| 12822 | Proper or improper part: x < y, 'x is (a) part of y' [Simons] |
| Full Idea: A 'proper or improper part' is expressed by 'x < y', read as 'x is (a) part of y'. The relatively minor deviation from normal usage (of including an improper part, i.e. the whole thing) is warranted by its algebraical convenience. | |
| From: Peter Simons (Parts [1987], 1.1.02) | |
| A reaction: Including an improper part (i.e. the whole thing) is not, Simons points out, uncontroversial, because the part being 'equal' to the whole is read as being 'identical' to the whole, which Simons is unwilling to accept. |
| 12823 | Overlap: two parts overlap iff they have a part in common, expressed as 'x o y' [Simons] |
| Full Idea: Two parts 'overlap' mereologically if and only if they have a part in common, expressed by 'x o y', read as 'x overlaps y'. Overlapping is reflexive and symmetric but not transitive. | |
| From: Peter Simons (Parts [1987], 1.1.03) | |
| A reaction: Simons points out that we are uncomfortable with overlapping (as in overlapping national boundaries), because we seem to like conceptual boundaries. We avoid overlap even in ordering primary colour terms, by having a no-man's-land. |
| 12824 | Disjoint: two individuals are disjoint iff they do not overlap, written 'x | y' [Simons] |
| Full Idea: Two individuals are 'disjoint' mereologically if and only if they do not overlap, expressed by 'x | y', read as 'x is disjoint from y'. Disjointedness is symmetric. | |
| From: Peter Simons (Parts [1987], 1.1.04) |
| 12825 | Product: the product of two individuals is the sum of all of their overlaps, written 'x · y' [Simons] |
| Full Idea: For two overlapping individuals their 'product' is the individual which is part of both and such that any common part of both is part of it, expressed by 'x · y', read as 'the product of x and y'. | |
| From: Peter Simons (Parts [1987], 1.1.05) | |
| A reaction: That is, the 'product' is the sum of any common parts between two individuals. In set theory all sets intersect at the null set, but mereology usually avoids the 'null individual'. |
| 12826 | Sum: the sum of individuals is what is overlapped if either of them are, written 'x + y' [Simons] |
| Full Idea: The 'sum' of two individuals is that individual which something overlaps iff it overlaps at least one of x and y, expressed by 'x + y', read as 'the sum of x and y'. It is central to classical extensional mereologies that any two individuals have a sum. | |
| From: Peter Simons (Parts [1987], 1.1.06) | |
| A reaction: This rather technical definition (defining an individual by the possibility of it being overlapped) does not always coincide with the smallest individual containing them both. |
| 12827 | Difference: the difference of individuals is the remainder of an overlap, written 'x - y' [Simons] |
| Full Idea: The 'difference' of two individuals is the largest individual contained in x which has no part in common with y, expressed by 'x - y', read as 'the difference of x and y'. | |
| From: Peter Simons (Parts [1987], 1.1.07) |
| 12828 | General sum: the sum of objects satisfying some predicate, written σx(Fx) [Simons] |
| Full Idea: The 'general sum' of all objects satisfying a certain predicate is denoted by a variable-binding operator, expressed by 'σx(Fx)', read as 'the sum of objects satisfying F'. | |
| From: Peter Simons (Parts [1987], 1.1.08) | |
| A reaction: This, it seems, is introduced to restrict some infinite classes which aspire to be sums. |
| 12829 | General product: the nucleus of all objects satisfying a predicate, written πx(Fx) [Simons] |
| Full Idea: The 'general product' or 'nucleus' of all objects satisfying a certain predicate is denoted by a variable-binding operator, expressed by 'πx(Fx)', read as 'the product of objects satisfying F'. | |
| From: Peter Simons (Parts [1987], 1.1.08) | |
| A reaction: See Idea 12825 for 'product'. 'Nucleus' is a helpful word here. Thought: is the general product a candidate for a formal definition of essence? It would be a sortal essence - roughly, what all beetles have in common, just by being beetles. |
| 12830 | Universe: the mereological sum of all objects whatever, written 'U' [Simons] |
| Full Idea: The 'Universe' in mereology is the sum of all objects whatever, a unique individual of which all individuals are part. This is denoted by 'U'. Strictly, there can be no 'empty Universe', since the Universe is not a container, but the whole filling. | |
| From: Peter Simons (Parts [1987], 1.1.09) | |
| A reaction: This, of course, contrasts with set theory, which cannot have a set of all sets. At the lower end, set theory does have a null set, while mereology has no null individual. See David Lewis on combining the two theories. |
| 12831 | Atom: an individual with no proper parts, written 'At x' [Simons] |
| Full Idea: An 'atom' in mereology is an individual with no proper parts. We shall use the expression 'At x' to mean 'x is an atom'. | |
| From: Peter Simons (Parts [1987], 1.1.11) | |
| A reaction: Note that 'part' in standard mereology includes improper parts, so every object has at least one part, namely itself. |
| 12844 | Dissective: stuff is dissective if parts of the stuff are always the stuff [Simons] |
| Full Idea: Water is said not to be 'dissective', since there are parts of any quantity of water which are not water. | |
| From: Peter Simons (Parts [1987], 4.2) | |
| A reaction: This won't seem to do for any physical matter, but presumably parts of numbers are always numbers. |