Logic – Kant & Hegel
Kant & Pure Reason
Immanuel Kant (1724 – 1804 CE) was a German philosopher who was born in what was once Konigsberg, Prussia, today part of Russia. While Kant grew up in a devoted Pietist family, Lutherans particularly into moral purity, he found himself drawn to philosophy, science, logic and rationalism rather than religion, particularly Aristotelian metaphysics and logic, like al-Farabi, Averroes and Aquinas. It is said that Kant never traveled more than fifty kilometers from his hometown in his entire life. He was known for being obsessively punctual, and legend has it that he would take his daily walks after lunch so routinely that housewives would set their clocks as Kant passed by their houses. Kant would always walk alone, as he believed it proper and healthy to breathe through one’s nose in the open air and so kept his mouth closed outside. It is said that the only morning Kant broke from his usual strict routine was to purchase a newspaper announcing the outbreak of the French Revolution, a somewhat important event that, like the work of Kant, had a great impact on Hegel, the next figure we will study.
While we will focus on Kant’s use of Aristotelian logic, it is useful to say that Kant’s ethics, epistemology and logic all follow a similar method, one that centers on Aristotelian categories and Leibniz’ Principle of Non-Contradiction. In ethics, Kant believed in strict, rule abiding morality, which he considered the true means of Christian salvation, not religious ritual. Using our universal faculty of reason, Kant argued that we can come to understand absolute principles, morals to which we should always adhere no matter the consequences. The example Kant gives is Do not lie, ever, for any reason. John Stuart Mill, who we will study with Russell, argued the opposite position, that the ends justify the means, and morality is only for the purpose of achieving happiness, more specifically preventing suffering and allowing freedom for others to be happy.
Kant thought about truth much the same as he thought about ethics, categorically even. In some matters, as Plato and Aristotle argued, there is mere opinion, but in other issues there are categorical, correct and objective answers that we possess as true knowledge above mere opinion. One of my favorite examples, and one I enjoy debating, is two and three making five, which many, like Kant, would argue has a single, objective and absolutely correct answer, unless we do math differently, which we easily can. Kant was primarily concerned with metaphysics, the laws of being, which he justified with Aristotelian logic, and Kant wrote that Aristotle’s logic was almost unchanged and unsurpassed since it was written down, as Leibniz before him thought.
Sadly, neither Kant nor the Cambridge Companion to Kant, a source of serious scholarship updated every several years, make any mention of either Avicenna nor Averroes, in spite of their Latinized names and millennial European reception, but Kant does follow both far more than most seem to know, including Kant himself, in his idealism. Greek idealists such as Plato and Aristotle argued that logic and the metaphysics of the universe is alive and physically above our heads, but German idealists such as Kant and Hegel, like Avicenna and Ockham, put logic and metaphysics in representations that are psychologically inside our heads. All Kant says in his lectures on logic about Muslims is, “The Arabs brought forth Aristotle’s doctrine again, and the scholastici (of Europe) also followed them.” Hegel similarly gives Islam credit for preserving Greek ideas for Europeans to truly use and develop later, and nothing more.
Kant wants to understand how reason works when it is pure, abstracted away from physical objects, emotions and conversational situations into pure, functional and categorical ways. Kant was well aware of ancient Greek Pyrrhonian skepticism, as well as the challenge that skepticism posed to metaphysics and Aristotelian logic. Aristotle hated ancient Greek skeptics, arguing that they were mere destroyers and no better than plants when it came to philosophy, as he thought thinking in terms of some and some not rather than all or none does not lead us to absolute, which he thought was the goal. Upon reading the work of the Scottish philosopher Hume, who argued that all truth is assumption, habit and prejudice, Kant famously wrote that he was awoken from his dogmatic slumbers, now tasked to prove that there is objective truth beyond mere assumptions given that all beliefs are acquired through experience.
