After covering Averroes, it is time to make the short trip from Spain to Italy and France to cover the beginnings of modern European logic, focusing on three figures: the Italian Neo-Platonist Aquinas, the English Nominalist Ockham and the German Idealist Leibniz. It seems like a short trip around Western Europe, but we will be covering entire universes of discourse, such as the set of all cows, horses and sheep. Then we will cover the basics of formal logical connectives, illustrating them with Wittgenstein’s truth table diagrams.
Aquinas & Subalternation
Thomas Aquinas (1225 – 1274 CE) was born in the family castle of Roccasecca, between Rome and Naples, minor nobles of the house of the Counts of Aquino. European castles are based on Persian questles, quite literally in Iberia, and Italians got castles from Persian and Islamic lands, the same place they got much of their Neo-Platonism and the latest in Aristotelian logic, particularly the commentaries and studies of Avicenna and Averroes, which Aquinas was more influential than any at popularizing in Europe and the Catholic Church. Here to the left is Gozzoli’s classic painting of Aquinas of 1471, between Aristotle on the inferior left, Plato on the superior right side, and Averroes lying at his feet, as if the Islamic enemy of Christianity has been defeated.
At six years old, Thomas, the youngest son of nobles, was given as an offering to the Church, specifically the Benedictines, and even more specifically the Abbey of Monte Cassino, which isn’t anywhere near Vegas, such that their youngest son would one day be a powerful abbot, which would have made Aquinas the Abbot of Monte Cassino, somewhat like the famous Count of Cristo Mountain. Unfortunately for his folks, Aquinas wasn’t interested in worldly power, but rather ideal universals found in philosophy, theology and logic, which likely made his family name far more famous than his family had hoped for.
Aquinas was taught to read and write in Latin, particularly to read the Latin author Augustine, the original Christian Neo-Platonist, like al-Kindi was for Islamic philosophy. Like al-Farabi and Avicenna, Augustine was a committed Neo-Platonist and quite committed to the logic of Aristotle’s Organon, but unlike Farabi, Avicenna and Islamic philosophy, Augustine followed Averroes, and Europe followed Augustine. Greek and Arabic philosophy were catching on in Palermo, Salerno and Naples, separate city-state principalities of Italy that were not united, such that Renaissance “Italians” would have identified with their state, not Italy as a whole, with continuous warfare between the states and their alliances and intermarriages with other European powers. Aquinas was sent to Naples, in 1239 at the age of 14 to study Aristotle, Averroes, and probably Maimonides, though antisemitism Aquinas shared with the Church and Naples was quite quiet about that.
Aquinas met and became a Dominican to further his studies, but his parents, who weren’t happy about Aquinas majoring in theology and philosophy, had him kidnapped and imprisoned in the family castle for over a year. One questionable story says that his family locked him in a prison tower with a prostitute hoping to discourage him from studying philosophy, logic and theology, but Aquinas drove her away with a flaming log from the fire, and then collapsed, after which two angels girdled him with a cord that removed all sexual temptation from him, which the angels told him no man can obtain except by God’s grace, not man’s own efforts. I’m impressed he could wield a flaming log with such chaste hands. Over a hundred years before, the logician Abelard of Paris was castrated for getting his student and boss’ daughter Heloise pregnant and shipping her off to a convent for safekeeping, which is somewhat the opposite of Aquinas’ problem. Both spurned sex for logic, though unintentionally on Abilard’s part, as well as the rest of him.
Aquinas’ family finally saw the scholastic light and sent him to Paris to study with Albert the Great, who gave him Pseudo-Dionysius, Fake Dennis, to study. The Neo-Platonist Dionysius argued that the mind works dialectically, positively and negatively through belief and doubt, to come to a greater vision of the angelic and demonic forces of the cosmos, both psychology and angelology. Aquinas earned his masters in theology in 1256, the year after the Pope lifted the prohibition on teaching and studying Aristotle in Paris which many Parisians were doing anyways through the Arabic works and translations into Latin. In Aquinas’ early Paris writings, which show more influence by Avicenna, he argued for Catholic theological doctrines such as the trinity in terms of Aristotelian and Islamic logic. He wrote commentaries on the Bible, Boethius, and Lombard, taught students in the morning and debated scholars in front of students in the afternoon, for three hours a day, about truth, knowledge and theology.
