised: Are modus ponens and its counterpart, modus tollens, universal laws of logic? Consider a definition of modus ponens: „the rule of logic which states that if a conditional statement (if p then q) is accepted, and the premise (p) holds, then the conclusion (q) may be inferred.‟ In an instance of modus ponens we assume as a premise that p→q is true and p is true. For example, „All fish have scales. This salmon is a fish. Therefore, this salmon has scales‟. This is an example of universal modus ponens. In binary propositional logic a formula is given, (p→q).p → q, and a truth table may be constructed for modus ponens. Table 1. Modus ponens binary propositional logic p F F T T
q F T F T
p→q T T F T
(p→q).p F F F T
(p→q).p → q T T T T
Modus ponens is a tautology for binary propositional logic, meaning that is it is always true regardless of the values attributed to its variables. Modus ponens by definition, however, only relates to the final line of the truth table in which p and q are true. The truth table for modus ponens can be deconstructed. If the premise, p, is false then the outcome of (p→q).p, will be false and the implication, (p→q).p → q, will be „vacuously true‟. A false conjunction will lead to false outcome and a false premise will be valid for any conclusion, termed „vacuously true‟. It should also be noted that modus ponens does not relate to the case when p is false, that is when the premise is not affirmed. Nevertheless, the first two lines of the truth table are true and modus ponens appears to be confirmed. Modus ponens is actually the case when p is true and q is true and this is confirmed in the final line of the truth table. In the third line of the truth table, the case when p is true and q is false is impossible for valid implication. When the premise is true and conclusion is false the implication is invalid and this corresponds with the false conclusion. Modus ponens is therefore valid for true premises. When the truth table for modus ponens is deconstructed it is found that the implications are vacuously true for a false premise and for a true premise is valid in both true and false cases. As there is a contradiction between the definition and the propositional logic formula it can be concluded that modus ponens is not well represented in the binary propositional logic truth table. Modus ponens is only valid for a true premise and conclusion, whereas in propositional logic, modus ponens is valid for any premise and conclusion.
As modus ponens is found not to be valid in relevance logic for first degree entailment and paraconsistent LP logic [1][2], it can be seen that modus ponens is not a universal law of logic. While modus ponens is seldom questioned as a rule of inference, Walton raises the question in his paper Are Some Modus Ponens Arguments Deductively Invalid?[3]. This article concerns the structure of defeasible arguments like: 'if Bob has red spots, Bob has the measles; Bob has red spots; therefore Bob has the measles.' The issue is whether such arguments have the form of modus ponens or not. By carefully examining arguments on both sides of the issue, Walton concludes that reasonable doubts are raised about the view that all arguments having a modus ponens form are valid. U4 is an extension of Boolean logic that is found when the Boolean truth values, T = U and F = , are separated into 4 distinct truth values { T, U, F, }. The truth table for modus ponens in U4 can be constructed and it will also be shown that modus ponens is not valid [4]. It is suggested that the truth table below reflects the true meaning of modus ponens. Table 2. Modus ponens in U4 p F F F F T T T T U U U U
q F T U F T U F T U F T U
p→q T T T T F T F T F T T F F F T
(p→q).p F F T T T F F F T
(p→q).p → q T T T T F T F T T T T T F T F T
An analysis of the U4 truth tables for modus ponens shows that: i) MP and MT are satisfiable for true and impossible premises ii) MP and MT are not satisfiable for false and uncertain premises In U4 modus ponens is not a tautology but it is semi-valid for true and impossible premises. In the truth table entry it is clear that the informal meaning of modus ponens is confirmed with a true premise and a true conclusion. Similar to Boolean logic, modus ponens is valid for a true premise and any conclusion. Hence, modus ponens is a semi-valid inference rule in U4, but it is not a theorem. Proofs of modus ponens employ the material implication formula, p→q ≡ p q. In U4 the material implication formula does not apply and so modus ponens is not a theorem. However, in U4 if the premise is true then the conclusion follows and modus ponens, for the limited case, is valid. Hence, modus ponens and its counterpart modus tollens remain inference rules in U4. References [1] A. Anderson, N.D. Belnap, Entailment the Logic of Relevance and Necessity, Princeton, US, 1975. [2] G. Priest, “The Logic of Paradox”, Journal of Philosophical Logic, 8, 1, 1979, pp 219-241. [3] D. N. Walton “Are Some Modus Ponens Arguments Deductively Invalid?”, Informal Logic Vol. 22, No. 1 (2002): pp. 19-46. [4] G. Heald “An Outline for a Universal Logic System”, Proceedings of the 7th Mediterranean Conference on Control and Automation (MED99) Haifa, Israel - June 28-30, 1999.
