Logic – Mill, De Morgan & Boole
Mill, Logic, Gathering & Dividing
John Stuart Mill (1806 – 1873) was educated by his father and Jeremy Bentham to be a genius for progress, and he is one of the most influential libertine thinkers in British history, who argued for the abolition of slavery, women’s liberation, individual rights, and freedom as the goal of modern enlightened society. He is known best and hated by some for Utilitarianism, the ethical philosophy that what is best overall is what should be done regardless of tradition or previous prejudice as to what was rational or logical.
Mill studied Greek, Latin and French as a boy, and then studied logic, chemistry and biology in France, where he met Henri Saint-Simon the socialist, who argued for meritocracy that benefited the working classes before Marx and Engels said that and much more in the name of communism, a more radical form of socialism. While Saint-Simon said society should be based on the maxim, From each according to ability, to each according to productivity, cutting off those on the top and bottom of society who do not contribute to the good of everyone, Marx and Engels argued the maxim should be, From each according to ability, to each according to need. His father wrote a history of India, and Mill was for a time involved with his father in the British East India company, the corporation that helped Britain maintain their economic hold over India. It was from Jeremy Bentham whom Mill learned about Epicurus of ancient Greece, who taught that the goal of individual and social life was not law or morality, but happiness.
Bentham called his philosophy Consequentialism, the progressive position that morals, laws and principles are merely tools for the obtainment of collective human happiness. However, it was Mill who found the word utilitarian in a Christian text that used the term negatively. The more conservative author of the text said that we should not be “merely utilitarian” in our actions, following principles only when they lead to happiness. Mill applied this progressive model of thought to logic, mathematics, economics and ethics. In all of these subjects, he advocated rethinking basic principles and assumptions based on the ongoing experiences of their usefulness. Political laws, ethical morals, mathematical rules, and scientific understandings are to be continuously examined and developed such that they are best A) for the greatest number of people, and B) over the longest period of time. It is wise and best to take the social view and the long term view. Neo-Confucians of the Song Dynasty debated about whether or not treating others the ways you want to be treated and not treating others the way you don’t want to be treated, two sides of the Golden Rule found across human cultures in ethical discussions (as if it is a continuous problem). The difference between Bentham and Mill reflects these positive and negative sides, and also parallels pro-active socialism and laissez-faire capitalism. Bentham argued that we should maximize happiness and do things for others we want done for us, such as collecting taxes from everyone to provide education for everyone, whether or not they can afford it.
Mill, who is quite influential in American law and legal theory, argued that we should minimize pain, doing things for others only when they prevent harm, and otherwise leaving individuals free. Consider that in Denmark, which is more socialist in ways than America, there are places funded by taxes that alcoholics can go to get treatment, and places they can go to slowly drink themselves to death, or better yet sheer boredom and recovery, places that many American tax payers and many Danish tax payers would not or don’t want to pay for.
In the text Logic and Mathematics, Mill asks: If we admit that all is induction as empiricists then why do we say there are “exact sciences”? We similarly say that there are hard sciences, such as math and physics. Mill argues that this is an illusion due to the fact that objects of math are conceptions and thus imaginary, hence they have perfect straight edges like an ideally straight line. A perfectly straight line, the example he uses, with no width, like a point, cannot exist outside of the imagination. Some, such as Russell, say without perfection of a sort there is no math, science or knowledge possible, but Mill argues this is silly as we have these things yet do not have an instance of a perfectly straight line in the real world. Our concept of a straight line is useful even if it is ideal.
Russell argued that we can strip down or “whittle” to the pure straight edged truth, but Mill argues that this merely helps us to focus our observation and thinking but it does nothing to guarantee that our knowledge is certain at all. We can ignore aspects of a thing to focus on particular aspects or parts, but this does not completely take these factors out of the picture, even as far as relevance to the parts that are in focus. Like Kant, Russell wanted to found science on a first principle, the Principle of Non-Contradiction, but Mill argues that this and the Principle of Bivalence are in fact general observations acquired from practice. We can see that belief and disbelief oppose one another, that they are “oppositional mental states” as Mill says, just as we can see that opposing stories often but not always lead us to see that someone is mistaken or lying. Mill argues that the two principles are merely useful generalizations, as are all concepts used by human beings whether scientists, philosophers or common folk.