Aquinas wrote many commentaries on Aristotle, like Farabi, Avicenna and Averroes before him, which would make him an excellent commentator, regardless of what the Pope, il Papa, the Big Potato or ‘Tater, would say. In 1265 Aquinas was sent to Rome to found a special school and teach much as he liked for the Dominican Order. Aquinas is still the crown jewel of the Dominicans and Catholic philosophy, second only to Augustine himself, whom Aquinas quotes more than anyone. Aquinas completed his Summa Theologiae, his theological masterwork, which is heavily Neo-Platonic, Aristotelian, Stoic, and Averroist, but is read by traditional Catholic theologians as completing and establishing the teachings of the early Christian Church Fathers, who were also quite Platonic.
Central to Aquinas’ theology is the Aristotelian concept of subalternation. If we know something that is completely true or not true, all or none on the top of the Square of Opposition, we necessarily know something else is true, what is directly beneath what we know on the square, such that if All A is B, then we also know Some A is B, and if we know No A is B, then we also know Some A is not B. This is often completely useless in daily life, as we certainly know that some elephants are nice if we are certain that all of them are, and certainly know that some reptiles don’t brush their teeth if none of them do, and this goes without saying. In context, if I know All A are B and tell you Some A are B, I could even be accused of lying, of not telling the full truth, even if according to Aristotle what we said is necessarily and entirely true, in the context of abstract Aristotelian logic and the Square of Opposition.
But subalternation, which is, strangely, true in a way we can understand, that if we get all of something we certainly get each and every some of it, was very useful for Aristotle and Aquinas, as intelligence itself and the order of things comes downward from above, the higher mind, so we, some and some mortal beings, can participate in degree with the higher mind, the thoughts and vision of God, which Aquinas the Catholic argued is by God’s grace and not the glory of man, in spite of the splendor of the Church. Humanity is thus subaltern to God, some and some not to the big All, somewhat of God but also fallen man, cursed by the acts of Adam and Eve.
This use of logic would surprise many Aristotelian-minded scholars who identify Greek thought with modernity and secularization, just as it surprised Aquinas’ fellow theologians who accused him of Aristotelian heresy. While Aquinas followed Averroes more than most, he also accused others of following Averroes rather than Aristotle and the Catholic Church. He also, in spite of being the primary Christian Aristotelian, sides with Plato and Neo-Platonists against Aristotle and the Peripatetics in many places, particularly where Plato suggests the One is beyond any particular substance.
Aquinas, like the Muslim commentators he read, focused on the logic, metaphysics, psychology and ethics of Plato and Aristotle that they had at the time. Much like Islamic philosophers, Aquinas argued that the soul is the source of the body, the immaterial the true essence and nature of the physical, even though Aristotle’s Unmoved Mover was quite physical. While Aristotle clearly argues that greater substances are above, Aquinas, like Avicenna and Averroes, argues there are potencies which are superior to substance and the material, which Plato doesn’t technically say. In the Renaissance painting by the Catholic Raphael The School of Athens, Plato points upward to the heavens above, and Aristotle points down to the world below, so Raphael, like Aquinas but not Aristotle, thought Plato was arguing for the immaterial above.
Aquinas argues similarly that speculative reasoning and practical reasoning, the mental abstractions and physical practices of thinking, are different, the same powers in us but employed in different ways. Speculative reason grasps truth in itself, for Aquinas the four perfect forms of Aristotle’s syllogisms, while practical reasoning grasps things for other ends. Elizabeth Anscombe, one of Wittgenstein’s most prized students, the one he trusted to edit his most important thoughts as what we now call the Philosophical Investigations, used the metaphor of someone shopping who is followed by a detective, both who keep lists, the shopper of what to buy and the detective of what mistakes the shopper makes, such as buying the wrong kind of butter. The shopper is judged on what mistakes are made on the list and made in what is bought and brought home, but the detective, more abstract than practical, is judged only on mistakes in the list, nothing else, like speculative reason is only concerned with truth, like concern with mistakes themselves rather than bringing home the butter. Wittgenstein wrote in the Philosophical Investigations that we strangely say meaning is a mental act, but we would not say rising butter prices are an act of butter, but rather a relationship in the situation as a whole.
