Please read Bertrand Russell’s essay *On Denoting*.

Ancient Greek philosophers such as Heraclitus, Plato and Aristotle thought about the world, early modern European philosophers such as Leibniz, Kant and Hegel thought about the mind that recognizes the world, and today’s Anglophonic analytic philosophers think about the language that expresses the mind recognizing the world, focusing on language use more than most other branches of philosophy. Hans Sluga, my favorite philosophy teacher at Berkeley, argues that Frege is the first analytic philosopher, a central influence on Wittgenstein, Carnap, and many others who sit between these two today, and that analytic philosophy has neglected Frege because it neglects historical questions in general, including questions about its own particular philosophical history, an anti-historical tradition that Frege himself started that ironically eclipsed his own historical existence.

Since Frege at the beginning, analytic philosophy has focused on the abstract and formal rather than the social and historical, separating what changes over time from the changeless and timeless. Frege wrote that the thought which *2 + 3 = 5* symbolizes when written on a wall and recognized is *completely impersonal* and it is *altogether irrelevant* who wrote it or how they came to understand it or feel about it, separating what pure reason can construct and reconstruct from what is subjective, historical, social, emotive and personal.

Frege had no idea how influential he would be, much like Boole before him, living out his final days disillusioned and frustrated with the lack of acceptance of his ideas by mathematicians, logicians and philosophers alike. Leibniz and those around him in Germany were conservatively Aristotelian in logic, much like Averroes and Aquinas, and then from Wolff to Kant and Hegel logic remained central to German idealism, with a focus on pure thought itself, which Kant structured categorically and Hegel dialectically, both calling logic the essence of true science and the rational mind. None of this changed Aristotelian logic much at all, with syllogistic forms working categorically, along with non-contradiction, and dialectically, with the joining of two together into a third. Kant coined the term *formal logic*, but in Frege’s day Wundt’s *Logik* and other popular textbooks avoided precise definitions, treating basic topics at length, intertwined with many other issues and speculations about thought.

In the years after Hegel’s death, much of the interest in German idealism shifted to anthropological naturalism, centering thought on the observed social world as the advancing sciences continued to take down long-standing metaphysical assumptions, drawing criticism of Kant’s *a priori* ideal categories. Darwin, Marx, Feuerbach and others brought this into the popular thought of Germany and Britain. British empiricists such as Hume and Locke argued we are a blank slate, with no ideas or ideals originally in us, which fit new developments in biology and psychology of the time. Frege wanted to prove Mill wrong about subjective, empirical logic and prove Kant right about *a priori* forms of certain, objective truth. Frege was devoted to both Leibniz and Kant’s position that reason is supreme and what is truly known is through reason alone, apart from all subjective experience and psychological conditions. Frege wrote to Russell that it isn’t your Pythagorean theorem, or my Pythagorean theorem, but *THE* one and only Pythagorean theorem, which could lead to a terrible analytic-inspired sitcom-soap-opera, *All Our Pythagorean Theorem*. In his *Foundations of Arithmetic*, Frege defended his new Kantian take on what he called *analytic* truth and logic:

*When a proposition is called *a posteriori* or analytic in my sense, this is not a judgement about the conditions, psychological, physiological, and physical, which have made it possible to form the content of the proposition in our consciousness; not is it a judgment about the way in which some other man has come, perhaps erroneously, to believe it true; rather it is a judgement about the ultimate ground upon which rests the justification for holding it to be true.*

Frege lavishly praised Leibniz, saying *hardly anybody can measure up to him*, and that the seeds in his works on logic took time to bear fruit. Ignoring Leibniz’s bizarre ideas that we are all separate monads and the world is a grand illusion, Frege says his idea of a *lingua characterica*, somewhat like Peano’s *Latin sin inflectione*, is Leibniz’s greatest idea, a universal logical language that works as a calculus for scientific truth. Kant, following Leibniz, insisted form is separate from content, and declared logic to be *formal philosophy itself*. Frege took the insights of Leibniz and Kant, ignored almost everything else in their philosophies, and stuck to this narrow focus with rigid designation, angering Neo-Kantians who knew him.

