What's the difference between 'affirming the consequent' and 'commutation of conditionals'?

Brilliant!

Many thanks Kaiden. Just what I was hoping for. 😀

I’ve been needing to say you’re welcome !

Although I agree with you that the fallacies are different, as your examples indicate, they are logically equivalent.  Meaning that in all four possible combinations of truth values (P true, Q true), (P true, Q false), (P false Q true), (P false Q false), these two statements have the same truth value:

1)  P → Q, Q, therefore P

2)  P→Q  therefore, Q → P

[To Ed F]

My short answer is that 1 and 2 do not express statements, and so are not logically equivalent. The word “therefore” gives away that they express arguments.

Kaiden, they are logically equivalent.  I can prove it using basic propositional symbolic  logic.  If you want me to send the proof, I can write it out.   

By the way, a statement is a sentence that is either true or false.  P then q is a compound statement consisting that connects two statements, P and Q. 

Ok. I see what you’re saying.  But an argument can be expressed as a theorem in which the premises are the antecedent and the conclusion is the consequent.  When expressed as theorems they are logically equivalent.  

Again, Kaiden, I do not disagree that they are two different fallacies; nor do I disagree with anything in your analysis, which was well written.
 
Let me clarify my point, which from the way you write, is something I’m guessing you’re already familiar with.
 
An argument is a series of one or more premises, which are claimed to prove another statement which is the conclusion.   A valid argument is one in which the premises prove the conclusion, whether the premises are true or not.   A valid argument with true premises is usually called “sound”.   So valid arguments can be sound or unsound, depending on whether the premises are true or not.  And unlike the other types of fallacies discussed in this website, formal fallacies, such as the ones we’re discussing here, refer to fallacies in the inference from the premises to the conclusion, not the truth of the premises.
 
Your description of the two fallacies correctly points out that in Affirming the Consequent unlike Commutation of Conditionals, there is a claim that Q is true.  Because of that, it is correct to describe them as different fallacies.  But my point is that it is impossible to find a case in which P and Q have the same truth value (true or false) and one fallacy is committed and not the other.
 
As I indicated in my other posting, every valid argument can be expressed as a theorem in the form of a conditional in which a conjunction of the premises of the argument is the antecedent of the conditional, and the conclusion of the argument is the consequent of the conditional.  I doubt if you’d disagree, but I support the point by citing the logic textbook,  Modern Logic by Graeme Forbes pp. 100-101.  He calls an argument a “syntactic sequent”, in which the premises are said to entail the conclusion.  He then goes on to say “for every provable sequent there is a corresponding theorem with embedded conditionals that have the various premises of the sequent as antecedents.”  He concludes that if we have a syntactic sequent (i.e., argument), and (if we label the premises p1. p2. p3 etc. and the conclusion q), then it is straightforward to prove a theorem which is a conditional consisting of the premises of the sequent p1. p2. p3 etc. as the antecedent and the conclusion of the sequent, q, as the consequent.
 
As indicated, the theorems for these two fallacies are equivalent which can be proven using symbolic logic.

[To Ed F]

Ok. I see what you’re saying. 

[quoted from your previous comment]

Good, I’m glad it struck you in the third reply. 

But my point is that it is impossible to find a case in which P and Q have the same truth value (true or false) and one fallacy is committed and not the other. 

It is possible to find a case, which my Answer to Trevor explains. And I finished by presenting some cases, in fact.

As indicated, the theorems for these two fallacies are equivalent which can be proven using symbolic logic.

There is  confusion in your posts about what a logical theorem is. But my short answer is that the fallacies are not expressible as logical theorems, let alone logically equivalent ones. A conditional statement whose antecedent is the conjunction of the premises of one of the fallacious arguments and whose consequent is the conclusion of it is a contingent conditional statement. And a contingent statement is not a theorem. 

Anyways, fallacies cannot be logically equivalent because they are not statements. You are trying to say that two statements have a relationship of logical equivalence. But next you stretch that to the fallacious arguments that the statements were built from. The relationship of logical equivalence doesn’t stretch like that. If you find that two statements are logically equivalent, that is fine and that is as far as that relationship applies. You don’t thereby find that two fallacies are logically equivalent, too, for fallacies are not statements. 

Since we’re talking about formal fallacies, you’re right in pointing out that precision in wording is important, and so below I’ll re-state my position to be precise in the ways you’re addressing.
 
As I previously indicated, I don’t disagree that they’re different fallacies, since only Affirming the Consequent makes the claim that Q is true.  What I mean when I say they’re logically equivalent is that if P and Q are assigned specific truth values, then the conclusion will always have the same truth value whether the argument is made in the form of Affirming the Consequent or in the form of Commutation of Conditionals.   They would not be the same fallacy but the truth table for both fallacies would be identical.
 