William of Ockham & The Razor
William of Ockham was born sometime around 1285, somewhere near the village of Ockham, southwest of London, ordained a Franciscan priest at 14, studied theology at Oxford at 15, and lectured and wrote commentaries as Aquinas did a few years before him. He was brought to Avignon for four years on trial for possible heresy, beat the rap, then fled to Pisa with several Franciscans who thought Pope John 22nd was heretical for rejecting Francis’ idea of absolute poverty for the followers of Christ. Ockham continued to work and died in Munich, under the protection of the German emperor and Franciscan order, but never reconciled with the Church.
Ockham wrote that if someone says they want to talk about things themselves, not words, signs or concepts, he says the only way we can talk about things themselves is using audible, visible and mental words. Ockham’s philosophy is called nominalism, based on the idea that words are names and mental acts and nothing more, without universal reality, similar in ways to Avicenna and unlike Averroes and Aquinas. Avicenna argued that our concept of a horse and unicorn are both real mental concepts, but one applies to actual horses, and the other doesn’t. While Ockham says in his Summa Logicae and commentaries on Aristotle that he is simply explaining Aristotle, he is actually doing something different than Aristotle, Avicenna, Averroes or Aquinas, the logic of Ockham alone, and quite modern in ways to many.
For Ockham, the written, visible word is inferior to the spoken word, which is inferior to the mental, internal word, which Ockham, like Heraclitus, identifies with concepts and understandings. The printing press had yet to arrive in Europe from China by way of Islamic lands, and so, like Plato, most considered direct, personal communication far more reliable for teaching others. Aquinas argued, in line with both Avicenna and Averroes by degrees, that words refer both to our concepts and physical things, but Ockham, going past Averroes, argues that words only refer to individuals, not even secondarily to concepts. While internal speech is a mental act of understanding, the speech does not refer to itself, but points to the world. For example, the term (terminus) human points to human individuals, and the broader, general term humanity refers to anything substantially and actually human.
Ockham argues, much like Wittgenstein, that philosophers, theologians and logicians get confused when they try to understand humanity in itself, as something mental by itself, and not as something mental that points to actual humans, indicating an object or rather than expressing a separate essence. Humanity does not do anything all together, so we can’t say humanity eats as a whole, like an individual does at a particular time, but human individuals regularly behave as humanity does, each eating alone, or hopefully in small, emotionally supportive groups, such that the word humanity refers to all, and we can say generally humanity eats, but this refers to the individuals eating, not the overall concept.
Ockham is also known for Ockham’s razor is often stated as, The simplest explanation is often the correct one, not always, a universal, but often, a generality. This fits with his Nominalist position: If our understandings are just convenient placeholders that point to things themselves, then most often it would not be mental complexification that would solve problems, but rather the most simple and practical mental moves that allow for reality to reveal itself and its workings, not the secret mental essential workings that only our ideas can grasp beyond our senses. Ockham’s razor is somewhat like and somewhat not like the movie Hellraiser, which sounds like a razor from hell in verbal words, but doesn’t look like one when written.
Leibniz & The Principle of Non-Contradiction
Gottfried Wilhelm Leibniz (1646 – 1716 CE) was a German philosopher and mathematician who invented calculus at the same time as Isaac Newton. Newton tried to confusingly name everything after himself, with many types of newtons, so science kept one kind, the newton as unit of force, and used Leibniz less self-serving system of notation for the rest of Calculus and physics instead. Leibniz’s first job was alchemist’s assistant, and then he became a lawyer’s assistant, perhaps with more worldly success. Leibniz published little during his lifetime, and to this day no definitive collection exists of his various and disparate writings. His most famous writings are his Monadology and his Discourse on Metaphysics.
