Logical Fallacies

In my previous post I introduced some valid rules of inference. In this topic I would like to discuss invalid inference and logical fallacies. I will begin by way of an example:

My dog has four legs. Cats have four legs. Therefore, my dog is a cat.
The above syllogism is obviously wrong, since although both the premises are correct, the conclusion is not. This is because the conclusion does not follow from the premises. If one of the premises had been "Everything which has four legs is a cat", the syllogism would be valid. The conclusion would still be false, since this new premise is not actually true, but the inference itself would have been correct. Indeed, if we imagine a world in which all four-legged creatures are cats and my dog was a four-legged creature, in that world he would be a cat. A valid conclusion from the given premises, might instead be "At least one four-legged creature is not a cat". To reiterate, the fallacy in the example above lies in the fact that the conclusion does not follow from the premises. This type of fallacy is known as affirming the consequent and is very common in cases where unidirectional implication is not understood.

A similar fallacy is known as denying the antecedent. An example of that would be as follows:
Cats have four legs. My dog is not a cat. Therefore, my dog does not have four legs.
Once again, this syllogism fails to properly connect the premises to the conclusion logically, so it constitutes a fallacy. And it is also a fallacy created from a lack of understanding of the rules of implication, just like the previous one.

The third most common logical fallacy arises from a misunderstanding of the logical OR relation, also known as "inclusive or" (to distinguish it from XOR, the exclusive or). Although in spoken language we often use the word "or" as an exclusive disjunction, the same is not true in logic. In logic, "p OR q" is a statement which is true when at least one of p and q are true; or even both. A misunderstanding of disjunctions leads to the following fallacy:
My pet is a dog OR it is not a cat. My pet is a dog. Therefore, it is not true that my pet is not a cat, so my pet is a cat.
In fact, my pet can be a dog and not a cat at the same time. This is, in fact, the case. The first premise allows for this situation and does not imply that only one statement is true.

The above are propositional fallacies and, as discussed, they arise from a misunderstanding of logical operators, like implication and disjunction. There are also other types of fallacies, which come about from misunderstanding the rules of inference themselves, the scope of the premises or the conclusion, the words and language used and so on. I will not discuss these more general fallacies for now, as they are abundantly available online and also because they are beyond the scope of what I am discussing in this post.