In his *Begriffsschrift* (1879, *Concept-Script* or *Concept-Notation*), Frege borrowed bits from Aristotle, Leibniz, Kant and Boole to create a formal logic which eventually replaced the logic of Aristotle at universities in the 1900s, much as Galileo’s physics had replaced Aristotle’s at the beginning of European modernity. Frege wrote that Aristotle’s syllogisms were inadequate and an obstacle to the pure reasoning Kant sought. Frege argued that such a logical system can determine the objective, conceptual content of any assertion, the bare bones of certain truth that can be found, if any, in any statement with meaning, if all content other than pure reasoning is stripped away. Russell later wrote, against Mill, that we can *whittle down to the hard data*.

Many who reviewed his work wondered why Frege had reinvented Boolean algebra with new notation. Unlike Boole, Frege said his new notation was like a *mechanical hand* more powerful than a mere human hand, just as* the microscope is to the human eye*, but that where Leibniz had failed to fully build his mechanical calculator, he would create a logic that would allow science far greater calculation than it had the ability to do before, more than a mere calculating tool that Boole’s algebra had been, but the tool that reveals structures of certainty themselves in scientific investigation. Where Boole and his followers argued about mere mathematics, Frege was applying Boolean functions to all certain and objective meaning in the world. While Kant said the pure *thing-in-itself* is never knowable and did not articulate it, separating it into parts with functions, for Frege all we know are justified articulations we can make about things with speech, and other than this we know nothing at all with certainty.

Frege argues that we can think a thought by articulating it truthfully, whether or not we then judge the thought to be true, showing we grasp the structural content independently of the truth of the content. It is not judgement of truth that makes the thought, but the structure of the truth, clearly understood, that allows us to determine the truth of the thought. Frege believed that this demonstrates objectivity beyond mere subjectivity, that it is not how we feel about a structure, but the components of the structure that determine whether things are true or false in cohering with other structures, much as there are three or four apples regardless of what we want to be true or how we feel about it. When we combine A and B with a Boolean operator, we are creating a structural connection that should be understood one truth-functional way.

Boole made the fundamental connectives of his algebra *AND, OR* and *NOT*, but Frege centered his logic on *negation *and the *conditional*, on *NOT* and *IF-THEN*. Wittgenstein’s truth table method uses all four of these, along with the biconditional, *IF-AND-ONLY-IF*, included in ways by both Boole and Frege in their systems, as the five basic connectives.

Frege knew that we use conditionals in more than one conventional way. In truth table logic, we treat *IF-THEN* as a promise that is true unless broken (TFTT), but we could treat it more exclusively, as a promise that is only true if kept, which would make it function like the exclusive *AND* of the truth tables, only true when everything is true, and false if any element is false (TFFF). Just as *and* and *or *can be inclusive and exclusive in everyday use, and in the middle are quite interchangeable, Frege knew he had to define the connective one rigid way, so he argues the *logical kernel* of the connection, separate from the *inessential accessories*, functions as Boole defined it, which is how we still do Wittgenstein’s truth tables today.

While Boole and Carroll were not certain about the truth of sets or statements that do or don’t contain members, such as Avicenna’s unicorns, which doesn’t correspond or refer to real animals at all, Frege said that *All men are mortal*, core to Aristotle’s first form of the perfect syllogism, is a functional statement about *if* anything is a man, whether or not there are, *then* those members of the set are mortal, or the statement is false, solving the *existential* problem, not affiliated with Sartre’s existentialists, of whether or not the form of truth depends on actual existing things. For Frege, the judgements of thought are formal, while the rest of the concept and content is beyond judgement, whether or not the judgement does or doesn’t apply to a thing, much as Aristotle argued that the statement *I am standing* is true when it is, and then isn’t when it isn’t, but it remains the same statement with the same structure. Thus, if we stick to judgements that are formally coherent and objectively certain we can discard everything else in thought as conceptual, psychological and subjective, irrelevant to genuine understanding.

Frege’s work was largely unrecognized, though Peano and Russell appreciated it, and Frege continued to attack other logics and systems, including the arithmetic of Peano, the infinite sets of Cantor and the phenomenology of Husserl. Unfortunately for Frege, Russell noticed a fatal contradiction in Frege’s system just as Frege was publishing the second half of the work. Russell’s paradox, which Frege admitted his logic never recovered from: if we say that a set contains all things except itself, does it contain itself or not? If it doesn’t contain itself, it qualifies as a member of the set, as it *contains all things except itself*, and if it does contain itself, then it doesn’t qualify, but it contradicts its original definition, as *a thing that doesn’t contain itself*. Russell tried to solve this problem by declaring sets of sets to be types and working on this as the foundations of mathematics with Whitehead, but this simply kicked the contradiction down the road, where he hoped Wittgenstein would solve it for him, which Wittgenstein tried to do with his truth tables before turning on them. Frege’s most important student, Rudolf Carnap, who knows if you are lying to him through simple force of truthful, serious gaze, took the work of Frege, Russell and early Wittgenstein and formed the Vienna Circle, who spread Frege’s analytic philosophy and Russell’s logical positivism to others between WWI and WWII.