Leibniz invented the binary system still used by computers today, which may or may not give way to something else like quantum computers in the future, which do not rely on two separate values such as 1 and 0. Leibniz was a sinophile, who loved Chinese culture, studied Chinese thought that was available to him, and invented his binary system inspired by the Yi Jing (I Ching), the ancient Chinese binary divination system that represents all possible situations with solid and broken lines just as Leibniz’s binary system represents all numbers with ones and zeros. Leibniz was communicating with Christian missionaries in China, and he, like some of the missionaries, believed that Europeans could learn much from Confucianism that was in line with Christianity. An admirer of the Chinese abacus, Leibniz was one of the most important innovators of the mechanical calculator, an early computer which employed his binary system.
Like Descartes and Spinoza, Leibniz believed that God created the world as a rational, mechanical apparatus. Because of this, Leibniz famously argued that this is the best of all possible worlds. As God is omniscient, God was aware of all possible worlds before creation, and chose this to be the created world, so it must therefore be the best. Of course, many who ponder the problem of evil, the theological problem debated for centuries about how suffering in a rational world is possible, would question this assertion. Leibniz came up with a pure deductive understanding of the world, which was quite unlike how we experience it. The infinite, the eternal, is for the mathematician Leibniz an infinite series of distinct points, the elementary particles of the universe, eternal and indivisible, like the atoms (“without cut”) of the ancient Indian and Greek atomists. This infinite plurality is entirely made of mind, and each is its entire universe, what Leibniz calls a pre-established harmony.
Leibniz is a strange, outlying philosopher, one I mention but don’t cover extensively for Modern European Philosophy, but he is very important to the history of logic, not only because he helped invent the binary computer, which is where many modern logics live, but he is also one of the first to articulate something Aristotle argues for but doesn’t draw into an explicit principle, what some logicians still call the Law of Non-Contradiction, but most others refer to as the Principle of Non-Contradiction, as it is more understood by secular scholars today as something psychological, but was, for Aristotle, Avicenna, Aquinas and others a law of the universe, one that Nagarjuna of India, Heraclitus of Greece and Hui Shi of China would certainly deny.
The Principle of Non-Contradiction, or PNC for short, can be stated as: If a statement is true, then its negation is false, and if a statement is false, then its negation is true. For example, if the statement Leibniz is a logician is true, then the statement Leibniz is not a logician is false, and vice-versa. Kant and Russell, advocates of logic and the Principle of Non-Contradiction, studied the work of Leibniz intensely, advocating this principle. Russell, who founded Logical Positivism, the basis of Analytic Philosophy, the dominant school of philosophy in the Anglophonic world, argued that we can base all logic, mathematics and certain, objective science on the single truth of the Principle of Non-Contradiction. In formal logic, Graham Priest at CUNY is one of the most famous critics of the principle, arguing, like Hui Shi and Heraclitus, that at least some contradictions are possible, such that Leibniz could be a logician but also not in different ways for different purposes which are both valid.
What is a contradiction? It is an argument, between sides, in its most visible and audible form. We can contradict ourselves, and argue with ourselves, which isn’t the same thing, but similar. A simple contradiction in speech, between two sides, whether or not there are one or two people, or more, is I turned the lights off and No, you didn’t, which is a difference of opinion. If we assume there isn’t a relative dimmer switch, and the lights could go more on or more off, then one side is right and the other side is wrong. Switches are designed as bifurcating devices, as a serious dilemma that could go one way or the other, like the contradiction over the switch. Checking to see if the lights are on or off can resolve the dilemma, debate and contradiction, unless there is a twist, and it turns out, given further evidence and experience, that the switch doesn’t work, or the lights were turned off by someone else later, or its the wrong switch, or the wrong set of lights that should never be turned off, unlike all the others. Even given the simple, bifurcating device of the switch there is a complex human situation that can easily involve contradiction between differently interested parties.
If this is what a contradiction is, then what is the principle of non-contradiction saying? That people can’t get into debates? Aristotle, like Kant long after him, and like Russell long after Kant, argued that not all questions can be solved correctly, but some can, and it is the task of the human mind to use reason to solve what can be solved completely. As for favorite flavor of ice-cream, there are some flavors that would turn heads in many cultures, but it is a matter of subjective taste and opinion. However, insofar as math and logic are supposed to ideally work, in these matters and questions there are true, objective and singular answers that are not relatively true, but absolutely true, much as many would say that two and three together making five is universally, objectively, absolutely and even ideally true, beyond practices or culture.