Specifically, Kant sought to rescue Leibniz’s Principle of Sufficient Reason and Principle of Non-Contradiction, and oddly not the Principle of Bivalence, from being considered mere assumptions, by Hume or any other skeptics. He argued that metaphysics could go beyond both dogmatism and skepticism to become critical, affirming what can be said in the face of what can’t after wrestling with how pure reason can be. A dogmatist would say there are truths in the world, and we should get these truths into our minds, and a skeptic would say there are not entirely truths in the world, or in the mind, but Kant said we should be neither, not say anything about the world, but show what true must be in the understandings of the mind.
Awoken by the skepticism of Hume, Kant spent a ten year “decade of silence”, from 1770 to 1780, working on the first of his three critiques, The Critique of Pure Reason. Originally, Kant thought the work would take three months. This first Critique focused on objective rational inquiry exclusively separate from the influence of experience. Kant’s second and third critiques, the Critique of Practical Reason and the Critique of Pure Judgement, focused on the use of reason in practical matters, the second on ethics, freedom and morality, and the third on aesthetics, beauty and art.
Kant argued that the regularity of the cosmos shows that it is intelligently designed and operates in a rational manner, which is how we can know metaphysics, physics, and logic, but sensation can always be false, while true logic must always be true. Kant argued that Hume was right about the world of experience, much as Plato argued Heraclitus was right about the mortal, changing world below, which can only be known subjectively and imperfectly, but not about the logical operation of reason, which we can know objectively and certainly. Strangely, in his old age Kant hypothesized that the use of domestic electricity caused strange cloud formations and epidemics of disease in cats, a speculative theory which might have survived if Kant had lived in the days of the internet.
Kant’s conception of understanding is often illustrated with the metaphor of eyeglasses. While we may not know what the world looks like without our glasses, we can examine the glasses to see the frame through which we view the world. Likewise, Kant argued that we can critically examine our faculty of understanding with reason to understand how our ideas must take shape, to understand both the basis of understanding and the motions of reason. This would put metaphysics, such as the Principle of Non-Contradiction, on the secure path of science according to Kant. Bertrand Russell follows Kant in his attempts to found formal logic on this single principle.
If we figure out elementary linear arithmetic in the dark, without physical objects, which is a big IF, we may never know about coconuts, or be able to predict how many coconuts we will gather next Tuesday, but we can be certain, if we somehow come up with arithmetic, that if we gather two of anything, including coconuts, and then three, we will have gathered five. We can then reason that if we gather six more, we will have eleven, synthesizing additional mathematical truths via reason apart from the experience of gathering any coconuts, following the logic itself, as an abstract form without objects, much like algebraic formulas.
Thus, Kant argues, while Hume is right that whatever we think we may gather on a Tuesday is merely an assumption, if we are being objectively rational, we cannot reason two and three together any other way than as the sum of five, giving us a synthesized assumption out of our analysis, a conclusion out of sure premises, that is objective and certain. This strangely makes the ideas of logic and mathematics certainties, while the existence of coconuts or Paris, France are merely assumptions, as long as we are in the abstract and not thinking in concrete situations.
Can we think about philosophy, mathematics or logic simply in abstract terms, without any actual objects, including symbols, variables and equations? Teaching any of these abstractions in a classroom requires a classroom, and students who have been trained many times to behave in a classroom as we do. Kant speaks as if a child could fully come to pure reason without interacting with anyone, as Descartes himself explicitly suggested over a century earlier in France.
The Arabic author Ibn Tufail, who lived around 1100 CE during the Islamic Golden Age, wrote a novel entitled Living Son of Wakening which tells the story of a child isolated from human society on a desert island who grows naturally by experience and reason to human adulthood. Some have called it the first philosophical novel in history, a thought experiment designed to demonstrate Avicenna’s empirical theories. As Muslims and Europeans traveled the world and encountered other cultures in search of trade and empire, they asked themselves questions about nature versus nurture, about whether cultures are innately different or different due to experience and education. Issues of the form of logic and reason itself are central to all this, as well as open issues about who is logical or reasonable and who isn’t and why.
