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## Modal Logic Notes

https://www.youtube.com/watch?v=sW0SQA386p0 "A Quick Introduction to Modal Logic, part 1" by eric pacuit.

History at 1:17:

Mathematical Modal Logic: a View of its Evolution. Robert Goldblatt. Centre for Logic, Language and Computation. Victoria University, P. O. Box 600, Wellington, New Zealand.

p26: Prior conceived the idea of using a logical system with temporal operators analogous to those of modal logic

p39: the algebraic study by Robert Bull, a student of Arthur Prior,38 of logics characterised by linearly ordered structures. Prior had observed that the Diodorean temporal reading of â·Î± as âÎ± is and always will be trueâ leads, on intuitive grounds, to a logic that includes S4 but not S5. ... In effect he was dealing with the complex closure algebra Cm(Ï, â¤), where Ï = {0, 1, 2, . . .} is the set of natural numbers viewed as a sequence of moments of time ... Prior called this logic D.

Sir Michael Anthony Eardley Dummett, FBA (/ËdÊmÉªt/; 27 June 1925 â 27 December 2011) was an English academic described as "among the most significant British philosophers of the last century and a leading campaigner for racial tolerance and equality."[5] He was, until 1992, Wykeham Professor of Logic at the University of Oxford. He wrote on the history of analytic philosophy, notably as an interpreter of Frege, and made original contributions particularly in the philosophies of mathematics, logic, language and metaphysics. He was known for his work on truth and meaning and their implications to debates between realism and anti-realism, a term he helped to popularize. He devised the Quota Borda system of proportional voting, based on the Borda count. In mathematical logic, he developed an intermediate logic, already studied by Kurt GÃ¶del: the GÃ¶delâDummett logic. https://en.wikipedia.org/wiki/Michael_Dummett Dummett retired in 1992 and was knighted in 1999 for "services to philosophy and to racial justice". He received the Lakatos Award in the philosophy of science in 1994.

R. A. Bull subsequently proved that a certain infinite class of modal logics between S4 and S5, including S4.2 and S4.3, all have the finite model property ('A Note on the Modal Calculi S4.2 and S4.3', Zeitschrift fuer mathematische Logik und Grundlagen der Mathematik, vol 10, 1964, pp53-55.) -- Michael Dummet, Truth and other enigmas.

â·(p â q) â (â·p â â·q) ---- where â· is necessity (or box)

S5 false for my temporal logic: S5 is S1 plus â¸p => â·â¸p (diamond p implies box diamond p)

U+25C7 WHITE DIAMOND (U+25C7) ( ◇ ) #9671

U+25A2 WHITE SQUARE WITH ROUNDED CORNERS (U+25A2) ( ▢ ) #9634

'WHITE MEDIUM SQUARE' (U+25FB) ( ◻ ) #9723

U+25EF LARGE CIRCLE (U+25EF) ( ◯ ) #9711

U+25CB WHITE CIRCLE (U+25CB) ( ○ ) #9675

U+26AA MEDIUM WHITE CIRLE ( ⚪ ) #9898

U+2192 RIGHTWARD ARROW ( → ) #8594

LOGICAL AND ( ∧ ) #8743

LOGICAL OR ( ∨ ) #8744

'NOT SIGN' (U+00AC) ( ¬ ) #172

▢(⚪R → ◇R)

◻(T = ¬F)