A late-night journey into formal logic

I’ve just worked 10 hours in Emergency, on one of our busiest days ever, with a record number of patients presenting. It’s 2am, and I’m completely exhausted.
And yet, there is this little problem that is keeping me awake.
You see, in formal logic, there are different types of arguments being made. We call them syllogisms if they are presented in a certain form. The ones that are of relevance to this post are the categorical, and the hypothetical, syllogism.
This is the form a hypothetical syllogism takes in formal logic, and we have to distinguish 2 subtypes :

1. Type 1, pure hypothetical syllogism

If P, then Q.
If Q, then R.
Therefore, if P, then R.

2. Type 2, mixed HS

If P, then Q.
Therefore Q.

This one is called modus ponens.There is another variety of it that is also valid, modus tollens :

If P, then Q.
Not Q.
Therefore, not P.

There are 2 forms of the hypothetical syllogism that are invalid, either affirming the consequent, or denying the antecedent, of the hypothetical syllogism :

If P, then Q.
Therefore P.

Example of affirmation of the consequent :

If Bacon wrote Hamlet, then Bacon was a great writer.
Bacon was a great writer.
Therefore, Bacon wrote Hamlet.

This is a fallacious argument. So is denying the antecedent :

If P, then Q.
Not P.
Therefore, not Q.


If I rob a bank, I am guilty of a felony.
I did not rob a bank.
Therefore, I’m not guilty of a felony.

Is invalid.

Now to categorical syllogisms in classical, deductive logic(ignoring the immediate inferences, and symbolisms, for the purpose of this post). There are 4 main forms :

A proposition : universal affirmative, “All S is P”, only S is distributed
E proposition : universal negative, “No S is P”, S and P are distributed
I proposition : particular affirmative, “Some S is P”, neither S or P are distributed
O proposition : particular negative, “Some S is not P”, only P is distributed

Categorical syllogisms have a figure and a mood, and are made up of a major, minor, and middle term, and a major and minor premise.
The mood is determined by the types(A,E,I,O) of propositions the syllogism contains, and is represented by three letters, first letter representing the major premise, second letter the minor premise, and third letter the conclusion.The major term is the predicate term of the conclusion, whereas the minor term is the subject term of the conclusion, and the middle term is that which appears in both premises, but not the conclusion.

Here’s an example(via Copi/Cohen) :

No heroes are cowards.
Some soldiers are cowards.
Therefore, some soldiers are not heroes.

This is is a categorical syllogism in EIO mood.(We leave figures out, again that’s past the scope of this post, suffice to say there are only 4 possible different figures, depending on where is the propositions the major, minor, and middle terms are located)

One of the fallacies you can commit here is that of the undistributed middle, it goes like this :

The middle term needs to be distributed in at least one premise, that means, it needs to refer to all members of the class designated by that term.If that’s not the case, the connection required by the conclusion cannot be made.

An example(again from Copi/Cohen) :

All Russians were revolutionists.
All anarchists were revolutionists.
Therefore, all anarchists were Russians.

In this example, “revolutionists” is undistributed in both premises, since they are A propositions, the syllogism, and hence the conclusion, is therefore invalid.

Now, and here’s where I would like some help from my readers.I came across this Wiki article on the fallacy of the undistributed middle, and it compares it to, and calls it similar to, the fallacy of affirmation of the consequent.
Now to me, these are different types of syllogisms, and not at all comparable, and I’d really like to see if any of my readers can shed some light on the matter !

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