Ibn Tufail speaks as if a child on an island would not necessarily turn into wolf-girl, but a civilized, well-mannered boy who would possibly speak the Ur-tongue spoken by Adam and Eve which disseminated into all languages after the Tower of Babylon, as the boy talks to himself, as he certainly could not have learned Arabic without interactions with others. This brings us to an issue with Kant: Can we think abstractly about reason, and how its operations purely work, if we have to think over time, in a space, and in a language of a culture? Kant argues that reason contains its pure, categorical forms regardless of experience, as the genetic form of logic, beyond all culture, much as Farabi’s teacher argued that logic is the same to a Turk, Indian or Persian, even if Aristotle is in Greek. In his early work, Wittgenstein said yes, and invented his Kantian truth tables to flesh out the pure motions of logic, but in his later work, Wittgenstein insisted that reason is always situational, and never purely mental, formal or categorical.
Much like the debates between Avicenna and Averroes about which internal senses we have and how they work when we think, Kant argued that we have two faculties, two internal senses, understanding and reason, which Hegel and Marx made much dialectical use of later. Kant argues in his first Critique that the mixing of understanding and reason is a major source of philosophical error as each has a proper, exclusive job to do. For Kant, experience requires two separate elements, sensation and understanding, what Avicenna might describe as outer and inner senses. Sensation is the raw content and understanding is the conceptual form that makes sensation coherent. Understanding takes various sensations and synthesizes them into categories, while also exclusively dividing the categories from each other. After we experience several objects, some of which are red, and some of which are apples, we form categorical conceptions of redness and apples as sorts of things.
Kant argues that as the self experiences the world through its understandings, it finds itself with pre-existing fundamental categorical understandings which Kant calls foundations. Kant says Aristotle was right to try to work out his fundamental ten categories in his Categories, but also says Aristotle haphazardly found them rather than work them out systematically. Oddly enough, Hegel, the next major German idealist and European philosopher, criticized Kant in the same way, saying Kant pulled his own twelve categories out of nowhere without describing their development, but Kant believed that the origin of the foundational categories was simply beyond human comprehension, and did not develop in the course of our childhood or in history.
One of these foundations is the category of causation, which Hume considered to be an assumption learned through experience. Kant, targeting Hume, argued that causation is a foundational category that is present in the mind before and apart from all experience, and that we find ourselves categorizing the world in terms of causation in a way that cannot be derived from experience. Another foundational category of Kant’s is substance. For Kant, the mind begins as an empty cabinet rather like Locke’s blank slate, but not completely empty, but rather with empty categorical compartments of causation and substance ready to be filled by experiences. Thus, no matter what our individual experiences are, we will all categorize them in terms of causation and substance. This would be why cultures would share two and three making five, whether or not they have coconuts.
Kant argues that reason serves as a higher level of understanding, just as understanding serves as a higher level of sensation. Just as understanding joins and separates sensations, reason joins and separates understandings, both fundamental categories like substances and categories learned through experience, like horses. Judgement performs both of these activities, dividing sensations into groups for understanding and dividing understandings into groups for reason. Reason infers similarities and differences of understandings, forming ideas. Ideas, such as freedom and beauty, central examples used by Kant and fleshed out in the second and third Critiques, are not directly experienced in the world but are formed through inferences drawn from the understanding.
Reason is separate from the sensible world, and thus free to form ideas. While the understanding is passive, categorizing sensation as it happens, reason is active, producing ideas as it sees fit. Experience is determined by the understanding, but ideas are formed from the free use of reason within the imagination, separate from, though derived from, experience and understanding. For Kant, reason is free to wander, taking a wider view, forming abstract ideas and speculating about what might be, however it may not contradict the understanding. For example, to use Avicenna’s example, I can see horses as a child, and then later, because of what I have seen with my eyes and understood horses with my mind, I can use my mind and my imagination, my inner sense and eye, to reason about what unicorns might logically do, if they existed, such that two and three unicorns make five, and unicorns eating substances such as magical corn is caused by hunger, even if I don’t believe or understand that unicorns exist.