The principle of non-contradiction applies to these truths, such that if someone contradicts what is absolutely true, they are necessarily wrong. If two and three make five, in all possible ways, then anyone who says two and three make four or six is simply and completely wrong, regardless of their upbringing, practices, cultures, or success, as it is not about what pragmatically works, but what is positively true. Much as Kant argues about morality, it isn’t what keeps the ship sailing, but what is true that we should believe, even if it means disaster and we all go down with the ship, because right is what is overall important, not results. This is the major issue we will talk through again in many ways between positivism and pragmatism.
Aristotle, Farabi, Avicenna, Aquinas, Leibniz, Kant and Russell are some of the central thinkers who made the principle of non-contradiction what it is today. Aristotle said that skeptics like Heraclitus are no better than plants, understanding nothing. Avicenna said those who say fire and beatings are and are not good, only relative evils, should be burned and beaten until they stop saying such things, which is hopefully a joke at the expense of skeptics like al-Ghazali, the Sufi mystic, who criticized Avicenna and was criticized by Averroes. Russell argues that Mill, Dewey and all other instrumentalists, utilitarians and pragmatists, who argue truth is not ideal but relative cannot say anything with certainty, and have to investigate all possibilities continuously to absurdity, such as wondering if we truly did have coffee with breakfast.
I suggest Wittgenstein was right, as a pragmatic person myself, when he said the trick is to see we can stop and start doing philosophy when we want to, as we may never hit bedrock and have the final understandings or answers to anything. Like Priest, Wittgenstein wrote that contradictions need not be false. Logic which excludes all contradictions as nonsense is only a small part of the ways we use language. Lewis Carroll certainly thought so. Aristotelian logic is not the genuine basis of all reasoning any more than trigonometry is the genuine basis of all geometry. (LWPP1 525) The logic of language is more complicated than it looks. (LWPP2 44)
Wittgenstein wrote that some believe in the excluded middle, that a statement cannot be both true and false, but it is rather that true and false divide the field of possibilities, but not always into exclusive parts. Wittgenstein gives the sad example, “Have you stopped beating your wife?” which is not simply a yes or no question, as if someone has never beaten their wife, it is true in one sense, as I am not currently engaging in domestic violence, and false in another, as saying yes implies that I used to, in a way that resembles subalternation. (RPP1 274) The comedian Mitch Hedberg joked, I used to do a lot of drugs… I still do, but I used to, also. Wittgenstein said contradictions are not catastrophes to be feared and avoided, but problems that require engagement with contrary judgements. (Z 685-9) This is likely why Wittgenstein had a deep appreciation of the nonsensical problems found in the arguments of Wonderland.
I I tell you A is true and false, am I telling you nothing, or am I telling you a great deal? If I say nothing I say nothing, even if the silence implies something significant, but if I say something contradictory, that Steve is good and Steve is not good, that the girl with the curl in the middle of her forehead is sometimes very, very good, but when she’s bad she’s horrid, I’m saying at least two things, if not implying more. An argument tells us a lot about both sides, and often shows us things both sides can’t fully see about each other. Humor and nonsense are directly contradictory, just as fantasy contradicts reality by degree, and they teach us much about us.
Leibniz has two other principles which are important in the history of formal logic, and which Kant and Russell both support. The second is the Principle of Identity of Indiscernibles: If two things are without any discernible difference, then they must be not two things, but identical, the same single thing. Of course, if two things are in different locations or exist at different times, this is a discernible difference, one that would shoot any instance of the principle down. Many illustrate this today with the example of two types of minerals labeled equally as jade in ancient China, before humanity had the technology to tell the difference.
The third is the Principle of Sufficient Reason: If something exists, there must be a reason why it exists the way it does. Leibniz believes that the world was teleologically created by God, who controls all in this best and most rational of all possible worlds, and so he assumes that each thing can be rationally explained because each thing was rationally created by an intellect superior but similar to our own. Many secular modern people hold this principle, but without teleology it is difficult to argue that humanity can come up with intelligible reasons that explain apples entirely, or anything else. Are there sufficient reasons apples exist and behave as they do? What is sufficient enough for this? Is it to our satisfaction, or are there objective reasons, in number, that exist independently?
