Editing Warren Sturgis McCulloch (section)

=== Mathematical logic ===
In 1919 he began to work mainly on mathematical logic, and by 1923 he attempted to make a logic of [[transitive verb]]s. His goal in psychology is to invent a "psychon" or "least psychic event" that are binary atomic events with necessary causes, such that they can be combined to create complex logical propositions concerning their antecedents. He noticed in 1929 that these may correspond to the all-or-nothing firings of neurons in the brain.<ref name="number" />

In the 1943 paper, they described how memories can be formed by a neural network with loops in it, or alterable synapses. These then encodes for sentences like "There was some x such that x was a ψ" or <math>(\exists x) (\psi x)</math>, and showed that looped neural networks can encode all [[First-order logic|first-order logic with equality]] and conversely, any looped neural networks is equivalent to a sentence in first-order logic with equality, thus showing that they are equivalent in logical expressiveness.<ref name="number" />

The 1943 paper describes neural networks operating over time, and [[Universal quantification|logical universals]] -- "there exists" and "for all"—for spatial objects, such as geometric figures, was further developed in their 1947 paper.<ref>{{Cite journal |last1=Pitts |first1=Walter |last2=McCulloch |first2=Warren S. |date=1947-09-01 |title=How we know universals the perception of auditory and visual forms |url=https://link.springer.com/article/10.1007/bf02478291 |journal=The Bulletin of Mathematical Biophysics |language=en |volume=9 |issue=3 |pages=127–147 |doi=10.1007/BF02478291 |pmid=20262674 |issn=1522-9602}}</ref>

He worked with [[Manuel Blum]] in studying how a neural network can be "logically stable", that is, can implement a boolean function even if the activation thresholds of individual neurons are varied.<ref>Blum, Manuel. "Properties of a neuron with many inputs." ''Bionics Symposium: Living Prototypes--the Key to New Technology, 13-14-15 September 1960''. WADD technical report, 60-600. (1961)</ref>{{Pg|page=64}} They were inspired by the problem of how the brain can perform the same functions, such as breathing, under influence of [[caffeine]] or [[Ethanol|alcohol]], which shifts the activation threshold over the entire brain.<ref name="number" />

He worked on triadic relations, an extension of the [[Algebraic logic|calculus of relations]] to handle relations that relates 3 objects, such as "A gives B to C" or "A perceives B to be C". He was convinced that such a logic is necessary for understanding brain activity.<ref name=":0" /><ref name=":1">{{Cite journal |last=Arbib |first=Michael A |date=2000 |title=Warren McCulloch's Search for the Logic of the Nervous System |url=https://muse.jhu.edu/pub/1/article/46496 |journal=Perspectives in Biology and Medicine |volume=43 |issue=2 |pages=193–216 |doi=10.1353/pbm.2000.0001 |issn=1529-8795}}</ref>