For Kant, understanding has jurisdiction over things, like horses you can see, whereas reason has jurisdiction over ideas, like unicorns that you can’t, or reason, or logic, or beauty, or any other ideas that are not seen as physical things, but rather conceptions and imaginations of the mind, beyond sensation but mostly of it, other than the twelve fundamental categories. While the understanding categorizes things, such as substances that cause and are affected, our ideas of substance and causation are part of our free, abstract reasoning, which judges the categories of the understanding to produce abstract ideas.
Kant used the metaphor of an island in a stormy sea to illustrate the rational mind amidst the flux of the sensual world, objective and rational in the sea of the uncertain. Schopenhauer, a Kantian himself, but an odd one, used a similar metaphor of a ship on a stormy sea, more skeptical than Kant as Schopenhauer’s boat is not fastened down by fundamental concepts but drifts with the current of the passions, which he argues oddly is what Kant truly meant all along, unlike almost all other Kantians. Schopenhauer, like Kant, were both central influences on Wittgenstein, the last logician we study.
Hegel & Dialectical Progress
Georg Wilhelm Friedrich Hegel (1770 – 1831 CE) argued that Kant’s exclusive division between understanding and reason, as well as the division between the thing-in-itself and our experience of it, were failures of Kant’s inability to synthesize the whole with reason above and beyond the divisions of understanding. While Kant thought that reason should ultimately serve understanding, maintaining exclusive distinctions, Hegel thought that reason should transcend while extending understanding, uniting all in the transcendental One.
For Hegel, reason’s job is to extend but also contradict understanding, to contradict accepted dogmas with opposite points of view and force progressively greater synthesis beyond exclusive boundaries. For Kant, contradiction with the understanding results in incoherence, an improper mixing of understanding and reason, not a greater synthesis of knowledge. For Hegel, understanding is extended by contradiction, transforming the incoherent into the coherent. Both believe that reason works through dialectic, by weighing both sides of a potential judgement and then extending the understanding, arriving at a greater understanding than before. Should reason never result in contradiction, or should it contradict itself and then overcome the contradiction? It depends on whether one accepts or opposes the Principle of Non-Contradiction.
Hegel took the ideas of Kant and his friends Fichte and Schelling and created an elaborate system that described the evolution of all human thought through the course of history in stages over time, including philosophy, politics, religion and art. This is quite unlike Kant’s fixed, ideal and timeless categories, but employs Kant’s ideas about the relationship between understanding and reason. Hegel was fascinated as a youth by the French Revolution, the one thing said to have broken Kant’s routine walks, for the left and rationalists a revolt of reason against traditional authority, dogmas and understandings, in an age overturning many old assumptions.
Like Fichte and Schelling, Hegel saw individuals, ideas, and cultures as positive assertions that spawn resistance, and that in each stage of history culture is resisted by counterculture, which then merge to become a greater culture, a greater synthesis, which then repeats the cycle. Hegel saw himself as lucky, living in the final stage of history, the age where Napoleon balanced order and freedom, authority and rights, in the modern European nation state. Similarly, Hegel saw his own philosophy and logical system as the final synthesis of Heraclitus, Plato, Aristotle, Aquinas, Hume and Kant.
After writing his Phenomenology, his history of the objectivity of the Egyptians and Babylonians leading to the subjectivity of the Greeks, leading to the objectivity and subjectivity, the science and art, the law and freedom, the understanding and reason of the Germans, and thus all humanity, uber alles, with German Protestant Lutheranism as the final form of culture. Hegel wrote his Logic (1816 CE), which further details the operation of understanding and reason that underlies the historical and cultural evolution of philosophy, politics, science, art and religion.
