The application of logic to knowledge seems problematic

Question

I was reading Dretske's text on 'Is knowledge closed under known entailment?' and I saw him using the material conditional while claiming entailment. But, in my head these two seem different. Since entailment is semantic and implication is syntactic. Of course, in logic, where every model is sound and complete, they seem to validify the same things.

However, if we have a model of knowledge, is it necessarily also complete? By which I mean, that if we have some statement A that entails B, does that mean the same as A infers B? To me it feels like, because we have a weak definition of knowledge, we try to fit some structure which gives us a proof theoretic framework which doesn't really fit. Meaning that perhaps knowledge doesn't behave in this proof theoretical way.

Or more generally, why can we assume that knowledge abides the proof theoretic laws of logic? Don't we need to make a model with axioms, relations, constants etc? I read Tarski who tried to solve the inconsistency problem of language by adding a metalanguage so that we can proof if an object language is consistent (Theory of truth). Is there something done like this for epistemology?

I tried to look this specific problem up in epistemology but could not find anything that discusses this question. Any help on this problem of mine would be helpful! I'd also like to hear if you think this problem is a misinterpretation of entailment and inference!

TL;DR I am looking for some papers or philosophers who have discussed the question if knowledge follows the rule of logic, in the sense that when we look at logical arguments applied to knowledge they can imply different results than when we are thinking about what knowledge in that statement means.

Answer 1

There is epistemic logic, for a more general example. With regards to the issue of A → B, then with a knowledge operator k, we have to distinguish between (among other things):

1. (kA & (A → B)) → kB
2. (kA & k(A → B)) → kB

Per the SEP article on epistemic closure, it seems that Dretske's position is the rejection of such closure: "Dretske made his point in the form of a rhetorical question: “... how is one supposed to get closure on something when every way of getting, extending and preserving it is open (2003: 113–4)?” To go back to the other linked-to SEP article, from its section on the problem of "logical omniscience" (the result of holding (1) to be true, and perhaps (2) as well (though in a different way/to a different extent)):

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(Addendum) erotetic logic and the logic of conditionals: the SEP articles on the logic of conditionals closes with a mention of erotetic logic:

Conditionals do not only appear in declarative sentences, but also in questions... In the case of questions, Isaacs and Rawlins (2008) treat questions as prefixed with a covert speech act operator, coming with a modal domain restricted by the if-clause. In a related way, Ciardelli, Groenendijk, and Roelofsen (2018) point out that the semantics of a question can be specified by resolution conditions relative to an information state. Again, conditional questions can be viewed as appropriately restricting the information state to the antecedent worlds, so as to resolve the question relative to those (see Ciardelli, forthcoming). ... Further connections have been explored between the semantics of interrogatives and the very structure of conditional sentences. Starr (2014b) in particular argues that all conditionals have a topic-comment structure, allowing to treat the antecedent as a covert question itself [bold emphasis added].

And, the closed/open-question dichotomy notwithstanding, it seems as if the process of inquiry is intrinsically in tension with closure, since e.g., "What is the first natural number after 2?" has the same answer as, "What is the first prime number after 2?" but the latter can't, as a question, be strictly deduced from the former by itself, but we must expand our semantics to the notion of prime numbers more specifically, and so on and on.

Answer 2

The application of logic to knowledge seems problematic

Sure it is. Logic comes from logos (shared concept among philosophy and religion); the enclosed concept of reasoning is only a part of it: teleology is fundamental in the way we think, operare; in nature too. Knowledge is an experiential property (awareness), the imprint of logos in our consciousness. Logic (as commonly used) is a self/common agreed-upon blueprint of what we perceive as reality, while reality is veiled behind logos.

Answer 3

Your question touches on the foundations of logic and epistemology, particularly the distinction between material implication and logical implication, and how knowledge conforms (or not) to the laws of proof theory.

First, let's distinguish between material implication and logical implication:

• Material Implication: This is a formal relationship between two propositions, where "A implies B" is true except when A is true and B is false.
• Logical Implication: This is a semantic relationship where A logically implies B if and only if in all models where A is true, B is also true.