So in your example:
 
Affirming the Consequent
1. If it is garbage day (P), then my neighbor has his garbage bin on the end of the driveway (Q) 
2. My neighbor has his garbage bin on the end of the driveway (Q) 
3. Therefore, it is garbage day (P) 

Commutation of Conditional
1.         If it is garbage day (P), then my neighbor has his garbage bin on the end of the driveway (Q) 
2.         If my neighbor has his garbage bin on the end of the driveway (Q), then it is garbage day (P).

An argument is invalid if it’s possible for the premises to be all true and the conclusion false.   In both cases, the only time the premises are all true and the conclusion is false is if it is not garbage day (not P) and the neighbor has his garage bin on the end of the driveway (Q).  

So that’s what I mean in saying they’re logically equivalent. 

Based on your point, which I acknowledge, I will re-word this quote:

it is impossible to find a case in which P and Q have the same truth value (true or false) and one fallacy is committed and not the other. 
 
To:

it is impossible to find a case in which P and Q have the same truth value (true or false) and the conclusion is true (or false) in one case and not the other. 
 
So you’re right—they’re different fallacies, but what I meant in saying they’re logically equivalent is that the result as to the truth or falsity of the conclusion is the same no matter whether the argument is stated in the form of Affirming the Consequent or Commutation of Conditional.

[To Ed F]

it is impossible to find a case in which P and Q have the same truth value (true or false) and the conclusion is true (or false) in one case and not the other.  

To see why this is mistaken, let me recycle my example. Let’s look at the two arguments below. 

1. If it is my birthday, then my family will bake a cake tonight
2. My family will bake a cake tonight
3. Therefore, it is my birthday 

The arguments commits the fallacy of affirming the consequent. Next is an argument committing the fallacy of the commutation of conditionals.

1. If it is my birthday, then my family will bake a cake tonight
2. Therefore, if my family will bake a cake tonight, then it is my birthday

Suppose it is false that it is my birthday and false that my family will bake a cake tonight. In which case, the P’s and Q’s of the arguments have the same truth value (they are both false.) However, the conclusion of the first argument is false and the conclusion of the second argument is true.

If you have your truth tables in front of you and correctly filled out, you recognize it that way, too. The row in which P and Q are both false is a row in which P and Q have the same truth-value, and P (the conclusion of the first argument) is false, while Q—>P (the conclusion of the second argument) is true. 

Hi Kaiden.  This has been an interesting exchange.  I agree with you in part and disagree with you in part, and for the part where I agree with you, I thank you for correcting me. 

My explanation of why the fallacies are equivalent was incorrect for the reasons you point out in your last posting.  You are correct.  I tip my hat. 

However, even though my explanation was wrong, I still contend that the two fallacies are logically equivalent (as expressed as theorems) since they have the same truth table, and as explained below, make the same error in thinking even though they’re stated differently and thus are different fallacies. 

This goes to the definition of validity and invalidity in deductive arguments. 

An argument is invalid if it’s possible for the premises to be true and the conclusion false.   If it’s not possible for the premises to be true and the conclusion false, then the argument is valid.   

In both argument forms, the premises are true and the conclusion is false if and only if - P is false and Q is true. (in your example, “if my birthday “ is false but “my family will bake a cake” is true.).

The underlying error in deductive logic is the same.   In Affirming the Consequent there is a claim that the premises are true, including that Q is true. but the validity of the argument depends not on the truth of the premises but whether the premises support the conclusion. Stated another way, if the premises are true, must the conclusion follow.?  That’s analogous to If A then B in a conditional. 

Affirming the Consequent is a fallacy because if “If P then Q” were true and if “Q”were true, it is still possible for P to be false.  In Commutation if Conditionals, we start with “ P then Q” but conclude that if Q is true, P is true. 

In both fallacies we start with P then Q.   Then, the question is—if Q is also true, does P follow?   That is the issue whether Q is listed as a separate premise as in Affirming the Consequent or the antecedent of the conclusion, as in Commutation of Conditionals.  Thus the fallacious reasoning in both is the same even though they’re different fallacies. 

[To Ed F]

I still contend that the two fallacies are logically equivalent (as expressed as theorems) since they have the same truth table,

As I said, logical equivalence is a relationship between statements and fallacies are not statements, nor are statements fallacious. Two statements built from the fallacious arguments might be logically equivalent, but not the fallacies. That is the shortest and simplest reply and it is independent of the rest of my post. And the fallacies cannot be expressed as theorems (a minor point that I also said.)