Many ancient philosophers, Greek but also Egyptian, Indian and Chinese, understood the world in terms of opposites that work together. In ancient India with the Jains and Buddhists, in Greece with Heraclitus, and China with Daoism, we find the idea of the union of being and non-being together as becoming. For Hegel, thought is exercised individually and historically, by individuals and cultures considering thesis and antithesis, belief and doubt, culture and counterculture together as a unified whole that can be viewed from opposite sides, and this progresses in stages into completion.
Like the Greek Pyrrhonian skeptics, Hegel considered each point of view to be an appearance or semblance (Schein), and this is why his logic, which is quite consistent and coherent, is in complete denial of the Principle of Non-Contradiction, as Hegel explicitly writes that everything around us can be understood Newtonianly, as composed of action and reaction, resulting in further action in stages. His system is an attempt to fuse all previous positions of philosophy together as contradicting opposites, the sum of which give the whole, just as Marx, a left-wing Hegelian, later attempted to do with politics. Just as for Newton, who argued that in every instance of force there is an action and an equal and opposite reaction, in human thought for every assertion there is an opposite assertion, for every logic, a counter-logic, for everything intuitive, something counter-intuitive. Just as Napoleon took the authority of Louis the 16th and balanced it with the rationalism and freedom of the French Revolution to become the next, greater authority, a great thinker is one who sees beyond the contradiction of thesis and antithesis to create a greater synthesis, which Hegel also calls sublation (Aufhebung), a preserving and transforming, some and some-not like the subalternation of Aristotle.
Just as Napoleon came on horseback at the right time in history to balance order and freedom in its latest form, Hegel saw himself as arriving just after the dogmatism of the rationalists and skepticism of the empiricists to synthesize the whole out of opposing parts. Similarly, if one looks at the front and the back of a thing, one has seen the entirety of a thing, and can put this together as an idea, as a view beyond what can be viewed with the eyes at one time, from one place as an individual, rather than from the communal point of view. The front and back views do not cancel each other out, but complement each other as a concrete whole, symbiotically.
While Hegel argued against pessimistic skepticism, what many would call nihilism, the position that there is no truth at all, an “excuse for non-philosophy,” he saw his own system as an optimistic system-building skepticism, as Marx did his own theories, that can fuse various sides into a full completion of possible perspectives and social positions. The problem with skepticism is that it can show both sides oppose each other, but it cannot build systems. The problem with dogmatism is that it affirms systems, but myopically, one-sided and without critical doubt. For Hegel, human thought as a whole must progress forward and persevere, but also contradict and criticize itself to progress. This tension requires geniuses to leap over the hurdle of each contradiction, of each stage in history. Hegel says Heraclitus was the first to do this, and sees Plato and Aristotle, and Hume and Kant on the path to himself.
Hegel calls Plato’s Parmenides the most “perfect and self-contained document and system of genuine skepticism”, the greatest masterpiece of ancient dialectic. In the dialogue, Parmenides shows young Socrates that debating dialectically shows our understanding of contradictory positions, and that if Socrates practices this over the years, arguing position and counter-position, he will become wise and possibly unstoppable, which is what the Oracle of Delphi said about Socrates and what he wandered around trying to prove until they killed him for it. Parmenides argues that being is both one and many, does and does not exist, is known and unknown, changes and does not change, and in all of this has and does not have contrary properties, does and does not contradict itself. You could look at existence from any of these perspectives, finding truth on whatever side you take.
Hegel argues that things are opposed to each other as well as the whole, and so it is opposition that defines a thing, making it into what it is. Hegel calls this determinate negation, and details how every way of being has particular ways of not being, much as a horse isn’t another white horse, nor is it a unicorn, but in another way, nor is it a car, but in another way, nor is it the idea of a horse or the memory of a horse, or the possibility of a horse, but all in different, interrelated ways that oppose each other. While many would consider non-being to be simply empty and nonexistent, each phenomena is intimately and essentially bound up with what it is not, to the thing or things it is opposed to, just as conservative and progressive, dogmatism and skepticism, define each other.