Henry Sidgwick wrote several days after Mill’s death that his influence had been declining lately because of his radical politics and interest passing on to other later thinkers, but that between 1860 and 1865 Mill’s philosophy ruled English thought as few have and Sidgwick didn’t think he’d ever see anything like it again. Sidgwick even said, “Mill will have to be destroyed,” but that, “when he is destroyed, we shall have to build him a mausoleum as big as his present temple of fame.” Balfour wrote that Mill presided over English universities much as Hegel did in Germany and Aristotle in the middle ages.
In reaction to Mill, first British idealism took hold, led by T. H. Green, and then Frege, Russell and the Vienna Circle, who called Kant’s a priori categories the analytic, founded logical positivism, the close ancestor of modern Analytic philosophy which dominates the Anglophonic world. From the time of the first World War to the liberation politics of the 1960s, Utilitarianism was roasted for “logical errors” at universities, and Mill’s empirical take on logic and mathematics was something serious empiricists were taught to avoid.
Mill’s naturalism assumes ethical and epistemological objectivity, but Mill and his tutor Bentham share much with Hume, and Bentham said Hume led him to understand the normative and the factual, the way that our practices and customs become what is objectively certain for us. Mill wrote: The most scientific proceeding can be no more than an improved form of that which was primitively pursued by the human understanding, while undirected by science,” and, “The laws of our rational faculty, like those of every other natural agency, are only leaned by seeing the agent at work.”
Mill argues that primitive normative dispositions are based on one sort: enumerative induction, and with accepting past statements based on memory or not, Mill builds his science of inductive logic and mathematics. In his System of Logic, the disposition that is primitively normative is generalization from experience. Deductive principles of reasoning are justified inductively, and “the inductive process” self-stabilizes, confirms, extends, refines, rather than undermines itself. Mill rejects the idea that there are simply intuitive beliefs, because, like Hegel, he argues we must know where the belief comes from, not just that we simply have it. For Mill, there is only happiness which is underived and not for any particular reason.
The most influential Kantian counter-attack was Frege in 1884, who had particular scorn for Mill. Frege turned his back on one of Mill’s central concerns: How do we get mathematical knowledge in the first place? Frege simply says they are a priori, analytic, and the laws of logic are, “boundary stones set in an eternal foundation, which our thought can overflow, but never displace.” Frege wrote that we cannot explain and must accept logical laws, such as identity, “unless we wish to reduce our thought to confusion and finally renounce all judgement whatever,” but Frege argues that our confusion itself isn’t a logical consequence, so it is not a logical reason to accept laws of logic, only the way we take things to be logical.
Criticizing the first positivist, as Frege came after, Mill scathingly says Comte is obsessed with consensus, and made no room for diversity or the individual, much as Mill thought of Bentham’s proactive communal society. Mill argued conflict and opposition in ideas is a precondition for the progress of society. Mill wrote Comte’s golden rule is to live for others, to be altruistic, and everything should be moral and the same for all, but society is not a system for a single end, and it makes everyone happier to pursue their own happiness. Mill wrote: The regimen of a blockaded town should be cheerfully submitted to when high purpose requires it, but is it the ideal perfection of human existence?
Mill’s System of Logic is meant to fully codify the methods of induction. The justification of logic and math is ultimately inductive. Mill says the “principle of exclusion” is “one of our first and most familiar generalizations from experience.” Mill argues arithmetic develops from rearranging pebbles and other ordinary objects, such that we see we can arrange them different ways such that one order isn’t another, and back again. Here, time CAN reverse, as arrangements can be put back the way they were with pebbles, unlike with mixtures of chemicals, which some have used to argue with entropy that time is irreversible. With two inter-rearrangeable arrangements of pebbles, we have one sensation, and then another, with the two different sensations identified and conceptualized as a constant number of pebbles, where quantity comes from in basic experience. As Mill says, the definition of 3 as 2 and 1 together is this sort of basic arrangement that strikes us two ways.
Mill emphasizes processes of joining and separating, giving rise to aggregates, another word for set or group, as we put them together or “withdraw” portions of them, and we can reproduce aggregates by reversing the process. Thus, bringing two things or three things together is “making 2” or 3, or separating one thing off from the group is “making 1,” which is different from simply finding one alone. At the end of Lewis Carroll’s Through the Looking Glass, the second and last of the two books about Alice, the queens sit on either side of Alice, a pawn who now has the chance at the end of the board to become a queen herself, and they ask her strange sums such as what results when you take a bone away from a dog. When Alice doesn’t answer that the answer is the dog’s temper is the result left over from this strange subtraction, the queens shake their heads and say Alice can’t do sums one bit. Alice, growing into an adult, is asked to gather and divide the way others do, which is strangely mathematical.
