# Do syllogistic and propositional formal fallacies overlap?

I am trying to find an exhaustive list of formal fallacies in propositional logic. So far, it seems to be only:

1. Affirming the consequent (basically incorrect application of MP)
2. Negating the antecedent (incorrect MT)
3. Affirming a disjunct (incorrect DS)

Everything else seems to just fall under the umbrella of "non-sequitur".

However, I recall the fallacy of the undistributed middle term from syllogistic logic.

All A are B
All C are B
Therefore, all A are C.


To me this looks like "bad hypothetical syllogism".

A -> B
C -> B
Therefore, A -> C


Is it okay then to classify an argument that commits this type of fallacy in propositional logic as "undistributed middle term"? Or is this reserved specifically for syllogistic logic since propositional logic does not deal with distribution?

• Trying to understand the nature of the question. Do you want to find "all" the possible logical fallacies in "propositional logic"? And then see which fallacies in syllogistic logic (the logic described in this link en.wikipedia.org/wiki/Syllogism ?) don't translate to a fallacy in propositional logic? Commented Dec 25, 2019 at 3:51
• As it happens, I can list "all" possible "fallacies" (including fallacies that don't have a "name" to describe them) in propositional logic, and make your work kinda easy in that setting. But it seems by the nature of Syllogistic logic (as far as I understand it from just looking it up), and particularly your first example, that you're interested in quantifiers, which propositional logic of course does not capture, though predicate logic does. However, trying to classify "all" fallacies in predicate logic (besides looking individually at the ones with a name) I'm pretty sure is a lost cause. Commented Dec 25, 2019 at 4:03
• syllogism is a term introduced by Aristotle which allows deducting a particular truth from a universal one. like "all men are bearded and I am a man, therefore I must be bearded". your example is not a syllogism. Commented Dec 25, 2019 at 5:02
• You cannot conflate the two, because syllogistic logic is (monadic) predicate logic: the concept of "term" is not available in propositional logic. Commented Dec 25, 2019 at 8:02
• @DarkLogician It would be sufficient to have a list of all the named formal fallacies in PL. The reason I asked if the two categories of fallacies overlapped is because sometimes they seem to do so (like in the example), and I was wondering if I could just re-use the names from syllogistic logic in terms of fallacies in propositional logic.
– Matt
Commented Dec 25, 2019 at 17:28

I have never seen any textbook classify that last propositional logic argument as 'undistributed middle' ... but FWIW I personally do label it as such :)

Also, given the following two valid inferences:

Strengthening the Antecedent

$$P \to R$$

$$\therefore (P \land Q) \to R$$

Weakening the Consequent

$$P \to R$$

$$\therefore P \to (Q \lor R)$$

You could consider the following two patterns as propositional logic fallacies (though I must say I have never had the occasion to point them out at any time in my experience):

Weakening the Antecedent

$$P \to R$$

$$\therefore (P \lor Q) \to R$$

Strengthening the Consequent

$$P \to R$$

$$\therefore P \to (Q \land R)$$

Oh, and one more:

Another valid inference is:

Exclusion

$$\neg (P \land Q)$$

$$P$$

$$\therefore \neg Q$$

But its invalid counterpart would be:

$$\neg (P \land Q)$$

$$\neg P$$

$$\therefore Q$$

... not sure what to call this one ... 'False Inclusion'?

• That is awesome. I think I have seen the last one of your examples being called "Negating a Conjunct". But yeah, I would love to have a list (as exhaustive as one can be) of all the fallacies in propositional logic.
– Matt
Commented Dec 25, 2019 at 17:30

Bram28 did the question justice and deserves the check mark but I'm going to answer the question from a more "creative" perspective that I hope everyone enjoys.

From a logic perspective, I see a "fallacy" in an argument as any kind of "false inference"; in other words, a conclusion $$B$$ from a collection of statements $$\Gamma$$ that isn't provable in classic logic. In other words, \begin{align*} \Gamma \\ \therefore B, \end{align*} when $$\Gamma \nvdash B$$.

If $$\Gamma:=\{A_1, \ldots, A_n\}$$ is finite, this is equivalent, using the deduction theorem and completeness theorem, to $$(A_1 \land \ldots \land A_n) \rightarrow B$$ not being a tautology, meaning that $$\not \models (A_1 \land \ldots \land A_n) \rightarrow B$$--i.e. there exists a truth table $$T$$ such that $$(A_1 \land \ldots \land A_n) \rightarrow B$$ is false.

So a logical fallacy can be equivalently be defined any pair $$(\Gamma, B)$$, with $$\Gamma$$ finite, such that $$(A_1 \land \ldots \land A_n) \rightarrow B$$ is not a tautology.

Turns out that such a collection is "decideable" (in order words, we can write a computer program that classifies whether or not $$(\Gamma, B)$$ is a logical fallacy), since the tautologies in propositional logic (the complement of what I'm calling a "logical fallacy") are decideable.

However, with most forms of predicate logic, and likely a similar issue arises with Syllogistic logic since that logic uses quantifiers as well, we are not so fortunate and the "logical fallacies" in those end up being "undecideable". But we can formalize "logical fallacies" using the same definition. We're just not able to compute them all.

I have a little more to write about how we can also look at logical fallacies from the perspective of inferring one connective from another connective (a generalized version of what Bram28 was doing), but will do so in an edit of this post. Thanks for reading and I appreciate feedback!