There is one last principle that should be mentioned and strangely isn’t as much, the Principle of Bivalence: A statement must be true or false, not neither true nor false. This is somewhat the inverse of the Principle of Non-Contradiction, that a statement must not be both true and false together, which is the third of Nagarjuna’s four things, and the Principle of Bivalence is the fourth. Leibniz, Kant, Russell and others are focused on non-contradiction, not bivalence. This could be because Aristotle himself wanders in his answer whether neither good nor bad is both or neither, and concludes it is more-so, relatively speaking, neither, which means Aristotle allows for Nagarjuna’s fourth but not third thing. It seems those who argue for non-contradiction think both sides can’t be right, but both sides could be wrong, as if correct is exclusive, but incorrect is inclusive, regardless of how much we all have common sense.
Symbolic Notation In Sentential Logic
Now that we have completed much of ancient and early modern logic history, we will begin doing assignments that introduce us to the basics of formal Sentential Logic. We will go over the basic ways of representing the relationships between propositions as formal connectives such as NOT, AND, OR, IF-THEN, and IF-AND-ONLY-IF, such that we can translate complex statements into symbolic logic.
In Sentential Logic, the basic formal logic taught in introductory classes, we use letters such as A, B, C and D to represent complete statements, such as I have an apple. A stands for the whole statement, a complete sentence and thought, not for the object apple. In predicate logic, which is sometimes taught at the end of a Sentential Logic class, is more complicated and has some controversial problems, this could be represented more algebraically by the function I(a), such that I(x) stands for I have an (x), with the I standing for I have a… not just the person me, and a standing for the object apple alone.
In predicate logic, we could represent the complex statement I have an apple, I have a banana and I have a carrot with the expression I(a,b,c), but in Sentential Logic we have to say each of these separately, such that A stands for I have an apple, B stands for I have a banana, and C stands for I have a carrot, but we do not simply say ABC, all together like early Egyptian numerals, however easy that is, because we use symbols such as AND (^) to connect the statements together to formalize it.
Negation, NOT, we will symbolize with the symbol ~, which you can produce by holding shift and pushing the key right below the escape key in the upper left corner of any keyboard. This is the same symbol used in Spanish above the letter n to give it a ‘nya’ sound. It is symbolized differently by various logic texts, but it is always a horizontal line and the only connective that modifies a single statement rather than linking two statements together. I am using the symbol ~ because that was the one I was taught, and it is easy to find on the keyboard.
If we say that A stands for, “I have an apple”, then not-A, or ~A, means, “It is not the case that I have an apple”, which is the same thing as saying, “I do not have an apple”. The simplest truth table is the one that defines negation, the only one that features a single statement with two possible truth values.
For any truth table, the cases of all possible truth values go on the left, and the truth values for the expression being evaluated go on the right. Notice that with one statement symbolized by a single letter, there are two possible truth values, the first case in which A is true (T), the second in which A is false (F). In the first case, when A is true, ~A is false, and in the second case, when A is false, ~A is true. There is only one other truth table we will study with only two cases and one symbol standing for a single statement, which is not-not-A.
Notice that first we flip and negate the original truth values for each case within the parentheses, and then flip and negate them again, placing them under the NOT sign outside the parentheses, showing us that in case one, in which A is true, ~(~A) is also true, and in the second case, when A is false, then ~(~A) is also false. Negating a truth value twice flips it twice, bring it back to its original truth value.
With conjunction, AND, we will need two symbols, A and B. We will symbolize the expression, “A and B” as A ^ B, using the symbol one gets by typing the number 6 while holding the shift key down. Some use the symbol & to symbolize AND, which is also acceptable, but I am using ^ because that is how I was taught.