Consider Hegel’s example of a possibility, a thing that is what it is insofar as it isn’t yet but could be, as opposed to a necessity, which isn’t yet like a possibility, but unlike a possibility will certainly be. Both are opposed to what is in not yet being, but opposed to each other as well. Jains and Buddhists of ancient India and Daoists of ancient China used examples such as a pot or room, which are what they are by not-being, being empty in most of their being and thus useful, as does Heraclitus, Hegel’s first true philosopher, but possibilities and necessities, unlike Kant’s categories, unfold over time, and unfortunately Aristotle argued that predictions of the future are neither true nor false, so we can’t say anything about them completely, even as we do say particular, determinate and opposed things about them. Opposed sorts of not-being determine how we talk about things that are not yet, and may or may not be, and may or may not need to be, such that it may or may not be that it may or may not be, or be necessary.
For Kant, reason extends the understanding through non-contradictory continuity and complete coherence when it is correct. For Hegel, Kant misunderstood reason as merely a higher understanding of understanding, holding on too tightly to Leibniz’s Principle of Non-Contradiction and not trusting his Principle of Sufficient Reason enough. For Hegel, reason does not operate like understanding, nor does it simply conform to it. Rather, reason is revolutionary. While understanding works by non-contradiction, the principle Leibniz that Kant wanted to salvage as central to metaphysics, reason must be free to oppose understandings in order to bring them to greater completion. If understandings were perfect, there would be no need for reason, just as if knowledge was complete and absolute, there would be no need for wisdom, and if authority and culture was simply good and just, there would be no need for innovation, counterculture or revolution.
Understanding’s job, as Kant recognized, is to create exclusive categories for experience, but reason has a different job, to create ideas that synthesize the understandings, resolving contradictions between them. For Hegel, the great leaps in human thought are unifications of contradictory understandings through reason. If reason merely operated like understanding, as Kant believed, it would be incapable of revolutionary insight. For Hegel, Kant’s synthesis of reason fails and falls short because Kant, afraid of skepticism and contradiction, failed to understand the cooperation of non-contradiction and contradiction, the cooperation of dogmatism and skepticism. In his Logic, Hegel writes about Kant’s antinomies, which he thinks can serve as the basis for Aristotelian syllogistic logic, but which have many, confusing problems:
Kant, as we must add, never got beyond the negative result that the thing-in-itself is unknowable, and never penetrated to the discovery of what the antinomies really and positively mean. That true and positive meaning of the antinomies is this: that every actual thing involves a coexistence of opposed elements. Consequently to know, or, in other words, to comprehend an object is equivalent to being conscious of it as a concrete unity of opposed determinations. The old metaphysic, as we have already seen, when it studied the objects of which it sought a metaphysical knowledge, went to work by applying categories abstractly and to the exclusion of their opposites.
However reluctant Understanding may be to admit the action of Dialectic, we must not suppose that the recognition of its existence is peculiarly confined to the philosopher. It would be truer to say that Dialectic gives expression to a law which is felt in all other grades of consciousness, and in general experience. Everything that surrounds us may be viewed as an instance of Dialectic. (118)
Thomas Kuhn’s idea of paradigm shift is useful for understanding Hegel’s dialectical process. A model always has problems, and so there are counter examples that arise as any model is used. This creates culture and counterculture, dogmatists who support the model and skeptics who attack its weaknesses. For a time, culture supports the continued use of the model and the skeptics and problems can be ignored, but then when the problems become unavoidable, the paradigm is changed, shifted, to resolve the problems. This resolution brings an end to the old paradigm, and now the new paradigm, with new problems, begins to generate a new contradictory situation.