But let me stop repeating myself and answer this new argument that you defend your position with.

In both fallacies we start with P then Q.   Then, the question is—if Q is also true, does P follow?   That is the issue whether Q is listed as a separate premise as in Affirming the Consequent or the antecedent of the conclusion, as in Commutation of Conditionals.  

And from this you conclude: 

Thus the fallacious reasoning in both is the same even though they’re different fallacies. 

Two fresh examples will be instructive for seeing what is wrong with concluding that. Consider the two formalized arguments.

1. P—>Q
2. P
3. Therefore, Q

And 

1. P—>Q
2. Therefore, P—>Q

It seems, though correct me if I’m wrong, that you would argue about these two arguments that “We start with P—>Q. Then, the question is—if P is also true, does Q follow? That is the issue whether P is listed as a separate premise as in the first argument or the antecedent of the conclusion, as in the second argument. Thus, [you would conclude] the reasoning of both these two arguments is the same.”

But, the reasoning of both arguments is not the same. For the second argument reasons circularly while the first does not reason circularly. And reasoning circularly is not the same reasoning as reasoning non-circularly. The lesson should now be obvious. It does not follow that two arguments reason the same just because they share a conditional and differ only with respect to whether the next variable appears as a second premise or as the antecedent of the conclusion. 

Sorry for this long posting, which is necessary in order to properly respond to Kaiden’s last posting and to correct some sloppiness in my prior postings that Kaiden keenly and correctly pointed out. 

Initially, Kaiden objected to the use of the term “logically equivalent” in referring to arguments.  So as not to get caught up with semantics, the expression “logically identical reasoning” can be substituted for “logIcally equivalent” in the following, particularly in reference to whether arguments can be said to be logically equivalent. 

1.  Affirming the Consequent and Commutation of Conditionals are two different fallacies. 

On that point we are in complete agreement.  The former fallacy has two premises whereas the second fallacy has only one.  Also, the second fallacy has a conditional as the conclusion whereas the first has only the antecedent of the first premise as the conclusion.   So because the form of the fallacies are different, they are, and should be considered different fallacies.   

2.  Every Deductive Argument Can Be Expressed As A Statement 

To properly respond to Kaiden’s last posting, I need to digress a little into the nature of deductive logic systems.    My main source for what follows is the textbook Modern Logic by Graeme Forbes, but the explanations (and any deficiencies) are mine except for the part quoted.   

There are numerous formal logic systems; many textbooks and university courses employ different symbols and rules for determining validity and often use different terminology  (I’m limiting my discussion to Propositional and one-place Predicate Logic).   The word “statement”, for example, (meaning a sentence that is either true or false), is often called a “formula”, “sequent”, or simply “sentence.”   

The term “theorem” is a statement or formula that is always true no matter the truth values of its components (I used the word “theorem” when I meant “conditional” in some of my prior postings). 

A logic system starts with various symbols, including symbols for names, statements, connectives, variables and predicates.   A proper logical system needs to be able to determine logical truths; in other words, be able to prove that theorems and only theorems are necessarily true.   

There are two types of methods used for determining validity: semantic validity and syntactic validity.   Semantic validity determines truth based on what the statement actually expresses, and does so via a truth table rather than by deduction using rules of logic.   If rules of logic are added to the system, then validity can also be shown by putting the statement into the form of a corresponding argument and determining whether the conclusion can be proven from the premises.  This is called “syntactic validity”, where derivations are used to determine validity. 

A logic system that is fully functional to determine truth will provide for both semantic and syntactic validity.  If the system, including its rules of logic, can syntactically derive the corresponding argument of any theorem, then the logic system is said to be complete.  If the system allows derivations only of semantically valid statements (I.e., theorems), then the system is said to be sound.   Except for obscure theoretical systems, all logic systems of the types described, including those found in introductory logic textbooks, strive to be complete and sound.  The discussion herein assumes that the system is  complete and sound. 

The point of the above is that every argument can be expressed as a corresponding semantic statement.   If the argument is valid, it can be proven with a derivation and the corresponding statement can be proven to be a theorem using truth tables. 

3.  The Form of the Corresponding Statement To An Argument Is A Conditional 

The corresponding semantic statement for a given argument is in the form of a conditional, with the premises of the argument as the antecedent and the conclusion as the consequent. 

A conditional statement is an assertion that if the antecedent is true, then the  consequent is true.   It is not an assertion about whether the antecedent and conclusion are in fact true, but rather an assertion that if the antecedent is true, the conclusion cannot be false. 

