Ramble, Part 53: An Algebra of Logic
Jun. 26th, 2008 01:21 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowThe name most often thought of in connection with symbolic logic is that of George Boole. Justly so: though other writers, at least as far back as Leibniz, had presented one or another symbolic representation of logic, it was Boole who first proposed a strictly algebraic presentation of the subject. Some discussion of his innovations, and of the differences between his version of symbolic logic and the one generally accepted today, is under the cut.
Boole presented his ideas most fully in his An Investigation into the Laws of Thought, published in 1857. In that work, he put forward an algebra involving variables such as x and y and operations which he represented as addition, subtraction, and multiplication. These operations, he proposed, would have certain properties, such as the following:
One very important point is this. Boole insisted that any work with this algebra must be founded on those properties, and not on any specific interpretation of the symbols involved. In particular, he recognized two equally valid interpretations, and saw the value of a presentation that captured what those two interpretations had in common - in a word, of abstraction.
The first interpretation, following in the footsteps of Leibniz and DeMorgan, took the variables as representing classes of things; multiplication becomes what we would now call intersection, and subtraction is relative difference - i.e., x-y is the class of things in class x and not in class y. x+y represents the class consisting of the members of x and the members of y, when those two classes have no members in common. (I'm not entirely sure what it is to mean when the classes do share members; I suspect that it is to comprise those elements which belong to one of the two classes but not to the other.) 1 represents the universal class, of which everything is a member, and 0 the null class, containing no members.
Alternatively, the variables could represent statements. xy is then the assertion that x and y are both true, and x+y that either x or y is true. 1 and 0 become standard true and false propositions, respectively. In this case, each statement x can be assigned a "truth value" - 1 if it is true, 0 if false. The operations have the following property: the truth value of any combined expression depends only on the truth values of the component statements. Thus, xy has truth value 1 if x and y both have truth value 1, and truth value 0 otherwise. That is, the operations are "truth-functional". This will be important later.
Boole's version of symbolic logic differs from the modern version in two significant respects. First, his "addition" operation is (in the interpretation as statements) the so-called "exclusive or": x+y is true if, and only if, one of x and y is true and the other is false. William Jevons, soon afterward, suggested that the "inclusive or" - allowing for the possibility that x and y are both true as well - is more useful, and his arguments carried the day; "or", in logic, is always interpreted as inclusive, unless it is explicitly indicated otherwise. I'll discuss this issue at some length in the next Ramble.
Second, Boole's logic did not explicitly include a representation of the important "if-then" relation. It is possible that he excluded it because it fits only one of the two interpretations - the interpretation as statements - well. It also turns out that this relation can be constructed using Boole's operations; nonetheless, it is extremely convenient to have an overt representation. It was Charles Sanders Pierce, the American philosopher/engineer, who introduced this bit of notation. There are complexities here as well, which I will talk about in another Ramble.
Regardless of the imperfections of Boole's work, it was a giant step forward, in its recognition of the essential similarities of the logic of classes and the logic of statements, in its insistence on the independence of the work from any specific interpretation, and in its purely algebraic character. Much of the subsequent work in logic has been set within the framework Boole established, and it has proven very fruitful. (Among its fruits, in one sense, are the very machines which allow these words to be written and read.)
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Boole presented his ideas most fully in his An Investigation into the Laws of Thought, published in 1857. In that work, he put forward an algebra involving variables such as x and y and operations which he represented as addition, subtraction, and multiplication. These operations, he proposed, would have certain properties, such as the following:
- For all x and y, xy=yx
- For all x and y, x+y=y+x
- For all x, y and z, x(y+z)=xy+xz
- For all x, y and z, x(y-z)=xy-xz
- For all x, x(1-x)=0
One very important point is this. Boole insisted that any work with this algebra must be founded on those properties, and not on any specific interpretation of the symbols involved. In particular, he recognized two equally valid interpretations, and saw the value of a presentation that captured what those two interpretations had in common - in a word, of abstraction.
The first interpretation, following in the footsteps of Leibniz and DeMorgan, took the variables as representing classes of things; multiplication becomes what we would now call intersection, and subtraction is relative difference - i.e., x-y is the class of things in class x and not in class y. x+y represents the class consisting of the members of x and the members of y, when those two classes have no members in common. (I'm not entirely sure what it is to mean when the classes do share members; I suspect that it is to comprise those elements which belong to one of the two classes but not to the other.) 1 represents the universal class, of which everything is a member, and 0 the null class, containing no members.
Alternatively, the variables could represent statements. xy is then the assertion that x and y are both true, and x+y that either x or y is true. 1 and 0 become standard true and false propositions, respectively. In this case, each statement x can be assigned a "truth value" - 1 if it is true, 0 if false. The operations have the following property: the truth value of any combined expression depends only on the truth values of the component statements. Thus, xy has truth value 1 if x and y both have truth value 1, and truth value 0 otherwise. That is, the operations are "truth-functional". This will be important later.
Boole's version of symbolic logic differs from the modern version in two significant respects. First, his "addition" operation is (in the interpretation as statements) the so-called "exclusive or": x+y is true if, and only if, one of x and y is true and the other is false. William Jevons, soon afterward, suggested that the "inclusive or" - allowing for the possibility that x and y are both true as well - is more useful, and his arguments carried the day; "or", in logic, is always interpreted as inclusive, unless it is explicitly indicated otherwise. I'll discuss this issue at some length in the next Ramble.
Second, Boole's logic did not explicitly include a representation of the important "if-then" relation. It is possible that he excluded it because it fits only one of the two interpretations - the interpretation as statements - well. It also turns out that this relation can be constructed using Boole's operations; nonetheless, it is extremely convenient to have an overt representation. It was Charles Sanders Pierce, the American philosopher/engineer, who introduced this bit of notation. There are complexities here as well, which I will talk about in another Ramble.
Regardless of the imperfections of Boole's work, it was a giant step forward, in its recognition of the essential similarities of the logic of classes and the logic of statements, in its insistence on the independence of the work from any specific interpretation, and in its purely algebraic character. Much of the subsequent work in logic has been set within the framework Boole established, and it has proven very fruitful. (Among its fruits, in one sense, are the very machines which allow these words to be written and read.)
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