AN EVEN SIMPLER DEFENSE OF MATERIAL IMPLICATION Draft of July 30, 2019 Matheus Silva ABSTRACT Lee Archie argued that if any truth values are consistently assigned to a natural language conditional to which Modus Ponens and Modus Tollens are valid argumentative forms, and affirming the consequent is an invalid argumentative form, this conditional would have the same truth conditions than a material implication. This argument is simple and it requires few assumptions that are relatively uncontroversial. We show that it is possible to extend Archie's argument to three-valued logics and five-valued logics and vindicate a slightly weaker conclusion, but that is still important: even if we would not believe in bivalence and in the classical negation operator, we would still have good reasons to accept that natural language conditionals and the material implication share truth conditions. 1. ARCHIE'S ARGUMENT Lee Archie (1979) held that natural language conditionals with the form "If P then Q" are logically equivalent to the material implication "P ⊃ Q". His argument relies on circumstance surveyors and the following assumptions: when the antecedent of a conditional is true and the consequent is false, the conditional is false; Modus Ponens and Modus Tollens have at least one substitution instance with true premises and a true conclusion; affirming the consequent have at least one substitution instance with true premises and a false conclusion; a conditional is truth-functional without any suppositions concerning the assignment of its truth values. He then arbitrarily assigned truth values in the circumstance surveyor as follows: the value "1" is assigned when both the antecedent and consequent are true. Due to assumptions, the value "F" is assigned when the antecedent is true and the consequent is false. "3" is assigned when the antecedent is false and the consequent is true, and the value "4" is assigned when both antecedent and consequent are false. According to him, we can justify material implication's truth table with these assumptions alone. Consider Modus Ponens' circumstance surveyor: P Q If P then Q P Q T T 1 T T T F F T F F T 3 F T F F 4 F F !1 The first line of the circumstance surveyor above is the only case where Modus Ponens has true premises and true conclusion. Therefore "1" must be true. Now consider Modus Tollens' circumstance surveyor: The fourth line is the unique circumstance in which Modus Tollens can have true premises and true conclusion. Therefore, "4" must be true. However, to fill up the circumstance surveyor we still need to justify the third line. Now consider the circumstance surveyor of affirming the consequent: Since the third line is the unique circumstance in which Affirming the Consequent can have true premises and false conclusion, "3" must be true. By doing this he shows that all the lines of material implication's truth table are the same than "if P then Q". This argument is quite simple and it requires few and relatively uncontroversial assumptions, namely the validity of Modus Ponens and Modus Tollens, the invalidity of affirming the consequent, the thesis that a conditional is false when its antecedent is true and the consequent is false, and the truthfunctional assumption. However, it can be objected that the last assumption is circular because some critics of the truth-functional hypothesis deny that natural language conditionals have truth values when their antecedent is false. In that case, we cannot establish the third and fourth line of the truth table. Nevertheless, Archie could reply that the truth-functional assumption in this case is indirectly justified by the intuitive process of filling up the truth values that follows from the other assumptions. If a conditional does not have a truth value in the third line, we must deny the supposition that affirming the consequent must have one instance with true premises and false conclusion. In addition, if a conditional does not have a truth value in the fourth line, we must deny the supposition that Modus Tollens must have at least one instance with true premises and true conclusion. Since both assumptions are plausible, the truth-functional assumption is justified. In fact, if a conditional can be true only on the first line of the truth table affirming the consequent turns out to be valid, but this cannot be the case, since it is a well-known fallacy. Notice, though, that to be taken seriously, Archie's argument assumes the bivalence and the classical negation operator. In spite of this, we can extend the circumstance surveyor argument and vindicate the same conclusions even if we abandon both assumptions in three and five-valued circumstance surveyors. P Q If P then Q ¬Q ¬P T T 1 F F T F F T F F T 3 F T F F 4 T T P Q If P then Q Q P T T 1 T T T F F F T F T 3 T F F F 4 F F !2 2. EXTENDING THE ARGUMENT TO MANY-VALUED CIRCUMSTANCE SURVEYORS Consider a truth table from Lukasiewicz's three-valued logic L3 (Lukasiewicz, 1920). Following the same arbitrary assignment of truth values from Archie's argument, we can add the following truth values: "i" for "indeterminate", "5" when the antecedent is true and the consequent is indeterminate; "6" when the antecedent is indeterminate and the consequent is true; "7" when both the antecedent and consequent are indeterminate; "8" when the antecedent is indeterminate and