An argument does the same thing.   To say an argument is valid is to say—if the premises are true, then the conclusion must be true (emphasis on the word “if”).   So in this respect the truth value of a conditional statement is analogous to the truth analysis of the corresponding argument. 

4.  Converting Arguments To Conditional Statements 

The next issue is the exact form of the conditional corresponding to an argument. 

The most straightforward form, as stated in a prior posting, is a conditional written as a conjunction of the premises as the antecedent and the conclusion of the argument as the consequent.   So an argument of the form P1, P2 therefore C would be written as 

(P1 & P2) → C.    

In the Kalish, Montague, & Mar textbook- Techniques of Formal Reasoning- for example, logic rules such as Modes Ponens, are listed both as rules in their argument form and as theorems in the conjunction of premises form: 

((P →Q) & P) →Q). 

There is another way in which arguments can be reduced to conditionals.   The other form uses a rule of Logic called Exportation.   It says that these two statements are equivalent: 

(P & Q) → R  and  P → (Q → R).    

So an argument of the form P1, P2 therefore C , could be written in the conjunction of premises form as 

(P1 & P2) → C 

or its logical equivalent by Exportation, 

P1 →(P2 →C).   See for example, https://en.wikipedia.org/wiki/Exportation_(logic)

This is actually the way Forbes translates arguments into conditionals.   Per Forbes, “for every provable (syntactic) sequent there is a corresponding theorem with embedded conditionals that have the various premises of the sequent as antecedents; in general, if we have constructed a proof of P1……therefore Q then it is straightforward to give a proof of P1 → (P2 → (Pn → Q)).”

In English, what Forbes is saying is that every argument can be re-stated as a statement of the form “if premise 1 is true, then if premise 2 is true, then if (etc for more premises), then the conclusion is true.   

5.  Converting The Two Fallacies To Conditional Statements

In the case of Affirming the Consequent, the corresponding conditional statement using the conjunction of premises form  is ((P → Q) & Q)→ P.   Forbes would write it using exportation, in which case Affirming the Consequent would be (P →Q) →(Q →P)  (look familiar?). 

As previously stated, the Rule of Exportation tells us that 

(A & B) →C is logically equivalent to A →(B →C).   If we substitute P →Q for A, Q for B, and P for C, then we get ((P →Q) & Q) →P is logically equivalent to (P →Q) →(Q →P). 

In other words, if we start with an argument in the form of Affirming the Consequent, the corresponding semantic formula is actually Commutation of Conditionals. 

6.  The Claim of Fallacious Circular Reasoning Does Not Apply

Kaiden in his last posting used the example of Modus Ponens where, he pointed out that using my approach, if we have an argument:

P →Q.   P  therefore Q

we would derive from this:

P →Q   therefore P →Q. 

He then claims this is faulty circular reasoning, since the conclusion is the same as the premise. 

Exportation, as previously stated, tells us that a statement of the form (A & B) →C is logically equivalent to A →(B →C).   In Modus Ponens, also called Affirming the Antecedent, we again substitute P →Q for A, but this time substitute P for B and Q for C. The result is (P →Q) → (P →Q).   He argues that this is circular reasoning.  But we did not get to this conclusion by repeating the premise, but rather by the rule of Exportation.  And the only reason the conclusion is the same as the premise is because he chose the one case of Affirming the Consequent where the antecedent of the first premise was also the second premise, so of course the resulting conclusion would be the same as the premise. 

Moreover, Circular Reasoning is not fallacious in Deductive Logic.   It may make the argument trivial or unpersuasive, but not fallacious.  To the contrary, the rule “P therefore P” is a fundamental rule in almost all deductive logic systems, called “Repetition“.

Circular Reasoning is a fallacy in Inductive Logic, where the concern is not just whether the premises support the conclusion, but whether the premises are probably true.  If the conclusion is merely a restatement of the premises, then the premises are no more probable than the conclusion and thus results in the fallacy of circular reasoning.  .   

Conclusion 

2 Key Points:

1.  Every argument can be restated as its corresponding semantic formula in the form of a conditional with the premises as the antecedent and the consequent as the conclusion. 

2. The two corresponding conditionals for statements in the form of Affirming the Consequent and Commutation of Conditionals are logically equivalent.  In fact, using the Rule of Exportation, the two conditionals are the very same expression: (P →Q) →(Q →P). 

Both fallacy forms start with P then Q.  They both go on to say that if Q is also true, then so is P.   In Affirming the Consequent, the “if Q” is premise 2.  In Commutation of Conditionals, the “if Q” is the antecedent of the conclusion.  They sound different because we don’t usually think of arguments as being conditionals, but they are saying exactly the same thing in different forms.    They both fallaciously claim in their own way that the combination of  P→ Q and Q is sufficient for P. 

