Page:Mind (New Series) Volume 9.djvu/89

VI.-SYMBOLIC REASONING (III.). 1 Br HUGH MACCOLL. FBOM some helpful criticisms which I owe to different logicians I gather that my recent contributions to this magazine contain certain obscurities ; and the editor has kindly placed a few of its pages at my disposal in order to dispel them. First comes my assertion (MiND, January, 1899) that a statement, no matter how high its degree, can be spoken of, not only as true or false, certain or uncertain, but also as probable or improbable within more or less exact limits according to our data. A difficulty appears to be felt as to the exact meaning I should attach to such a statement as, for example, A* 1 '", and espe- cially as to how I could, under any conceivable circumstances, express the exact chance of its truth or falsehood. Before giving any concrete examples by way of illustration, it will be as well, and perhaps not unamusing, to translate the above symbolic statement into ordinary language. The symbol A is short for A T , and may be read " A is true " ; the exponent T being often left understood, like the sign + in common algebra. The symbol A 11 may be read " It is im- possible that A is true ". The symbol A 1 ' may be read "It is false that it is impossible that A is true "; which may be abbreviated into " It is possible that A is true," or, more conveniently still, into "A is possible". The symbol A 19 " may be read " It is certain that it is false that it is impossible that A is true"; which may be abbreviated into "A is certainly possible". The symbol A""" may be read "It is certain that it is certain that it is false that it is impossible that A is true " ; which may be abbreviated into " It is. certain that A is certainly possible ". Probably no reader at least no English reader, born and brought up in England can go through the full unabbre- viated translation of this symbolic statement A 1 "'" into ordi- 1 (For II. see MIND, October, 1897.)