If we try to represent tossing a coin or a die, or picking a card out of a deck at random, in logic, how should we do it?<\/p>\n
Tossing a coin might look like:<\/p>\n
Toss(coin) \u2192 (Heads or Tails)<\/p><\/blockquote>\n
Tossing a die might be:<\/p>\n
Toss(die) \u2192 (1 or 2 or 3 or 4 or 5 or 6)<\/p><\/blockquote>\n
Picking a card:<\/p>\n
Pick(52 card deck) \u2192 (1\u2663 or 2\u2663 or … or k\u2665)<\/p><\/blockquote>\n
This begs asking, do these statements make sense? For instance look what happens if we try to abstract:<\/p>\n
\u2200x Toss(x)<\/p><\/blockquote>\n
such that ‘Toss’ represents a random selection of the given object.<\/p>\n
But this is weird because Toss is a randomized function and x is not selected randomly in this formula. Perhaps if we added another variable, we could generate the right sort of function:<\/p>\n
\u2200y \u2203x Toss(yx)<\/p><\/blockquote>\n
Then x would be a function of y: we would select x with respect to y. The problem is still that a Toss involves randomness. So this setup is incorrect because treating x as a function of y is not randomized, because y is not random.<\/p>\n
How can we represent randomness in logic?<\/p>\n
As noted, functions alone will not work. Variables and interpreted objects cannot invoke randomness. Perhaps we can modify some part of our logic to accommodate randomness. The connectives for negation and conjunction haven’t anything to do with randomness either.<\/p>\n
But, if we use the game theoretic interpretation of logic, then we can conceive of each quantifier as representing a player in a game. Players can be thought of as acting irrationally or randomly.<\/p>\n
Therefore, let’s introduce a new quantifier: \u042f. \u042f is like the other quantifiers in that it instantiates a variable.<\/p>\n
\n
- \u042fx T(x)<\/li>\n
- Tb<\/li>\n<\/ol>\n
However, \u042f is out of our (or anyone’s) control. It does instantiate variables when it is it’s turn (just like other quantifiers) but it instantiates randomly. So we have three players, Abelard, Eloise and Random (or the Verifier, Falsifier and Randomizer).<\/p>\n
But more is still needed. We need a random selection between specific options, be it between heads and tails, 1-6, cards, numbers, or anything else. One way of doing this would be to create a special domain just for the random choices. \u042f would only instantiate from this domain, and if there are multiple random selections, we will require multiple indexed domains.<\/p>\n
Hence, given Di<\/small><\/sub>(Heads, Tails),
\n\u042fi<\/small><\/sub>x
\nrepresents a coin flip since \u042f randomly instantiates out of the domain containing only Heads and Tails.<\/p>\n(aside:
\nI prefer to use an artifact of Independence Friendly logic, the dependence indicator: a forward slash, \/. The dependence indicator means that the quantifier only depends on those objects, variables, quantifiers or formulas specified. Hence<\/p>\n\u042fx\/(Heads, Tails)<\/p><\/blockquote>\n
means that the variable x is randomly instantiated to Heads or Tails, since the only things that \u042fx is logically aware of are Heads and Tails. Therefore this too represents a coin flip, without having multiple domains.)<\/p>\n
Now that we have an instantiation rule for \u042f we also need a negation rule for it. If some object is not selected at random, then it must have been individually selected. In this case the only other players that could have selected the object are \u2200 and \u2203. Hence the negation rule for \u042f is just like the negation rule for the other quantifiers: negating a quantifier means that a different player is responsible for instantiation of the variable. If neither player is responsible, it can be considered random: ¬\u042fx ↔ (\u2200x or \u2203x). We can leave the basic negation rule for \u2200 and \u2203 the way it is.<\/p>\n
Therefore, given the additions of the new quantifier and domain (or slash notation), we can represent randomness within logic.<\/p>\n
———<\/p>\n
See “Propositional Logics for Three” by Tulenheimo and Venema in Dialogues, Logics And Other Strange Things<\/em> by Cedric Degremont (Editor) College Publications 2008, for a generalized framework for logics with 3 quantifiers. Since the above logic requires either indexed domains or dependence operators, \u042fandom Logic is a bit different, but it is a good discussion.<\/p>\n","protected":false},"excerpt":{"rendered":"
If we try to represent tossing a coin or a die, or picking a card out of a deck at random, in logic, how should we do it? Tossing a coin might look like: Toss(coin) \u2192 (Heads or Tails) Tossing a die might be: Toss(die) \u2192 (1 or 2 or 3 or 4 or 5 or 6) Picking a card: Pick(52 card deck) \u2192 (1\u2663 or 2\u2663 or … or k\u2665) This begs asking, do […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11,16,34],"tags":[251,256,293,124,125],"class_list":["post-2334","post","type-post","status-publish","format-standard","hentry","category-game-theory","category-logic","category-science","tag-independence-friendly-logic","tag-logic","tag-measurement","tag-random","tag-random-logic"],"_links":{"self":[{"href":"https:\/\/www.noahgreenstein.com\/wordpress\/wp-json\/wp\/v2\/posts\/2334","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.noahgreenstein.com\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.noahgreenstein.com\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.noahgreenstein.com\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.noahgreenstein.com\/wordpress\/wp-json\/wp\/v2\/comments?post=2334"}],"version-history":[{"count":0,"href":"https:\/\/www.noahgreenstein.com\/wordpress\/wp-json\/wp\/v2\/posts\/2334\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.noahgreenstein.com\/wordpress\/wp-json\/wp\/v2\/media?parent=2334"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.noahgreenstein.com\/wordpress\/wp-json\/wp\/v2\/categories?post=2334"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.noahgreenstein.com\/wordpress\/wp-json\/wp\/v2\/tags?post=2334"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}