Mill’s empiricism rejects both Kant’s synthetic a priori and analytic a priori, the idea that we have basic concepts through experience or our minds. Mill argues that scientific definitions serve as landmarks, much as Frege later says logic is like boundary stones, but Mill’s are conventions that could change, while Frege’s boundary stones are set in stone, not by humanity but by the mind, a priori, beyond convention or experience. Mill argues that classifications and definitions in science change, using the example of redefining the term acid after the discovery of hydrochloric acid, HCl, what Mill calls muriatic acid, because it, unlike acids previously studied, doesn’t combine with alkalis to form salts or contain oxygen, which scientists though all acids did, and defined them so. Quine, who balances between positivism and pragmatism, makes a similar point at the end of his influential Two Dogmas of Empiricism (1951), that definitions can always be redefined with new discoveries, so any definition can be taken as basic to thought, as if it is set in stone and the basic frame of analysis, thus analytic, until we reframe things.
Like Kant, Mill doesn’t say what the conditions for legitimate definitions are, but both suggest that the language should pick out objects in the world. Like Avicenna argues with unicorns, Mill argues that centaurs are not real, so the word refers to imaginary objects of mind that do mean something but that don’t apply to actual objects as the word horse does. Ideal triangles, which no triangles perfectly are, are similar imaginary things, though certainly triangle-like triangles are plenty everywhere. However, against Whewell and his followers, Mill argued that parallel lines do not enclose a space not because of rational projections along axioms, but because we do not experience them as we draw them out enclosing space, finding these efforts at following patterns frustrated in experience again and again. Properties are found in actual objects, not abstractions of pure mind or reason, Mill argues against Kant.
Mill does not show, however, how advancements in his own time can be legitimated, as he falls back on any way of gathering a dividing is how math forms, with no other basic structures. German logicians, culminating in Frege, made advancements with symbolic logic that Mill’s simple basis does not enlighten much at all, which is why Frege and others before him resulted in set theory, in theoretical relationships between abstract sets, to account for how mathematics had evolved in super-algebraic ways, from algebra to calculus to Boolean operations, and then, after Frege, Wittgenstein’s truth tables, which is a powerfully useful way of arranging these sets visually, like Venn diagrams were for Aristotle’s syllogisms. These give us abstract ways of gathering a dividing things, as Mill says, but ways that vastly out-power physical gathering and dividing, such as considering the group of all horses with three legs throughout the universe and history, or all Democratic presidents, or any other things it would be physically impossible to collect into a single room. Does this show Mill is wrong, or super-right, more than he could know before Frege’s set theory?
Phillip Kitcher argues while discussing Mill that mathematicians are more into a priori forms than most who claim “scientific” and objective knowledge, because mathematicians do not do experiments and so rely on the “felt necessity” of mathematics, that a symbolic manipulation or mathematical operation can be used the way it must be. Oddly, this would imply that we do arithmetic regularly not because we can’t do it another way, but we feel we must do math the way it is done by others. We could take one away as a type of tax for practical purposes, such that one equals nothing, and one and one are one, as it also oddly is for Boolean algebra, in which true and true is true, by way of the logical connective AND and two values, one as true and zero as false. In such deviant mathematical practice, two and three make four, which we can do, and adjust to such that it feels we must do things this way, to fit the pattern and make the system work as it does for everyone, but there is something in taking one away that can feel wrong, whether or not this feeling is based on prior forms set in the mind before any experience or posterior forms learned through experience and practice.
Mill does say that mathematics is the language of nature, as if we feel it more than other things as basic quite naturally through experience, but the feeling of necessity is an illusion, particularly felt about mathematics that makes them feel particularly certain, and they are consistently confirmed by experience, with two and three constantly making five and not other things, neither four nor zebras. Whewell argued that a proposition must be false if it is inconceivable that it is true. Mill points out that this shows we are certain in what we imagine to be true. Channeling his fellow British Empiricist Hume, Mill wrote:
Now I cannot wonder that so much stress should be laid on the circumstances of inconceivableness, when there is such ample experience to show that our capacity or incapacity of conceiving a thing has very little to do with the possibility of the thing in itself, but is in truth very much an affair of accident, and depends on the past history and habits of our minds. There is no more generally acknowledged fact in human nature than the extreme difficulty at first felt in conceiving anything as possible which is in contradiction to long-established and familiar experience, or even to old familiar habits of thought.
