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The statement $X\vDash Y$ means "every assignment to the variables that makes $X$ true also makes $Y$ true." Or to put it another way, "There is no assignment of variables that makes $X$ true but fails to make $Y$ true." Well, there's no assignment of variables that makes $A\land\neg A$ true, so there's certianly no assignment that makes $A\land\neg A$ true ...

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There are many related ways you can mechanise your logic. Deep embedding into one of the well-developed provers such as Isabelle/HOL, Coq or Agda. This is (almost) always possible, but makes using the embedded logic awkward. Shallow embedding into one of the well-developed provers such as Isabelle/HOL, Coq or Agda. This is only possible when the embedded ...

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Most logical systems in use are automated enough in principle. That is, there are algorithms which find the proof of a statement, if such a proof exists. The problem is computational complexity. Such algorithms are very, very, very, very, very slow. We only know efficient algorithms for special, easy cases. More generally, knowledge representation is not ...

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In §3.2 they distinguish between the following four grammatical categories, with some examples: Predicates: Woman, Rich, Beautiful, Bankrupt Relations: LivesAt, HadAnAffairWith, Loves Functions: fatherOf, bestFriendOf, ceoOf Individual constants: maryJones, johnQSmith, tomsHouse In the background they have first-order logic with identity, which means ...

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Because $\neg\forall x\,(\text{Comp}(x) \wedge \text{Study}(x,l))$ means "It is not true that every student is both a computing student and studying logic." In particular, that would be true if there is at least one student who is not a computing student, regardless of whether all computing students do or do not study logic.

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Consider the expression $+(x,\times(y,y))$, which is the FOL way to write $x+(y\cdot y)$. If I substitute $x=3$ and $y=5$ and do the arithmetic over the integers, what do I get? I need to evaluate the expression recursively. In order to evaluate $+(x,\times(y,y))$, I need to evaluate $x$, to evaluate $\times(y,y)$, and then to combine them using the ...

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There are plenty of papers on fuzzy logic, and applications of it. Fuzzy logic models a piece-wise control function, and is regarded as a non-linear controller. Generally, fuzzy logic can approximate any non-linear function if you divided the state-space into enough fuzzy sets. Because it is non-linear, it is generally useful if it is controlling a non-...

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You describe what is called expert system in AI. As such, they are artifacts of articial intelligence research. As far as the Wikipedia definition goes, a reasoning system is a software system that generates conclusions from available knowledge using logical techniques such as deduction and induction, which is vague enough to cover expert systems. Is ...

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You could use an SMT solver inside a counterexample-guided inductive synthesis loop to learn an expression that fits. The paper below does something similar. If you aren't interested in finding the weakest expression, you could save on computational costs by using templates. Albarghouthi, Aws, Isil Dillig, and Arie Gurfinkel. "Maximal specification ...

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In elementary logic, $A \supset B$ is notation for the formula $\neg A \lor B$ (for the implication "$A$ implies $B$"). In other words, $A \supset B$ is a formula; it is true if $A$ is false or if $B$ is true, and false otherwise. https://math.stackexchange.com/q/1106001/14578 How you could have figured this out on your own: Read the entire paper. For ...

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This question is a bit strange. I see it's based on the Wikipedia article, but that article is also a bit strange. I will try to clarify some concepts. First, we have a language that can be used to make claims whose meaning are formally defined. Such a language may allow us to say, for example, that if $x$ is a Person, then $x$ is a Human, which can be done ...

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There is no exhaustive list of reasoning tasks, as the phrase "reasoning task" does not have a precise definition. Don't put too much weight on categorizations. They're a descriptive tool that might help you get a high-level overview of a field, but they're inherently subjective, incomplete, and imperfect. There's tons of work on artificial reasoning, and ...

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The theory is not inconsistent and trivially admits two different models : a first one in which dave is rich but not poor; a second one in which dave is poor but not rich. You're maybe confusing the "empty clause" which contains no literal, thus being always false, with the "true clause" which contains both a literal and its negation, thus being always ...

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