(These are excerpts from my book "Intelligence is not Artificial")
Footnote: Logic and Probability
Unifying probability theory (useful to describe a world that is fundamentally uncertain) and mathematical logic (that works pretty well at describing relationships between the objects of the world) is an old ambition.
In 1946 Richard Cox, a physicist at Johns Hopkins University, proved a theorem that was widely believed to have shown that probability theory extends mathematical logic beyond the realm of true and false.
Another physicist, Edwin Jaynes at Washington University in St Louis, interpretated probability theory as an extension of logic in his influential book "Probability Theory" (which he started writing in 1952).
Later, Artificial Intelligence scientists, faced with the desire to save both logic and probability theory, attempted practical fusions of the two forms of reasoning, for example Fahiem Bacchus at the University of Alberta in Canada ("Representing and Reasoning with Probabilistic Knowledge", 1988) and Joseph Halpern at IBM's Almaden Research Center in San Jose ("An Analysis of First-Order Logics of Probability", 1989).
Perhaps the most intriguing attempt at unifying logic and probability at the dawn of deep learning was the work on "Markov Logic Networks" by Pedro Domingos and his student Matt Richardson at the University of Washington ("Markov Logic Networks", 2006), later expanded by Domingos' student Jue Wang as "Hybrid Markov Logic Networks" (2008). These representations combine first-order logic and Markov networks and use the Markov Chain Monte Carlo method for (approximate) inference.
However, Stuart Russell at UC Berkeley pointed out that a marriage of logic and probability requires that we think of objects as uncertain too ("Unifying Logic and Probability", 2014). One thing is to tell a machine that there are apples in the world and asking the machine to recognize apples, and another thing is to ask the machine what objects it sees in the world: the machines sees pixels, billions of pixels. There is a "difference between knowing all the objects in advance and inferring their existence and identity from observation".
Finally, David Chapman showed that Cox and Jaynes fundamentally misunderstood what logic is. Probability theory is an extension of propositional calculus but propositional calculus is not logic: it says nothing about objects. Logic starts with predicate calculus, which can describe relationships among objects, and logic can do things that probability theory cannot do ("Probability Theory does not Extend Logic", 2016).
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