Mill argues that the history of science is full of examples of traditional beliefs that were mistakenly thought to be true because anything else was inconceivable, and it was only experience that surprised the believers and revised the beliefs. This reminds us of the Princess Bride, and how inconceivable may not means what those who use it think it means. Derrida, the later postmodern French philosopher, said the impossible becomes possible all the time. Mill shows Whewell himself admits our ancestors thought many things “inconceivable” that we can conceive of today because what was unknown is now known, and so the impossible now possible. Mill says it is right to imagine what we think is impossible, and say it isn’t the case, but we can see that this is imaginary, and not necessarily what is necessarily the case. This is fine for Mill because we constantly have our beliefs confirmed in experience, so we are not left in an existential long night of doubt with Kierkegaard or Sartre.
Mill suggests our failures of imagination shouldn’t be taken too seriously. Like Quine later, he sees the inability to imagine breakdowns in arithmetic and felt necessity to be akin to pre-Copernicans saying the earth can’t move, like Aristotle, or pre-Newtonians that there can’t be action at a distance, without contact of some visible kind, such as gravity, an invisible force, that can affect two objects at a great distance. Each exhibit in the Exploratorium is a case of this, of reasoning and theorizing that explained why what is normally felt to be impossible is seen to be the case, in front of our eyes, contradicting what we normally reason and think can be possibly the case, which is why each exhibit has a description following the phrase, What’s going on?
Mill says a satisfactory explanation, scientific or otherwise, is a structure of propositions that expresses relevant causal relationships. An explanation is a series of worded statements, which are also meaningful ideas, that point out particular things and how they interact that is useful to us, relevant to our purpose in our situation and context. An explanation does not point out everything that exists, which would be a meaningless act, like suggesting someone do everything is a very meaningless utterance, unless you are using it as a philosophical example of a meaningless utterance. An explanation does not point out what is useful or important universally, either. Rather, explanations are contextual and situational, and we mistakenly imagine and demand that they be universal rather than general or particular, about most things or some things here or there, but not everything.
Mill gives the example that we notice all men die if they drink more than the smallest bit of arsenious acid it, so we can consider it a poison to most, most solidly and justifiably, as it is confirmed in general experience. For a political philosopher whose libertine ideas were dangerously in fashion in the 1960s, including women’s liberation and that reason and practice can always be sought beyond convention, Mill seems suspiciously into acid. We now have to explain why this acid is poisonous to everyone so far, and Mill’s explanation is, as far as science of his day knew, that acid comes into contact with cells and prevents them from decomposing as they naturally do, which are not our current explanations to my knowledge, but they were consistent and practical in Mill’s time, for their purposes as they understood them.
The science Mill engaged in was botany in France, which he studied first-hand. William Whewell (1794-1866) was the greatest English historian and philosopher of science of Mill’s time, one who knew far more about the history and philosophy of science than Mill himself, and master of Trinity College, Cambridge, where Wittgenstein would later live and work with Bertrand Russell. Sydney Smith, the reverend with revered wit who once joked that his friend thought Christian heaven is eating foie gras to the sound of trumpets, said about Whewell, “Science is his forte; omniscience is his foible.”
Whewell took the Kantian tack that there must be basic categories of understanding connected to sensation for there to be knowledge produced at all. Mill drew on Whewell’s work in history of science while arguing against him overall, writing that German Idealism, the a priori view of basic human understanding inspired by Kant, is an enemy to British Empiricism, the a posteriori view that all knowledge comes from experience. Oddly, this would mean Mill would have thought Russell, who called himself an Empiricist by name, was actually betraying empiricism and scientific inquiry by following Kant’s idealistic insistence on basic, unchanging mental structures.
Mill argued that such a Kantian position was incapable of explaining scientific objectivity, as the categories cannot be explained as caused over time if they are simply given and immutable, just as Hegel had argued against Kant, and that there is no way of validating them, of confirming them with new experience. If scientific objective knowledge is both explained in the past and confirmed in the future, Mill argues Kantian ideal categories cannot be considered scientific or objective, even if they are considered fixed and fundamental. Mill wrote, again channeling Hume: There never was such an instrument devised for consecrating all deep-seated prejudices. Whewell and Mill wrote criticisms of each other back and forth over the years.
