Science: the Fallacy of an Undistributed Middle Term

QUEST. Is the fallacy of affirming the consequent a type of inductive reasoning; or is inductive a type of the fallacy of affirming the consequent; or are the two completely unrelated? Induction is defined as arguing from a particular to a universal.
Affirming the consequent: P ⊃ Q; Q; ∴ P.

ANS. Affirming the Consequent and inductive reasoning are similar or comparable, if we define inductive reasoning as “having more information in the conclusion than what the premises contain.”

In essence, the informal fallacy called, “non-sequitur” – “does not logically follow from the premises”—is what all inductive reasoning is.

Deduction: Conclusion has information only contained in the premises.

Induction: Conclusion has new additional information the premises do not contain.

For example

E1. All [things that comes to pass] are [determined by God]. B is C
E2. [Man’s moral acts] are [things which come to pass]. A is B
E3. Thus, [man’s moral acts] are [determined by God], & [not responsible]. A is C & D

The conclusion “man’s moral acts are determined by God,” is obviously already contained in the original premise, “All that comes to pass are determined by God.” If all things are determined by God, then so is man. Simple enough. However, the term “not responsible” and the necessary connection to it are not in the premises. This the essence of all inductive reasoning, it a non-sequitur.

As for affirming the consequent, depending on the terms and its simplicity many of them can be interchanged with categorical logic. Be forewarned not all can be interchanged like this.  It needs to be a simple,  If A then B is C. (Example, “If A is B, then C is D,” type of arguments will not work. 

The thing to remember is if one does truth tables in Natural Deduction, one will see that the simple forms (modus ponens, modus tollens) do not become invalid with complexity (for example with multiple conjunctions). Thus, the key is to master the basic forms, and realize they will continue to be valid, even in complexity, long as one keeps the form. Since scientific experimentation uses the form of affirming the consequent, and denies theory’s with a modus tollens, all one needs to do is understand these basics. Also, keep in mind, basic propositional logic like modus ponens, focus on the necessary connections, while basic category logic will focus on necessary category realities. If you have one, because these are “necessary,” then you have the other, but they are not the exact same thing. 

This simple modus ponens is stating the B and C terms, the third term, which is missing is an implied fill-in-the-black, ‘A’ subject.

If a mammal, then warm blooded. (B is C)
Is a mammal. ( B )
Thus, warm blooded. ( C )

The argument is based on the presupposition that mammals are warm-blooded (B is C) is a given truth.

M.1 If [Bats] are [mammals], then they [warm-blooded].  A, (B is C)
M.2. [Bats] are [mammals]. A is B
M.3. Thus, they [Warm-blooded]. A is C.

Even though the first line of this Modus Ponens, M.1., has all three terms (A is B is C), the main emphasis is that B is C, like the major premise of a Category Syllogism. Next, M.2. is A is B, which is similar to the minor premise of a Category Syllogism. Finally, the conclusion is A is C.

B is C
A is B
Thus, A is C.

This Modus Ponens is hypothetical in form only. The essence of this argument is the comprehension and extension of the terms, not mainly about the necessary connection from B to C. Thus, we will put this into a bullseye syllogism.

N.1. All [Mammals] are [Warm-blooded]. B is C.
N.2. All [Bats] are [Mammals]. A is B
N.3. Thus, All [Bats] are [Warm-blooded]. A is C.

Now, let us review Affirming the Consequent, which is the structure for scientific experimentation. We will use a simple enough form that it can be used in categorial logic.

H.1. If [Jack] eats [lots of bread], then his [belly gets full]. A, (B is C)
H.2. [Jack’s] [belly got full].  A is C
H.3. Thus, [Jack] ate [lots of bread] A is B

B is C
A is C
Thus A is B.

This of course is a fallacy. It could be that Jack ate lots of durian rather than bread. Let us put this into categorical logic to see the fallacy.

Y.1. All [who eat lots of bread] are [those who belly’s get full]. B is C
Y.2. All [Jack] is [he who belly got full]. A is C
Y.3. Thus, [Jack] is [He who ate lots of bread]. Thus, A is B

If you noticed, the information in the conclusion has more than what the premises provide. This is the fallacy of an undistributed middle term. The picture below will help show a visual of this logical fallacy.

Thus, the fallacy of scientific experimentation, if restated in a category fallacy, is the fallacy of an undistributed middle term.