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I came across a comment recently "reads like sentences strung together with no logic." But is this even possible?

Sentences can be strung together randomly if the selection process is random. (Random sentences in a random sequence.) Stochasticity does not seem logical—it's a probability distribution, not based on sequence or causality.

but

That stochastic process is part of an algorithm, which is a set of instructions that must be valid for the program to compute.

So which is it?

  • Is randomness anti-logical?



Some definitions of computational logic:

The arrangement of circuit elements (as in a computer) needed for computation also: the circuits themselves Merriam Websters A system or set of principles underlying the arrangements of elements in a computer or electronic device so as to perform a specified task. Logical operations collectively. Google Dictionary The system or principles underlying the representation of logical operations. Logical operations collectively, as performed by electronic or other devices.
Oxford English Dictionary

Some definitions of randomness

Being or relating to a set or to an element of a set each of whose elements has equal probability of occurrence. Lacking a definite plan, purpose, or pattern. Merriam Websters. Made, done, happening, or chosen without method or conscious decision. Google Dictionary Having no definite aim or purpose; not sent or guided in a particular direction; made, done, occurring, etc., without method. Seeming to be without purpose or direct relationship to a stimulus. Oxford English Dictionary

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    $\begingroup$ A simple counterexample to your statement would be an error-proof logical circuit built out of noisy components. $\endgroup$ – naive Aug 16 at 6:39
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I might misunderstand your question, but there seem to be different levels of logic at play here.

  1. Computing logic, whereby any computational process is based on processor logic. In this case, any computing is involving logic, as boolean logic drives any processing.

  2. Linguistic logic, where there is a logic in the sequencing of sentences within a text. A random collection of sentences is not a text, as there need to be certain principles behind the structure to make it a narrative.

While you can easily generate a sequence of random sentences, they will not mean anything; there won't be any logic behind selecting a particular sentence to follow on from another one. So this is linguistic logic rather than processing logic. Note that where the linguistic logic is makes it a bit vague: I can read a randomly selected sequence of sentences and ascribe meaning to it by building a mental model that treats it as a logically constructed text. This principle is what made ELIZA so successful: even though the program's answers were based on simple pattern matching rules with no understanding, many users assumed there was logic/meaning behind it and interpreted it as such, papering over the cracks in the conversation.

In summary: there is logic involved in random sentence combining, but it is the low-level computing logic, not the higher-level linguistic interpretative logic, which is generally absent from randomly generated data.

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I think the answer here lies in that the dictionary definition of randomness you have is not the one used in statistics, ML, or mathematics. We define randomness to mean there exists a distribution with generally greater than 0 uncertainty.

Depending on who you talk to, we live in a random universe (the way we define quantum mechanics depends on a wave function (essentially a probability distribution)

So why if a sequence is drawn from a distribution is it illogical? First, even as humans we can make a strong argument that what we say is random. I mean we speak to convey some form of message or context, but there exists multiple ways to deliver this, but we choose a single one. Our brains inherently model $p(\vec w|c)$ where $\vec w$ is the sequence and $c$ is our context or message we want to convey.

Takeaway: Generating a sequence in an ergodic or uniform manner would be illogical, but that is not what is being modeled or done in practice. Normally its drawn from some complex distribution.

Sidenote: My above claim could make it seem that being uniformly random implicates something illogical, and I want to emphasize that is not the case. It is domain to domain, sometimes that is the most logical solution, just in the case of sentence generation it normally isnt. I would define a logical algorithm as one that given the information at hand acts in a sensible manner towards achieving some goal, and so if something purely random does that, I don't see the problem.

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  • $\begingroup$ Good answer. I intentionally restricted my definitions largely to computational logic. There has been some inquiry into the role of quantum phenomenon (probability distributions in general) and "mind", so I'm pondering the link and role of stochasticity. For instance, can there truly be choice in a purely deterministic system? (Yes and no?) $\endgroup$ – DukeZhou Aug 2 at 17:19
  • $\begingroup$ For instance, two deterministic algorithms can be playing a non-trivial abstract game. Where the decision making process is of the competitor is unknown, indeterminacy arises. Yet the decision making process is fully deterministic, so the algorithms don't have "choice" in the sense that the decisions will be the same for a given game state. $\endgroup$ – DukeZhou Aug 2 at 17:26
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In certain games, random selection is the optimal strategy. See: Matching Pennies

Strategy is essentially a plan of action utilized to achieve a goal.

  • If random choice can be a strategy, it seems that it must be a form of logic, even if the nature of the stochastic process is counter to all forms of formal logic.

This seems paradoxical, in that the random strategy is to have no strategy (random choices.)

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    $\begingroup$ As I mentioned in my answer being purely random is not illogical in certain cases— where are you seeing the paradox to formal logic? $\endgroup$ – mshlis Aug 2 at 17:11
  • $\begingroup$ @mshlis it seems like stochasticity has no role in formal logic, but is utilized in computational logic. I've been doing some linguistic and semantic research, trying to get at the roots of these concepts. $\endgroup$ – DukeZhou Aug 2 at 17:20
  • $\begingroup$ why do you say no role? Where is that coming from? $\endgroup$ – mshlis Aug 2 at 17:21
  • $\begingroup$ @mshlis Formal logic seems to involve truth statements and inference. Randomness seems to throw a wrench into these gears. $\endgroup$ – DukeZhou Aug 2 at 17:23
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    $\begingroup$ distribution in leads to distribution out. Only when you observe do you get a definitive output (similar to collapse in quantum mech) $\endgroup$ – mshlis Aug 2 at 17:30
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Previous answers are very well written. I just wanted to supplement the thread by giving a simple example. The example shows how a logical function can be computed without errors using noisy components.

Taken verbatim from Neural Networks by Raul Rojas. An excellent book: enter image description here

an example of a network built using four units. Assume that the first three units connected directly to the three bits of input $x_1, x_2, x_3$ all fire with probability $1$ when the total excitation is greater than or equal to the threshold $\theta$ but also with probability $p$ when it is $\theta − 1$. The duplicated connections add redundancy to the transmitted bit, but in such a way that all three units fire with probability one when the three bits are $1$. Each unit also fires with probability $p$ if two out of three inputs are $1$. However each unit reacts to a different combination. The last unit, finally, is also noisy and fires any time the three units in the first level fire and also with probability $p$ when two of them fire. Since, in the first level, at most one unit fires when just two inputs are set to $1$, the third unit will only fire when all three inputs are $1$. This makes the logical circuit, the AND function of three inputs, built out of unreliable components error-proof.

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