the consequent is false; "9" when the antecedent is false and the consequent is indeterminate. Since this logic has an additional truth value-the indeterminate-we also have a different negation operator that works as indicated in the table below: Having this in mind, consider the following circumstance surveyors: Modus Ponens Modus Tollens P ¬P T F i i F T First premise Second premise Conclusion P Q If P then Q P Q T T 1 T T T i 5 T i T F F T F i T 6 i T i i 7 i i i F 8 i F F T 3 F T F i 9 F i F F 4 F F First premise Second premise Conclusion P Q If P then Q ¬Q ¬P T T 1 F F T i 5 i F T F F T F i T 6 F T i i 7 i i i F 8 T F !3 Affirming the Consequent Looking at the circumstance surveyors, we can observe that the conclusion is the same as in the two-valued logics: lines 1, 3 and 4 are true even if we abandon bivalence. The same results apply to other three-valued logics (e.g. Kleene's, Bochvar's). An identical conclusion obtains in five-valued logics. Consider a five-valued logic with the following truth values: unknown (U), possibly known but consistent (K), false (F), true (T) and inconsistent (I) (Ferreira, 2004: 134–140). With the assumptions that Modus Ponens and Modus Tollens have at least one substitution instance with true premises and true conclusion and affirming the consequent have at least one substitution instance with true premises and false conclusion we make the following additional arbitrary attributions of truth values: "1" when the antecedent is U and the consequent is U; "2" when the antecedent is U and the consequent is K, and so on, proceeding until the end of the truth table with the value 24. The negation operator is the following: Given that, we have the following circumstance surveyors: Modus Ponens F T 3 F T F i 9 i T F F 4 T T First premise Second premise Conclusion P Q If P then Q Q P T T 1 T T T i 5 I T T F F F T i T 6 T i i i 7 i i i F 8 F i F T 3 T F F i 9 i F F F 4 F F P ¬P U K K U F T T F I I !4 Modus Tollens First premise Second premise Conclusion P Q If P then Q P Q U U 1 U U U K 2 U K U F 3 U F U T 4 U T U I 5 U I K U 6 K U K K 7 K K K F 8 K F K T 9 K T K I 10 K I F U 11 F U F K 12 F K F F 13 F F F T 14 F T F I 15 F I T U 16 T U T K 17 T K T F F T F T T 18 T T T I 19 T I I U 20 I U I K 21 I K I F 22 I F I T 23 I T I I 24 I I First premise Second premise Conclusion P Q If P then Q ¬Q ¬P U U 1 K K U K 2 U K U F 3 T K U T 4 F K U I 5 I K K U 6 K U K K 7 U U K F 8 T U K T 9 F U !5 Affirming the Consequent K I 10 I U F U 11 K T F K 12 U T F F 13 T T F T 14 F T F I 15 I T T U 16 K F T K 17 U F T F F T F T T 18 F F T I 19 I F I U 20 K I I K 21 U I I F 22 T I I T 23 F I I I 24 I I First premise Second premise Conclusion P Q If P then Q Q P U U 1 U U U K 2 K U U F 3 F U U T 4 T U U I 5 I U K U 6 U K K K 7 K K K F 8 F K K T 9 T K K I 10 I K F U 11 U F F K 12 K F F F 13 F F F T 14 T F F I 15 I F T U 16 U T T K 17 K T T F F F T T T 18 T T T I 19 I T I U 20 U I !6 Once more, the truth table of the material implication was justified. Thus, even when we have a many-valued conditional connective that departs from the bivalent connective of classical logic we still maintain the same results. Apparently, we can keep adding new truth values and the lines of material implication's truth table will still be vindicated. My defence is simpler because it requires fewer assumptions: even if you do not believe in bivalence and the classical negation operator, you still have good reasons to accept material implication's truth conditions. 3. SOME POSSIBLE REPLIES AND CONCLUDING REMARKS Someone could object that a connective with more than two truth values cannot be a material implication since a material implication is by definition bivalent. In that case, we have offered an argument only for the thesis that an indicative conditional is a many-valued truthfunctional cousin of the material implication. However, the objection goes, since material implication's truth conditions are the only viable truth-functional option for a conditional connective with two values this should not be a surprising result. 1 The problem of this objection is that it ignores that the argument is an attempt to justify the lines of the truth table of the material implication and not of its many-valued cousins. Of course, under the assumption that we have discarded the bivalence and that Lee Archie argument's works, the natural language conditionals do not have exactly the same truth conditions than a material implication, since we are considering many-valued connectives that have more truth values. Nonetheless the fact remains that under these assumptions the many-valued connectives preserve the truth conditions of the material implication and that any natural language conditional with a false antecedent or a true consequent will be true. REFERENCES Archie, L. 1979. A Simple Defense of Material Implication. Notre Dame Journal of Formal Logic. 20(2), 412–414. Ferreira, U. 2004. A Five-valued Logic and a System. Journal of Computer Science and Technology, 4(3), 134–140. Kleene, S. 1967. Mathematical Logic. New York: John Wiley and Sons. Lukasiewicz's, J. 1920. On 3-valued logic. In: McCall, S (ed.), Polish Logic. Oxford: Oxford University Press, 1967. I K 21 K I I F 22 F I I T 23 T I I I 24 I I The alternatives validate fallacies such as denying the antecedent, and are counter-intuitive. See Kleene (1967: 1 10). !