Regarding the completeness of knowledge models:

• Ideally, a knowledge model should be both complete and coherent. However, human knowledge is often imperfect and partial.
• In epistemology, the formal implication that A implies B doesn't necessarily mean A deduces B in a practical or contextual sense.

Now, consider epistemological models:

• Epistemic Modal Logic: This branch of formal logic studies the modalities of knowledge and belief, providing richer frameworks than simple material implication.
• Model Theory: As Tarski's work demonstrates, this provides tools to understand the coherence and completeness of knowledge systems.

In conclusion, knowledge goes beyond simple logical formulas. More sophisticated models are needed for a comprehensive understanding. Philosophers like Edmund Gettier have shown that knowledge is not just justified true belief. To delve deeper into these issues, you might explore works in epistemic modal logic and model theory.

Answer 4

"why can we assume that knowledge abides the proof theoretic laws of logic?"

There are indeed many forms of knowledge that are not strictly speaking modeled well by grammatical constructions (see Knowledge How (SEP)). But for knowledge-that, it is expressed by natural language which can be modeled by formal semantics. Such proof theoretic laws have for more than 100 years been very productive at producing and automating the production of knowledge. So, to engage in a little wordplay, the proof is in the pudding.

To make clear, we are not assuming the reasoning in natural language can be conducted in formal systems that handle proof-theoretic and model-theoretic construction of knowledge; rather we have justified and thus know it by its efficacy. Consider one exhibit where such epistemic automation is productive: automated theorem proving. From the article:

First-order theorem proving is one of the most mature subfields of automated theorem proving. The logic is expressive enough to allow the specification of arbitrary problems, often in a reasonably natural and intuitive way. On the other hand, it is still semi-decidable, and a number of sound and complete calculi have been developed, enabling fully automated systems.[20] More expressive logics, such as higher-order logics, allow the convenient expression of a wider range of problems than first-order logic, but theorem proving for these logics is less well developed.

Therefore, many aspects of modern computation and logic rely on what with a broad brush be referred to as "epistemic automation". It is a fact that these systems produce reliable knowledge in the same way people do, albeit much more quickly and reliably but without any comprehensive understanding of the knowledge they process.

Answer 5

I took an engineering course called Knowledge-Based Systems back in 1990. These are also known as expert systems, but I think expert systems include implicit knowledge stored in artificial or biological neural networks. Knowledge-based systems are based on explicit knowledge and logic and are known to be brittle:

https://ksi.cpsc.ucalgary.ca/KAW/KAW96/compton/compton.html

This pdf is a slide presentation of expert systems where the system is a "black box" to the user:

https://cse.hkust.edu.hk/~dekai/600G/notes/KM_Slides_Ch08.pdf

Human expert knowledge does not reduce to closed logic systems and this attribute of human knowledge is called heuristic.

The movie I Robot () dramatizes some of the issues with knowledge as an attribute of so-called artificial and human intelligence. This link is a summary of Isaac Asimov's Three Laws of Robotics:

https://webhome.auburn.edu/~vestmon/robotics.html

1. A robot may not injure a human being or, through inaction, allow a human being to come to harm. 2. A robot must obey orders given it by human beings except where such orders would conflict with the First Law. 3. A robot must protect its own existence as long as such protection does not conflict with the First or Second Law.

How does a Knowledge-Based System, programmed with explicit knowledge and logic, avoid injuring a human being through action or inaction? How does a neural network system, with implicit brain activity, learn to recognize what it means to prevent harm to human beings through action or inaction of the robot system? In common law an animal or child cannot understand the meaning of harm to another person, and a child without the capacity to govern action by the use of reason, cannot act to avoid or prevent harm to another person. There is a logic to harm and benefit of others in the dramatic context but can we map that logic to an explicit knowledge-based system? So far I would characterize such efforts as Mission Impossible!

I think IBM Watson, the computer system that won Jeopardy, was a Knowledge-Based System with a Bayesian inference engine to rank possible answers as better or worse in the context of the question. But it may have had elements of large language model I have not done a study of the system engineering. Human knowledge is open in the general context and even human expert knowledge does not seem to be closed under any system of human logic.