Modus Ponens Cannot Be Expressed by the Conditional in Predicate Calculus
Why conditional statements with a false antecedent are true in predicate calculus
The law of detachment (A → B, A ⊢ B) is the best possible replication of modus ponens and causation there is in predicate calculus, and it expresses nothing about it.
The Existence of the Problem
The conditional (or, material implication) connective (→) is not a translation of “if, then” statements. In fact, A → B, A ⊢ B is not an expression of modus ponens either. Modus ponens is a form an argument can take, like so:
If A is true, then B is true. A is true. Therefore, B is true.
If I am alive, then I am not dead. I am alive. Therefore, I am not dead.
If I am a woman, then I can be empathetic. I am a woman. Therefore, I can be empathetic.
This is a valid form of argumentation, meaning it is true so long as the predicates A and B are true — this is how the reality of what makes modus ponens valid, its original meaning and intention before predicate calculus. In predicate calculus, the validity of A → B, A ⊢ B is founded on the premises A → B (which is said to translate to the “if, then” statement) and the predicate A. The following is not a valid form of argumentation:
If A is true, then B is true. A is false. Therefore, B is true.
If I am alive, then I am not dead. I am not alive. Therefore, I am not dead.
What an absurd position! Neither is this a valid form of argumentation:
If A is true, then B is true. A is false. Therefore, B is false.
If I am a woman, then I can be empathetic. I am not a woman. Therefore, I cannot be empathetic.
The standard use of the conditional connective dictates, however, that the expression A → B is only false if the antecedent predicate A is true, and consequent predicate B is false. Modus ponens expresses not a figment of this idea. To be sure, predicate calculus does not consider the poor replication of these examples into A → B, ¬A ⊢ B and A → B, ¬A ⊢ ¬B, respectively, as valid either. Yet, this is arguably even worse. Predicate calculus is committed to the truth of A → B given ¬A, and uses the conditional connective to “translate” modus ponens by adding additional ideas about the relationship between A and B, but these cannot even serve the original concept of argumentational validity. The conditional connective completely fails, then, as the “if, then” element in a translation of modus ponens, as such a schema would be unconditionally committed to the truth of the expression “if A is true, then B is true” where A is false.
There is a popular explanation for this, saying that the ability for this connective to return true results is “innocent until proven guilty”. The idea is that, until the expression A → B is a hypothetical formulation we should assume to be true, unless it is directly shown to be false by “the facts” — a case where B has not followed A. This is the opposite of the attitude deductive reasoning has toward truth, which is one of falsity until proven true. This explanation is often supported by examples with an impossible antecedent:
If I can time-travel, it is raining. I cannot time-travel. Therefore, it is raining.
If I can time-travel, it is raining. I cannot time-travel. Therefore, it is not raining.
Supposedly, this shows the “irrelevance” of a false antecedent. If A is false, there is no obligation for B to be anything. Or put differently, B might as well be anything. In short, the conditional expresses no meaning if the antecedent is false! In reality, the example relies on the irrelevance of the antecedent itself, which will never be true, taking the hit for the absurdity of the entire sequence. The existence of such an explanation only serves to be developed out of.
The Solution to the Problem
The Ambiguity of “If, Then” Statements
Connectives in predicate calculus, being truth-functional, must return a result given the truth-value of their predicates. Knowing this leads to what can be called the “soft error” of understanding for the conditional: “The connective returns ‘true’ to fill the void that ‘if, then’ statements have in the case of a false antecedent!”
On the otherhand, one might argue that the conditional returning true for all false antecedent predicates actively defends conditionals which ought to be assumed false. This “hard error” of understanding leads to active condemnation of the conditional.
The reality is that A → B is a proposition for all the information contained in it, and can be proven or disproven on those terms. For example, if we know that A → B is true, we know that either A is false or both A and B are true. If we know that B is true and A → B is true, we know that A is true, etc.
A → B works the same mechanically as “if, then” when the antecendent is true, but A → B is not the premise of “if, then”, just as A ∧ B isn’t.
What the Conditional Connective Is a Crystalisation of
The error of the false association between modus ponens and the conditional connective is expressed with the ambiguity of “if, then” statements, which only proposes a consequent’s truth-value on the condition of a certain antecedent.
Upon revisiting modus ponens, we may see it for what it is:
If A is true, then B is true. A is true. Therefore, B is true.
We have established that the first statement, “If A is true, then B is true,” can be taken as the actual connective itself, corresponding to A → B. The second and third statements, “A is true. Therefore, B is true,” relate more closely to a row on the truth-table for where A → B is true, than independent propositions. But, to fully define the operations of the conditional connective, we need to add both:
A is false. Therefore, B is false.
A is false. Therefore, B is true.
The analogy, however, has already been stretched too thin with the use of therefore between A and B. It should be mentioned, too, that the direction of the arrow is merely a bad analogy, and A → B ought to be reduced to what predicate calculus is about — an instantaneous, motionless, impression of the state of things. Here are some better translations than modus ponens of the type of argument the conditional connective can make:
A is true. B is true. Therefore, B is true on the condition that A is true.
B is true on the condition that A is true. Therefore, it is not the case that both A is true and B is false.
Modus ponens has existed long before the predicate calculus, and formalising its meaning using the conditional connective is a mistake.
Proofs Demonstrate the Impossibility of Defining Consequence With Predicate Calculus
To introduce the conditional connective in a proof, one must assume the antecedent and derive the consequent to be true. It is not possible, for example, to assume the antecedent to be false to introduce the conditional into a proof, as a false antecedent can imply both a true and a false consequent. The same goes for false and true consequents, which can imply both a true and a false antecedent. If the conditional connective were redefined to return falsity on a false antecedent, this would only make the resulting conjunction identical to the connective. Therefore, if predicate calculus wishes to have a connective proven by the emphasis of “B is conditional on A”, the conditional is the best it can offer, and it only works when appreciated as a snapshot.
Returning now to our absurd examples:
If A is true, then B is true. A is false. Therefore, B is true.
If A is true, then B is true. A is false. Therefore, B is false.
The validity of these statements is analogous to the validity of A → B, ¬A ⊢ B and A → B, ¬A ⊢ ¬B, respectively. They measure as invalid, precisely because the truth of the consequent cannot be known from a false antecedent according to our truth-values.
Furthermore, consider that there is no I could go on, but I will leave further contemplation for you to do. My conclusion ends here:
When one sees motion in logic, they may not only misapprehend what it represents, but also be prone to fatal errors within its practice. When one is unsure of the conditional as a translation of “if, then”, they are right. When teaching predicate calculus, I have seen how it is better to acknowledge this than sanctify the convention. But look deeper, as I have done in this article, and it’s clear why this is the convention.