With two symbols, we will need four cases of possible truth values. We will use this set of eight truth values in four cases many times, as we will be doing many truth tables with two symbols. With four cases for two symbols, the first case is both as true (TT), the second case is the first as true and the second as false (TF), the third case is the first as false and the second as true (FT) and the fourth case is both as false (FF). When we have a truth table with two symbols (A and B), to the left of the vertical line we will always see these four in this order.
Let us say that A stands for, “I have an apple”, and B stands for, “I have a banana”. This means that A ^ B means, “I have an apple and I have a banana”, or “I have an apple and a banana”. In the first case, when it is true that I have an apple and I have a banana, the statement, “I have an apple and I have a banana” is true.
In the second case, when it is true that I have an apple but I do not have a banana, as B is false, the statement, “I have an apple and I have a banana” is false. Notice that in everyday speak, we might say that I was half right, and my statement is a half-truth, but in formal logic we can only give absolute truth values to statements and sets of statements, so we must call my entire statement false.
In the third case, when I do not have an apple, as A is false, but I do have a banana, as B is true, the statement, “I have an apple and a banana” is false, for the same reason that it is false in the second case. It is only a half-truth, not fully true.
In the fourth and final case, when I am lying about having both an apple and a banana, as A and B are both false, the statement, “I have an apple and a banana” is entirely false.
Notice that the four truth values beneath the AND symbol (^) are T, F, F and F, meaning that the expression A ^ B is only true in the first case, when both A and B are true, and it is false in the second and third case, when one side is true but the other is false, and false in the fourth case, when both sides are false. This set of T, F, F and F down the middle is the final answer for the truth table, the truth values for each of the four cases for the expression being evaluated (A ^ B).
In the last chapter, we defined a contradiction as a statement that must be false or set of statements that must be false and inconsistent. As an example, we used the example, “It is both raining and not raining”. We can demonstrate contradiction with a truth table.
Notice that beneath the AND symbol there are two Fs, as “A and not A” is false when A is true and false when A is false, false in both possible cases.
Disjunction, OR, connects two statements, like AND and the rest of the connectives other than NOT. OR is symbolized with a lowercase V (v), the inverse of our symbol for AND (^). If we say that A stands for the statement, “I have an apple”, and B stands for the statement, “I have a banana”, then the expression A v B means, “I have an apple or I have a banana”, which can be shortened to, “I have an apple or a banana”.
Using the same four cases in the same order that we did for AND (TT, TF, FT and FF), we can show how OR functions. The first case (TT) presents us with an interesting case. If I say, “I have an apple or a banana”, and I have both an apple and a banana, we could say that what I say is false, as I have both, not simply one OR the other, OR we could say that what I say is true, as I have both, one AND the other. In everyday speech and judgement, we use “or” in both of these ways, exclusively and inclusively.
INCLUSIVE OR: You can choose more than one.
EXCLUSIVE OR: You can only choose one.
The interesting thing is that we use the word “or” without indicating whether or not we mean it inclusively or exclusively. We can imagine that in English, or any other language, we could use two different words for exclusive and inclusive OR, but we do not. It is the context of the situation in which “or” is used that give it its meaning and use. Before this is explicitly brought to our attention, we do not even notice that we automatically use and hear “or” appropriately without problems. This is because we are good at sharing frames of context, although there can be problems.
In order to construct a consistent system of logic, we have to choose which way we will consistently use OR. This is the first ambiguity in logic that shows us formal logic is one possible logic, not the simple bedrock of human reasoning. If formal logic was nothing more than the complete way we use reasoning, we would not have to make a choice as to how a particular element should be used. We will encounter another similar ambiguity, another place where we must make a choice as to how a thing is to be used with two disjunctive possibilities, with the next connective we will study, the conditional (IF-THEN). Before that, however, we must make a choice and continue to see how OR functions.
In formal logic we use OR inclusively. This means that if I say, “I have an apple or a banana”, and I have both, we would choose to say what what I say is true, as if I had said, “I have an apple AND a banana”. Thus, in the first of our four cases for the OR truth table (TT), the expression is true. In the second case (TF), if I say, “I have an apple or a banana”, and I have an apple, what I say is true, which is the same in the third case (FT), when I do not have an apple but I have a banana. It is only in the fourth and final case (FF), when I have neither an apple nor a banana, that what I say is false.
