De Morgan & Laws of Symbolic Transformation

Augustus De Morgan (1806 – 1871), who is most famous for early work on symbolic algebraic logic and for formulating his De Morgan’s laws, which we can express as neither A and B is equivalent to not A nor not B, ~ (A ^ B) = (~ A v ~ B), and not A or B is equivalent to not A and not B, ~ (A v B) = (~ A ^ ~ B), that strangely if we invert AND and OR it changes the relationship to NOT in basic combinations.  De Morgan was born in Madurai, India to a Lieutenant Colonel serving the British East India Company, the same of Mill’s father and Mill.

Speaking of fancy French words, such as Saint-Simon’s socialisme, lieutenant literally means place-holder, a minor officer, lieu–tenant, as in in lieu of, in place of, and tenant, who occupies a place until someone else takes it.  A colonel, pronounced kernel by English speakers, like popcorn, leads a column, a portion of a large marching army, much as a brigadier leads a brigade, a group of warriors of various size.

Augustus moved to England when he was seven months old.  His mother wanted him to become an Anglican minister, but De Morgan became a mathematician and outspoken atheist.  Attending Trinity College at 16, he became student and friend to Whewell, Mill’s opponent, and with him De Morgan got to work renovating algebra and logic to give each rigorous theoretical systematic foundations, to explain both using the fewest assumptions and ideas as possible to make them clear and useful.

At the time, Oxford and Cambridge, the two great English universities, required theological oaths that barred Jews and non-Anglican Christians from teaching.  Libertine-minded academics and others decided to establish London University in 1826 as a secular alternative. De Morgan was hired to teach mathematics, and he became a leading member of the London Mathematical Society and the Society for the Diffusion of Useful Knowledge.  In 1866, when a Unitarian minister was denied the chair of mental philosophy, De Morgan resigned, as he had before, in protest over lack of religious neutrality, as even though he was an atheist he was against the chair being given or withheld due to the religion or lack thereof of the holder.

In a letter, De Morgan’s friend Sir William Rowan Hamilton joked with him that his copy of Berkeley’s work, the Irish Bishop and idealist philosopher, was not his, because, like Berkeley and De Morgan, he is himself an Irishman, presumably who can’t afford his own books while holding a noble title. De Morgan wrote back that Hamilton’s book was certainly not his, but Berkeley’s, of course, but it is alright for Hamilton to use the word mine either way, to mean a book he doesn’t own or to mean a book he didn’t write, but to use the word both ways at once is not, much like the Irishman who was hauling up a rope, and after awhile with no end in sight, cried out that someone had clearly cut off the other end of it.

This fits with Nicholas of Cusa’s observation, treasured by Hegel, that we somehow know a circle is infinite, and goes on as a path forever just as it is in front of us, but we have to come to the conclusion, as we do when we are old enough, that there is no end at the end, and cut off our expectations of a cut off.  It is more than two clever, witty Irishmen mocking their fellow countrymen for being poor and stupid in clever ways. Both lived through the Irish Potato Famine of the 1840s outside Ireland. As the joke goes: How many potatoes does it take to kill an Irishman?  None.

De Morgan argued that symbols do not have inherent meaning, and we can use them as we like, using addition to mean reward and subtraction punishment, or virtue and vice, and those who hear us can believe us or contradict us as they please, but we can use these terms and symbols in regular ways if we agree to together.  Lewis Carroll, who read and followed De Morgan’s work, wrote similarly in his work on logic that anyone can use the word white to mean black and the word black to mean white, and Carroll doesn’t personally care one bit as long as the author lets us know what they mean by which words up front.

De Morgan centered everything on the equals sign, which in both algebra and logic signifies that what sits on each end has the same numerical or truth value, numerical for algebra and true or false for logic.  De Morgan published his Formal Logic in 1847, after correspondence with Hamilton in Dublin and Boole in Cork.  Aristotle and those who followed his syllogistic forms of logic argued that nothing follows from the combination of Some A is B and Some A is C, as we don’t know if some B is C or anything else necessarily from this, but De Morgan showed that if we know Most A is B and Most A is C, we also know necessarily Some B is C.  If we know that there were a thousand people on a ship, that most of them, over 500, were at the bar, and the ship sank, and most were lost, we also know that some who were at the bar had to be some of those lost to the sea.  De Morgan showed that a logic of relationships could replace Aristotle’s syllogisms, which led the way to Boole’s logical connectives, which relate two terms together.

In De Morgan’s later Syllabus Of A Proposed System Of Logic, he begins with Aristotle’s four forms of proposition, and proceeds to show that, while we do not know what we input into his systematic working of strings of syllogisms is true, the whole has to be true if all of the terms are, just as Boole and Carroll say in their works on logic, with similar language.