Is it philosophically difficult to adequately define “formal logic”?

Question

If so, why? If not, what do you consider a complete definition of “formal logic”?

Answer 1

John MacFarlane has written a whole PhD thesis on this topic which is available online here. It's a very well-known text which looks at the roots of the expression "formal logic" in Kant etc.

Answer 2

As an introduction to the topic of formal logic see the following definition from Logic

Formal logic is the science of deductively valid inferences or logical truths.

Mathematicians do not have problems to work with this definition since more than 100 years. Why should there be any philosophical problems?

There are several questions on this platform which deal with formal logic, e.g., Is formal logic unsuitable for philosophical reasoning?

Because you did not further specify your question, it is not clear what your point is.

Answer 3

Is it philosophically difficult to adequately define “formal logic”?

Philosophically define?! Isn't it good enough to just provide a definition in good English? I would hope so.

Mine is as follows:

Formal logic is the empirical science of logic.

To answer your question, as it turns out, yes, it is really hard. The history of formal logic seems to start with Aristotle... Ah, already we have a problem, because people produced texts which may be categorised as formal logic before Aristotle.

A crucial point about what Aristotle did with respect to how to define formal logic is that Aristotle didn't even try to explain why a perfect syllogism was perfect. Subsequently, logicians adopted the same perspective, presumably without even realising that they were. Imagine someone explaining that it is true that 1 + 1 = 2 and content to leave it at that! That could be called "formal addition", and, in a sense, it would be correct.

Allow me to borrow the definition quoted by Jo Wehler in his answer as it is indeed typical:

Formal logic is the science of deductively valid inferences or logical truths.

This is interesting but it has obvious flaws. What about invalid inferences? Surely, it is the job of formal logic to explain why an invalid inference is invalid.

And science? There is a clear distinction between Einstein's Theory of General Relativity, which is science but only one part of it, even if an essential one, and what Einstein actually did to produce it, which, presumably, is just as essential. Which is really science?

Let's agree that "science" is too broad a term to get anywhere, but, typically, in a science, people produce theories using the scientific method. Logic is obviously a very particular object and the scientific method has to be adapted to it, but there is no science without some scientific method adapted to its object. Or else, you expect to get correct results through sheer luck, or perhaps the power of divination, and this is not quite what we want to call science.

Before researchers seriously engage in science, there is usually a long period during which they just do something else, and this is what logicians have done for the last 2,500 years. They have often said interesting things, but they have also said absurd things. Just to mention one, Bertrand Russell's theory of descriptions which says that the sentence "The king of France is bald" is false if there is no king of France. How could that be a scientific theory?! A theory which says that a statement is false even though the thing of which it is said to be false doesn't even exist. Absurd. And more to the point, anti-scientific. When we say that a sentence is false, we mean that what the sentence says of the subject is false of the subject, and then how could that be possible when there is no subject?

And this is just one example.

So, if formal logic is a science, then what people do at the moment is not formal logic.

Formal logic is often thought of as something much like arithmetic. This is broadly the perspective of Gottfried Wilhelm Leibniz and George Boole. A calculus. However, a calculus is not science. It may be the output of science, but only if there is the method. Leibniz and Boole worked more as mathematicians than scientists. They tried to produce a formal system without really trying to understand what logic was. The result of Boole initial impetus is that today the specialists of what is widely considered to be formal logic prefer to ignore, and even dismiss, any empirical evidence on human logic. Thus, so-called Propositional Logic says that the disjunction (α → β) ∨ (β → α) is true, even though it is empirical data that humans think of it as false. So, if so-called Propositional Logic is formal logic, then formal logic is not an empirical science. That is, it is just mathematics, and then better define it as such.

The alternative is to go back to the scientific method. This would require to conceive of a method adapted to its object, logic, which is arguably a difficult proposition. Formal logic, according to this perspective, would be an empirical science. It would try to produce not a mathematical theory, but an algorithmic model of what the human brain does. This model would just output the valid conclusion of whatever premises we would input. The proper science, then, would be just as much in the model as in the process to arrive at it.

Thus, if formal logic is the science of logic, it has to start with what empirical sciences do, which is observe what its object does. To be able to do that, you have to pay attention to the empirical data. For example, one empirical data is that the disjunction (α → β) ∨ (β → α) is false, not true.

So formal logic can only be the science which tries to understand how logic really works, and to achieve this, it has to understand why logical truths are true.

So, yes, formal logic is a science, but it is the science of logic, and then you have to define logic itself as something which is not formal logic, as is anyway apparent. To define logic itself, you have to understand what it is, and this requires a scientific investigation. So, you can only properly define formal logic as the science of logic which has to define its object. This is as should be. If we could define the object before the science is done, we wouldn't need the science. All we need for now is an ostensive definition of logic, which is possible because we all recognise logic when we see it, just as we recognise the moon when we see it. We know humans recognise logic because the logical truths identified by logicians are self-evident: for example, the modus ponens (α → β) ∧ α ⊢ β. Thus, logic is just the cognitive capacity which makes us capable of recognising that the modus ponens (α → β) ∧ α ⊢ β is true. The question now is, how exactly does it work? To answer this question, we need the science of logic, namely, formal logic.