# Is affirmation of the consequent always invalid?

Examples that I've seen usually go something like

"If the lamp is off, then the room is dark. The room is dark, therefore the lamp must be off."

However, what about the following example

"If it is January 1st, then it is my birthday. It is my birthday, therefore it must be January 1st."

This statement affirms the consequent, yet it is a valid.

• The reason your second example is valid is because it is a biconditional. It is your birthday if and only if it is January 1st. With biconditionals, affirming the consequent and denying the antecedent are valid forms. Nov 14, 2023 at 5:22
• First, statements cannot be valid or invalid, they can be true or false, what can be be valid or invalid are inferences. And second, the inference about birthdays has a hidden premise folded into the meaning of "birthday" (one and only one day of the year is a birthday). So your inference does not affirm the consequent, it only looks like that superficially when the hidden premise and reasoning associated with it are omitted. Replace "birthday" with "happy day" and you'll see that it is invalid. Nov 14, 2023 at 5:28
• There's another place in which affirming the consequent can work - probabilistic reasoning. If you apply Bayes Theorem to most situations, affirming the consequent increases the probability of the antecedent most of the time, I'm pretty sure, maybe all the time?
– TKoL
Dec 14, 2023 at 10:24
• It is the January first <-> it is my birthday. It's my birthday / it's the January first. The room is dark -> The lamp is off. The room is dark / the lamp is off. This last is the true implication. Apr 12 at 12:15

Validity is a notion such that if it is even violated once, then it is necessarily violated. That is, an argument is valid iff the premises guarantee the conclusion by the form of the argument alone. So, even if you could state an instance of affirming the consequent that preserves truth, the reason that it preserves truth is different from the argument form used. For your example, you’re implicitly assuming “it is January 1st iff it is my birthday,” so you’re really using regular Modus Ponens/affirming the antecedent.

Another way of thinking of it is that since a contradiction (P&~P) can never be true, it must be always invalid to infer it, right? Wrong! If you have an inconsistent set of formulas A, then from A you may validly infer (P&~P). This is just an example to clear up that there is no such thing as a contingently valid argument within the constraints of a specific logic.

Probabilistically speaking, affirming the consequent is often evidence (though perhaps usually very weak evidence) of the antecedent.

Imagine someone says to you: If my husband has just come back from swimming, he'll be wet.

Then a few minutes later, her husband walks in and he looks quite wet.

Is he DEFINITELY wet because he went swimming? No. But is it fair to surmise that he's probably wet for that reason? Yeah, I think so, unless you take a peek outside and it's raining or something.

• Agreed. Also abductive reasoning generally tends to take the form of affirming the consequent. Such reasoning does not have deductive validity, but that is not the point of it. Apr 12 at 21:53

Logic is a set of rules, which apply to facts (either empirical or rational). So, there are two completely different domains: rules and facts.

• Affirmation of the consequent, AS A RULE, is always invalid. Period.
• As a FACT, affirmation of the consequent can be or not be valid, as your examples show.

If it is January 1st, then it is my birthday. It is my birthday, therefore it must be January 1st.

It is valid AS A FACT, and not because of the rule. In addition, it is a tautology; the RULE, applied over tautologies, is incorrect, even if FACTS appear correct.

• Should probably not say “it is valid as a fact” since in the context of logic, validity is only about inferences. Saying that might confuse people even more Apr 12 at 13:57
• @confusedcius precisely: the ACTION is valid as a FACT, it is not valid "in the context of logic". Read again. Apr 13 at 5:42
• What is the action? You can just say it is true, there’s no confusion in that. Also it is not a tautology. A -> B is not a tautology Apr 13 at 13:38

"If the lamp is off, then the room is dark." "The room is dark, therefore the lamp must be off."

If we expand the premise into "If the lamp is off, then the room is dark, otherwise (if the lamp is on) the room is lit", then affirming "The room is dark, therefore the lamp must be off" also is valid. And the point is, that "otherwise etc." that I have added is typically implied in occurrences of such an informal speech.

So, the word of caution is that those examples are paraphrases of the logical form of a fallacy, not to be taken as representative of the terms of an actual analysis of discourse, which has to preliminarily make explicit the relevant context as well as the rhetorical conventions.

In fact, for a direct answer to the question: while all Affirmations of the Consequent are indeed invalid (by definition!), most real world utterances that look like Affirmation of the Consequent are more possibly dubious elocution (and/or possibly dubious analysis) than they are bad logic.