Hegel argues that the development of logic and philosophy start with sense certainty, like seeing a rock and thinking it is simply real. The position is sometimes called naive realism: things just are as we see them. Consider the expression, “Right as rain”. Although rain cannot speak, and so cannot be right about anything, we consider rain to be self-evident, clear in itself and unmistakable. Hegel shows that what we call common sense is actually empty and abstract, and concrete understandings are the result of a process of rational reflection. While saying The table is solid seems to be nice, complete and objective, we can only know how solid the table is if we doubt and test its solidity, coming to a more complex and thus more accurate, defined and concrete understanding, so what we say can be easily contradicted, as the table is not completely solid, nor could it be made into a table if it was made of a simply solid substance.
Hegel says that if we write down obvious truths, such as It is now night time, and then look at them the next day, we find that they are now longer true but have, as Hegel says, grown stale, much as Aristotle says that if I say I am standing this is true and then false when it is or isn’t true, over time, unlike a truth which is universal or eternal we could say, such as Fire is hot. Our now, which was so immediate, changes with time, and so any statement about here or now is not universal, but rather contingent on time and place.
Overall, Hegel argues that first the mind looks for the answer in essences, fixed positions that are stable and static behind the ever-changing and becoming beings within the world. In the second stage of the Greeks, Hegel argues that this is Plato’s idealist response to Heraclitus’ never-ending flux. In the third European stage, which unites the mind and its world, Hegel argues that this is Kant’s Idealist response to Hume’s charge that all truth is assumption. Just as Heraclitus’ grasp of becoming leads to Plato’s eternal forms in the heavens, Hume’s Empiricism leads to Kant’s objective categories in the mind. Unfortunately for Kant, the categories are disconnected and their origin unexplained.
Hegel sees his own position as the superior synthesis, as the gathering up of Kant’s categories into a living, singular Idea. Subjectivity and objectivity are understood as one continuous thing, unified while opposed to itself. Similarly, necessity and freedom are one and the same. This is the totality of all previous positions, unified together, which goes forward as philosophy and science. In order for philosophy and science to be authentic, they must be understood as dynamic, as evolving and open, not as dead and closed. Hegel considers Kant to have been a great thinker, but his inability to tolerate contradiction made his system a dead corpse, not a real living grasp. Living, authentic thinking must be ready and willing to contradict any part of itself, breathing the life of skeptical opposition into any stale, dogmatic conception. Strangely, Hegel thinks that Western, German, Lutheran, Scientific civilization is now simply capable of this, unlike any previous culture, and never to be surpassed or eclipsed.
Complex Truth Tables
Now that we have the basic truth tables for the five connectives, which you can find at the end of the last lecture, we can examine complex truth tables, truth tables that involve combinations of connectives, sometimes requiring parentheses. The truth table method allows us to evaluate expressions made of several statements one step at a time until we reach the set of truth values for the expression in all possible cases.
We will now learn how to express complex sentences in Sentential Logic, and how to evaluate those sentences with truth tables to give us a final set of truth values for each possible case. For example, if we say I have an apple and I don’t have an apple, that sentence can be expressed in Sentential Logic as A & ~A, much the same as A and not A, and the final set of truth values when we construct a truth table with two cases for this sentence is FF, as there is one symbol (A), with two cases, TF on the far left, and it is false in both the first case (T) and the second case (F). This is the truth table that shows in Sentential Logic the Principle of Non-Contradiction is true in all possible cases, as it can never be the case (F) that A is both true (T) and false (F) at the same time in the same case.
We have seen that the expression A ^ B is true only when both A and B are true. What about the negation of this expression, ~ (A ^ B), or Not (A and B). We use parentheses to show that the NOT applies to the entire A and B, much like in algebra. It is easy to see, particularly when we create a truth table, to see that we get the inverse or opposite of the truth values for A ^ B, FTTT, rather than TFFF. Notice that the final truth values for the whole expression are now on the left, underneath the NOT symbol. First, we figure out the truth values for the part of the expression inside the parentheses (A ^ B), and then, for our second step, we take those truth values to figure out the truth values for the whole expression.