Notice that in the first and fourth final case, OR functions the same way as AND. In the first case, when both A and B are true, A ^ B and A v B are both true. In the fourth case, when both A and B are false, A ^ B and A v B are both false. It is in the second and third cases that AND and OR are different. When one side is true but the other is false, AND is false but OR is true. AND is only true in the first case, when both sides are true, and OR is only false in the fourth case, when both sides are false. When we begin to combine connectives and work with complex truth tables, we will see that AND and OR, in accord with DeMorgan’s theorem, are inverses of each other.
A conditional, IF-THEN, is trickier than NOT, AND and OR. We will use the symbol > to symbolize a conditional. As with AND, IF-THEN is sometimes symbolized differently, some using an arrow, and others using a capital U on its left side like a finger pointing. We use > because, like ~, it is easy to find on any keyboard.
It is useful to think of IF-THEN as a promise, such that if I say, “If A then B”, I am promising that whenever A is true, B will also be true. Let us say that A stands for the statement, “You give me an apple”, and B stands for the statement, “I give you a banana”, such that A > B stands for “If you give me an apple, I will give you a banana”. What happens in each of our four cases?
In the first case (TT), if you give me an apple and I give you a banana, what I say is true, and my promise is unbroken. Just like for AND and OR, IF-THEN is true when both sides are true. In the second case (TF), if you give me an apple and I do not give you a banana, what I say is false and my promise is broken. So far, no problems.
Things get tricky in the third and fourth cases (FT and FF), when you do not give me an apple. Clearly, if you do not give me an apple, I do not owe you a banana. I can give you a banana if I feel like it, but I am not obligated to give you a banana unless you give me an apple. The question is, is my promise true? We could say we do not know whether my promise is true or not unless you give me an apple and we see whether or not I give you a banana. However, according to the Principle of Bivalence, in logic things must be either true or false, not neither true nor false, so we cannot leave the third and fourth cases open without assigning a truth value, so we must decide what to do.
We could decide that a promise is false unless it is proven true. This would make A > B false in the third and fourth case, giving it the same function as AND, only true in the first case when both A and B are true. We could also decide that a promise is true unless it is broken and proven false. This would make A > B true in the third and fourth cases. This is another ambiguity, another choice we must make, like the one we encounter with OR, which we choose to use inclusively. In formal logic, we make the choice to say A > B is true in both the third and fourth cases. This means that if you do not give me an apple, my promise is considered unbroken and still true whether or not I give you a banana.
Many have problems with the fourth and final case, thinking that it should be false when both sides are false as it is with both AND and OR, but this is not the way IF-THEN functions. It is useful to remember that A > B is only false in the second case (TF), when you give me an apple but I do not give you a banana, when my promise is explicitly broken.
A biconditional, IF-AND-ONLY-IF, is a conditional that works both ways. We will symbolize it with the symbol we use for conditionals along with its inverse, <>. Some symbolize it with an arrow with points on both ends, others with an equals sign with three instead of two bars, but again we will use < > because it is easy to find on any keyboard.
Let us again say that A stands for the statement, “You give me an apple”, and B stands for the statement, “I give you a banana”, as we did for the conditional. Consider that saying, “You give me an apple ONLY IF I give you a banana” (A > B) means the same thing as saying, “IF I give you a banana THEN you give me an apple” (B > A). This would mean that A <> B stands for, “You give me an apple if and only if I give you a banana”, making the condition work both ways.
In the first of our four cases (TT), if you give me an apple and I give you a banana, the biconditional statement holds true. In the second and third cases (TF and FT), when one of us gives the other something but does not receive what they were promised in return, the biconditional statement is false as our arrangement is broken. In the fourth and final case (FF), our arrangement is unbroken if neither gives the other anything, and the statement is considered true because it has not been shown to be false, like in the third and fourth case for conditionals. This is the truth table which shows how biconditionals function.
Fifth Assignment: Explain each of the two cases of the negation and double-negation truth tables, and each of the four cases of the conjunction, disjunction, conditional and biconditional truth tables, saying which cases are true or false and why, using your own examples of statements to explain each case.