Mill says the uniformity of the course of nature is the major premise of all induction, and the law of causation stands at the head of all observed uniformities, as an un-contradicted universal, a general that doesn’t seem to be particular or qualified with counter-examples at all. We can conceive of everything dissolving into chaos, but this doesn’t happen, as things seem to regularly cause other things all around us. It is difficult to conceive of how we would have experiences without having causation, whether or not it is a “principle”, constantly confirmed in countless ways that we consistently experience in ways that are quite general, with counter-examples but not as equally as it seems experiences and objects are quite regular. Apples are each individual, but we wouldn’t often say each apple behaves in an individual, unique way to the same extent. Identity is oddly irregular compared to causality, with objects considered and thought of as each individual, compared to the regularity of causation in general, with objects thought to behave as most do.
However, Mill thought that reality is only generally regular, not universally, and said that it is acceptable in popular talk to say nature is uniform, but this is not precise enough for philosophical explanations. We do not expect rain or clouds to ever be exactly the same, nor do we expect the same dreams every night. Mill wrote, “The course of nature is not only uniform, it is also infinitely various,” and, “The order of nature, as perceived at first glance, presents at every instant a chaos followed by another chaos.” Anyone who wants to seek a general theory of science should explain how a chemist can discover a new substance that breaks the old expectations and theory and then know several new things about it from a single sample. We should be skeptical that generalities hold far from cases distant from those that are generalized.
The law of causation is the axiom of induction. General uniformity is secondary, to discovering uniformities that are particular in context. The most important of experimental methods are those of agreement and difference. The method of agreement is that if two things have a circumstance in common, something in a particular sort of situation which is the same for both in that sort of situation but not in others, then the situation is the cause of the similarity. Like Ockham, this can be a maxim, not an absolute rule but something often the case, like the simplest explanation, like the similarity of the situations and things.
The method of difference is the converse, that if there is something that behaves in two different ways in different circumstances, then the difference of the circumstances is likely the cause of the difference. Mill’s other methods derive from these two, involving causation and other aspects of situations. For example, if apples are red and pears are yellow unless there is a heat wave, and both grow green whenever there is a heat wave, then it is justifiable to assume and proceed as if there is something about heat waves that cause apples and pears to be similarly green and different than they usually are.
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 in both ways at once, to mean a book he doesn’t own and a book he didn’t write, but to use ideas in two 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 the rope.
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.
Boole & Boolean Algebra
George Boole (1815 – 1864) was an almost entirely self taught logician, who taught at Queen’s College, Cork, Ireland. He is best known for his Laws of Thought (1854), which contains the seeds of Boolean Algebra, as it was called by later logicians but not Boole himself. Boole, who died with little success to his name the year the American Civil War ended would have been astonished at how his name is so well known today as a fundamental founder of the information age, as Boolean Algebra, and Boolean Logic, are fundamental to how logic was formalized in modern times as it became the language of telephone systems, electronic circuits, and computer languages. To show the difference between Mill’s empiricism and a more Fregean analytic mindstate, we need only quote the opening paragraph of Boole’s Laws of Thought:
The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the symbolic language of a Calculus, and upon this foundation to establish the science of Logic and construct its method; to make that method itself the basis of a general method for the application of the mathematical doctrine of Probabilities; and, finally, to collect from the various elements of truth brought to view in the course of these inquiries some probable intimations concerning the nature and constitution of the human mind.
Boole assumes there are fundamental laws of operations of the mind, presumably universal and unvarying between situations, like Kant and unlike Mill, which are how reasoning is performed. Hegel would take exception to this, showing Boole is a very Kantian logician in Cork, Ireland, just as Mill says is wrong, but widespread in his British Isles. Boole used two truth values for logic, 1 as true and 0 as false. Thus, A and B (A ^ B) can be expressed A + B, and if A and B are both true, then in Boolean algebra 1 + 1 = 1, true and true being true as well, not twice true, but just as true. Boole also focused on the basic connections of NOT, OR and AND, just as Frege and Wittgenstein’s truth table work did. From these basic three operations, there are 16 possible connectives that result in truth operations, each of which can be symbolized and used algebraically with others to determine truth value as answer and output.
Seventh Assignment: Determine the final four truth values for each expression.
1) ~(A v B)
2) ~A ^ ~B
3) De Morgan’s A: ~ (A v B) <> (~ A ^ ~ B)
4) ~(A ^ B)
5) ~A v ~B
6) De Morgan’s B: ~ (A ^ B) <> (~ A v ~ B)