Does ~ A ^ ~ B, not A and not B, have the same truth values as ~ (A ^ B), not (A and B)? We can create a truth table to show that it does not. First, we negate the truth values from our original four cases underneath ~ A and ~ B, such that underneath ~ A we have FFTT, and underneath ~ B we have FTFT, the opposites of what are always on the far left side under two valued, four case truth tables. Then, taking the new truth values and remembering that AND is only true when both sides are true, we evaluate the expression to find the truth values for the entire expression, which sits under the AND. Notice that the final truth values for ~ A ^ ~ B are FFFT, but the values for ~ (A ^ B) is FTTT, the same in the first case (F) and fourth case (T), but not the same in the second and third cases. This shows us that we cannot distribute the NOT within the parentheses without changing the expression, as we can in ways in algebra. With DeMorgan, we will learn ways of distributing NOT which flip AND into OR, and OR into AND.
We have now used NOT inside parentheses and outside parentheses. What about expressions in which NOT is both inside and outside of parentheses? Let us construct a truth table for the expression ~ (~ A ^ ~ B). Within the parentheses, it is the same truth table as our last, for ~ A ^ ~ B. Then, as we did for ~ (A ^ B), we will negate the expression within the parentheses to arrive at the final truth values for the whole expression, TTTF. Notice that ~ (~ A ^ ~ B) has the same final truth values as A OR B, and the AND has changed to an OR. This means that ~ A ^ ~ B is equivalent to ~ (A v B), such that saying, Not A and not B is the same thing as saying, Neither A nor B. We can show this, one half of DeMorgan’s Theorem, using truth tables, which is what made Wittgenstein famous, as logicians had to assume DeMorgan’s Theorem was right until Wittgenstein’s truth tables gave them a method to prove it.
Now, let us construct a truth table in which two expressions that are joined together with connectives are themselves joined together with connectives. Let us say we want to know in what cases the expression, “Either I have an apple and a banana or I have an apple or a banana”, which we can symbolize as (A ^ B) v (A v B). First we must find the truth values for both of the expressions within the parentheses. Then, as our second step, we use the truth values for each of the expressions within the parentheses to arrive at the final truth values for the expression. With negation, we worked from the inside to the outside, but here, we work from the outside inward. Interestingly, the entire expression also has the same truth values as and is the equivalent of A v B. That means that if you put the entire phrase in brackets, and put a biconditional between it and A v B, or ~ (A ^ B), it would be logically equivalent, with the biconditional true in all four cases, with TTTT as the final set of values.
Let us construct one final complex truth table. Let us say we want to evaluate the somewhat overly complicated expression If I do not have both an apple and a banana, then I do not have a banana, which we can symbolize as ~ (A ^ B) > ~B. First, we put the truth values TFFF under the AND within the parentheses, as well as negate the original truth values for B and place them underneath the ~B. For our next step, we negate the truth values for the expression inside the parentheses, and then for our final step we use these negated values along with the truth values underneath ~B to find the final truth values for the entire expression, remembering that IF-THEN is true unless the truth value to the left is true and the truth value to the right is false.
Assignment: For each of the following sentences, symbolize the sentence in Sentential Logic using A, B, and logical connectives, and then evaluate the sentence using the truth table method, giving the four truth values that are your final answer. You do not need to turn in the truth table you draw or construct to get the final values, but you can. For example, if the sentence is “I have an apple and a banana,” then you would symbolize the sentence as A ^ B, and include TFFF as the final truth values. This first complex set of truth tables does not need parentheses, as these deal with negations of A and B, such as ~A^B, but not negations of combinations of A and B, such as ~(~A^B).
2) I have an apple and I do not have a banana.
3) I do not have an apple or I have a banana
5) If I do not have an apple, I have a banana.
6) If have an apple, I do not have a banana.