On probability logics.
Rolf Schock · Notre Dame Journal of Formal Logic · 1965
Our language contains the following symbols:(1) the (individual) variables ζ υ 1 9 , ζ v 2 9 , and so on;(2) the sentential connectives ' N ' ('not'), ' -' ('only if'), ' Λ ' ('and'), ' v ' ('or'), and ' ++' ('if and only if');(3) the variable binders ' 1' ('the'), 'P' ('the probability that any -is a .. .'),'Q' ('the probability that any -which is a ... is a ---'), ' Λ ' ('for all'), and ' V ' ('for some');(4) the individual constants Ό 9 , Ί', 'c 3 ', 'c 4 ', and so on;(5) the I-place operation symbols '-' ('minus'), Όl', c θ\\ and so on;(6) the 2-place operation symbols '+' ('plus'), '.' ('times'), '-' ('minus'), '/' ('divided by'), 'Ί' ('to the power'), ' Γ ' ('the -th non-negative root of), Όγ', '0l' f and so on;(7) the 3-place operation symbols Of', *0f', and so on; and so on for any greater number of places;(8) the i-place predicates 'R' ('is a real number'), 'N' ('is a positive integer'), ζ p\\ 'P\' 9 and so on;(9) the 2-place predicates Ί' ('is identical with'), 'α' ('is less than'), Φ 2 3 ' ζ P\\ and soon; and (10) the 3-place predicates ζ Pf 9 , ( Pl\ an d so on; and so on for any greater number of places.We use the symbols '' and '{', '}' in the metalanguage to mark the boundaries of non-empty finite sequences and sets respectively.The letter t m 9 will be used as a metalinguistic variable ranging over positive integers.Terms and formulas will be understood as follows:(1) all variables and individual constants are terms;(2) for any m-place operation symbol o and ra-term sequence of terms t, is a term;(3) for any variable υ and formulas/ andgy, , and are terms;(4) for any m -place predicate p and ra-term sequence of terms t, is a formula;(5) for any formulas / and g, , ' g >, , , and ' g > are formulas; and