[To Ed F]

I skimmed this and focused on your concluding two key points. I appreciate the better accuracy and precision of the two key points. I would tweak a few things, but I have better plans today. But let me leave with corrections on criticisms that I saw you make of my post.

Kaiden in his last posting used the example of Modus Ponens where, he pointed out that using my approach, if we have an argument:  P →Q.   P  therefore Q  we would derive from this:  P →Q   therefore P →Q. 

I did not point out that by using your approach we would derive the second argument from the first. I simply posted the two argument forms, one after another. After posting them, what I did do is point out that they have a similar first premise and differ with respect to the location of the second P, point out that the second argument reasons circularly and the first does not, point out that an argument reasoning circularly is not reasoning the same way as an argument reasoning non-circularly, and conclude that two arguments do not reason the same just because they have a similar first premise and differ with respect to the location of the second P.

He then claims this is faulty circular reasoning, since the conclusion is the same as the premise. 

I did not claim that deriving the second argument from the first, or applying exportation to the sentence form of the first argument to get the sentence form of the second argument, is faulty circularly reasoning. First, because I didn’t derive the second argument from the first and didn’t say or do anything concerning the sentence forms of the arguments or the application of exportation. Second, because I did not at any point state that circular reasoning is faulty or fallacious. What I did call circular is the argument from the premise P—>Q to the conclusion P—>Q.

That’s all. From what I saw, good work on the two key points.

In a nutshell, the argument committing the fallacy of affirming the consequent does not state that Q is sufficient for P, whereas the argument committing the fallacy of the commutation of conditionals does. The argument committing the fallacy of the commutation of conditionals does not state that the consequent is true nor that the antecedent is true, whereas the argument committing the fallacy of the affirming the consequent does. 

This is an important observation - I was hitherto ignorant of the distinction between the two fallacies. Thanks, Kaiden.

[To Rationalissimus of the Elenchus]

It is true that Affirming the Consequent makes a claim that Q is true.  But I think it's important to remember that the fallacy is in the form of the argument, not whether the statements are true or false.  

So Affirming the Consequent (P →Q, Q  Therefore P) is a fallacy whether Q is true or not---the conclusion (P) doesn't follow from the premises, so to claim it does is a fallacy.  That's why Affirming the Consequent and Commutation of Conditionals are logically equivalent.

[To Ed F]

I don’t intend to be nosy, but your reply to Rationalissimus makes me think that I should clarify myself. I am not saying that Q’s veracity or falsity matters to whether the argument is fallacious. Rather, I am explaining that affirming the consequent is a different fallacy from the commutation of conditionals because it affirms the consequent of the conditional that they share in common—affirming it through having  a premise stating Q .

I don’t disagree that they’re different fallacies.  But they’re logically equivalent.  As mentioned in my other comment, this can be proven using propositional symbolic logic.  

You’re welcome, ROTE.

It is important to remember that in a deductive argument, determining validity (and assessing for fallacies) does not depend on whether the premises are true or false but whether the premises entail the conclusion.  So it is not correct to say in Commutation of Conditionals that there is a claim that the conclusion is in fact true (Q is sufficient for P).  Rather, the claim is that if P is sufficient for Q, then Q is sufficient for P.  Similarly in Affirming the Consequent, although Q is a premise  and P is the conclusion, the claim of the argument is:  if P is sufficient for Q and if Q were also true, then P would be true.  

Here are the argument forms again:

Affirming the Consequent says that if P →Q is true (p1) and if Q is true (p2), then P is true (c). 

Commutation of Conditionals says that if P → Q is true (p1), then if Q is true (antecedent of (c)), then P is true (consequent of (c)). 

As you can see if you take away the parts in parentheses, these two statements are virtually identical (the only difference is changing the word “and” to “then”).   As I explain in great detail in my lengthy response to Kaiden’s last posting, there is a rule of Logic called Exportation that says these two statements say the same thing, even though the forms are different.      

This is really helpful. Thanks Jorge.

[To Trevor Folley]

No problem! I think others already answered your question. Just wanted to give my take. I'm glad it helped!

Many thanks for this. I can't see the reply icon for your answer so apologies for responding through the comments.

Are you able to provide an example of when the affirming the consequent fallacy has been committed but not the commutation of conditionals fallacy?

and/or

An example of when the commutation of conditionals fallacy has been comitted but not the affirming the consequent fallacy.

I'm interested in the distinction between them.

Many thanks again.