>Arguably Russell saw the limitations of “technical philosophy” more clearly than those who thought of themselves as following in his footsteps. The analytic tradition has produced some great work but too many of its practitioners have conflated rigour with technical argument. I would wager that there is not a single major work of political or moral philosophy which depends on a formal logical proof. What endures of Russell’s logic is of interest only in logic. If Russell wanted to address the problems of real life, he had to leave behind the symbols and numbers that had so captivated him in his youth.
What are the shortcomings of logic? Just incompleteness, or is there more?
For a fun logical proof that proves why you can't use logical proofs for philosophical questions, see the "Tractatus Logico-Philosophicus" by Ludwig Wittgenstein.
Spoiler: logic is inherently limited by language. To quote Russell,
"In order to understand Mr Wittgenstein’s book, it is necessary to realize what is the problem with which he is concerned. In the part of his theory which deals with Symbolism he is concerned with the conditions which would have to be fulfilled by a logically perfect language. There are various problems as regards language. First, there is the problem what actually occurs in our minds when we use language with the intention of meaning something by it; this problem belongs to psychology. Secondly, there is the problem as to what is the relation subsisting between thoughts, words, or sentences, and that which they refer to or mean; this problem belongs to epistemology. Thirdly, there is the problem of using sentences so as to convey truth rather than falsehood; this belongs to the special sciences dealing with the subject-matter of the sentences in question. Fourthly, there is the question: what relation must one fact(such as a sentence) have to another in order to be capable of being a symbol for that other? This last is a logical question, and is the one with which Mr Wittgenstein is concerned. He is concerned with the conditions for accurate Symbolism, i.e. for Symbolism in which a sentence “means” something quite definite. In practice, language is always more or less vague, so that what we assert is never quite precise."
I'd say the major shortcoming of logic is its inability to express uncertainty.
It's well-suited for mathematical proof as practiced, where axioms and definitions are precisely defined, and there is no reliance on empirical observation with potentially noisy data.
However, most of real-life is not as clear-cut. Deriving the truth of a statement may depend on multiple potentially faulty pieces of evidence which must be taken into account together. For this, one needs to assign probabilities.
This is useful even when applied back to mathematics. In practice, mathematicians form conjectures "likely to be true" long before they are formally proven. Additionally, they must narrow the search space in their minds in order to try the most likely avenues of proof, a process we refer to as "creativity".
Even using probability is only one more step towards solving the question of formally codifying general reasoning. We must also consider factors such as use of language and forming concepts (what precisely IS a "chair", after all?), and further aspects which form a basis for human action and which cannot be logically derived, namely our morality and base goals. Not to mention the entire plethora of such questions with which the field of philosophy concerns itself.
(These are the types of questions to which we will need to find some answer if we are ever to construct a useful generally reasoning AI)
Much as classical Newtonian mechanics is a useful approximation of physics at large scales and low speeds, formal logic is a useful approximation of reasoning at high certainty and low flexibility of interpretation.
Indeed! I was speaking somewhat imprecisely - I was referring to logic in the sense of Bertrand Russel's work. I should rather say the major shortcoming of CLASSICAL logic is its inability to express uncertainty.
That being said, there are many flavours of non-classical logic and (paraconsistent, multivalued etc.) but their usage remains scarce outside of work in logic itself. Some intuitionistic, constructivist, and computational logics seem to be gaining popularity, especially in computer-related circles (computer-aided proof, numerical methods etc.)
> Handling uncertainty is difficult regardless; attaching probabilities to judgements did not ensure the social sciences avoided the replication crisis.
Indeed! I'd say that using probabilities is necessary (or at least very useful), but not sufficient for handling uncertainty.
> I'm not sure what you mean by "low flexibility of interpretation": purely logical proofs are supposed to assume nothing about interpretation.
Yes. I found this part hard to phrase. What you are saying is correct.
I meant it in the sense that there are some assumptions made in logic which do not necessarily hold in normal reasoning. As a simple example, classical logic requires no contradictions whereas the average person may hold several contradictory beliefs without going insane (human compartmentalization is there for a reason, after all!). Paraconsistent logics aim to address this. Classical logic also does not take into account the passage of time.
But by "flexibility of interpretation" I meant something like in logic to derive the truth or falsehood of a statement P(X) about some element X, we can only use known facts about X (i.e., previous statements Q, R, S). We pin down a very specific definition which we can interpret any way we want, but we pick axioms Q, R, S to match our interpretation. This is what I meant by the "interpretation of X is inflexible" (poor phrasing). I mean that the properties / axioms are decided at the beginning and do not shift.
However, when we reason in every day life about e.g. chairs, you and I don't start by pinning down an exact definition of what a "chair" is - we assume some shared knowledge and then debate despite starting from different worldviews. During the debate, we might decide to start pinning down a definition of a chair for the purpose of the debate (is it "something that has been created with the explicit purpose of being sat on"? What is "created"? What if I come across a log that I use to sit on every day? What if I take a dining room chair and stick it to the roof so no-one can sit on it?). If you are convincing enough, the way I use the word "chair" in every day life might change. This is what I mean by the "interpretation of the word chair is flexible".
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My own background is in mathematics and programming, with interest in mathematical logic. I'm afraid my philosophical background is rather lacking. I'm sure such concepts have been described in some depth by various philosophers, but I'm not sure which, or I would just reference the relevant concepts by their common name / link the relevant Wikipedia / Standford Encyclopedia of Philosophy articles.
> However, most of real-life is not as clear-cut. Deriving the truth of a statement may depend on multiple potentially faulty pieces of evidence which must be taken into account together. For this, one needs to assign probabilities.
This what the fuzzy people want you to believe. The logicians have a better answer. For this you need more context. E.g in programming you would add types, pre- and post conditions. And not this statement will be 85% true. As the current AI hype is pretending.
I'm all for using types, and pre- and post-conditions where applicable, but I don't see how they would be a useful replacement to the situations in which probabilities would apply. Could you elaborate?
To give an example where I think probabilities would be used: consider a recognition AI that should figure out who someone is. You have a phone, on which you have some photos of its owner, some voice recordings, and some text messages. For each of those, the AI can assign probabilities that e.g. my voice matches the recordings, my face matches the photos, and my writing style matches the texts. Then it could combine these into an aggregate estimate probability that the phone belongs to me.
How would you use types and pre- and post- conditions to solve this problem?
Oh, okay... are you referring to fuzzy logic where statements have a partial truth value?
I'm (mostly) referring to the case where the truth value is either true or false, but where you aren't sure, so you can say "80% probability this is your phone".
There are also cases where truth values aren't as clear cut, which I also mention, such as the question of whether or not something IS a chair. (Is a chair taped to the ceiling still a chair? Is a log I sit on out in the middle of the forest a chair? Etc.)
What are the shortcomings of logic? Just incompleteness, or is there more?