Lord Bertrand Russell (1872-1970 CE) was born into one of the most prominent aristocratic families of Britain, and retained the title ‘Lord’ even when arrested and jailed for demonstrating as a pacifist and socialist against the first World War. Russell distinguished his own *Logical Empiricism* or *Logical Positivism* from that of the earlier British Empiricists Locke, Hume and Berkeley, arguing that empiricism could learn with Boole and Frege to analyze using the method of mathematics and logic and arrive at pure certainty, not just in the ideals of the mind but in the physical world that was mechanizing. Russell argued against Hegel and Mill, who had strong followings in Britain, and claimed that Boole was a modern Copernicus.

Bertrand Russell opened his 1929 essay *Mathematics and the Metaphysicians* saying the 1800s prided itself on the steam engine and evolution, but it might have more legitimately claimed fame from Boole’s *discovery of pure mathematics*. Russell also famously said *Mathematics can be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true*, which is true to Boole and nonsense worthy of Lewis Carroll. Russell was known jokingly as the Mad Hatter of Trinity College, Cambridge, with his friends McTaggart and G.E. Moore, the March Hare and the Dormouse, together known as the Mad Tea Party of Trinity for speaking of Boolean forms as empty and formal as the glass jars and shelves Alice falls past down the rabbit hole into Wonderland. Carroll was mocking Aristotle and Boole, and now Cambridge used Carroll from Oxford to mock Russell’s trio at Trinity.

An excellent book on the life and thought of Russell is the graphic novel *Logicomix*. As a boy, Russell feared going mad like his uncle, and believed that reason alone would save the world and himself from madness. He found what his Grandmother had hoped to give him with strict Christian faith in Euclid’s geometric proofs, *delicious*, absolute certainty. Later, when he discovered that Euclid relied on *axioms*, unproven principles merely *assumed* to be true, Russell was deeply disappointed, but later said that this was a defining moment of his life. He would attempt to completely ground mathematics in deduction from certainty, which for many years he believed the truth table logic of his young associate Wittgenstein would eventually do.

Russell found that mathematicians were all in agreement, afraid to contradict each other but afraid also to ask the deeper philosophical questions about the certainty of mathematics, questions that had to be asked to give mathematics and thus science a firm grounding. However, when he turned to philosophy to ask the deeper questions, Russell found that the philosophers were all in disagreement, contradicting each other at every turn. Russell and G. E. Moore were both utterly confused and angered by the needless obscurity and abstractions of the British Hegelians. Russell believed that philosophy needed a Euclid to clarify the basic principles and lay the foundation, and he would be that Euclid. After discovering the work of Leibniz and Boole, Russell dedicated himself to being a logician.

Russell met Peano at the 1900s Second International Congress of Mathematicians in Paris, and went straight home as the congress continued to study Peano’s notation, convinced that Peano and his students were on to the next step in mathematics and logic. Russell completed three hundred pages of his *Principles of Mathematics* before discovering Frege’s work, and it was then Russell had a major breakthrough. One of the problems of Boole and Frege’s work is the problem of designation and denoting. If we say *All men are mortal*, do we mean all men today, all men *ever*, even if they ain’t now or ain’t yet, or *never were*, fictional men like *Unicorn-man*, *Superman*, your unborn sons, who could be real but ain’t yet nohow, or who?

Descartes argued that God has all states of perfection, and actual existence is such a state, so God must exist, which modern analytic philosophers have, thinking of Russell, compared to the argument that Superman has x-ray vision, whatever has something must be to have something, so Superman must be real, as the statement seems to apply to one actual person, even if Superman is fictional. Boole erred on one side, saying empty sets are simply nothing, but the class of *men* could refer to one or more. Carroll was in the middle, and said we can do logic either way, and logicians simply have to be consistent with any method to be effective, saying empty sets are still sets or saying a set must include and refer to one or more individuals.

Frege fell far on the other side of Carroll from Boole, saying that truth is purely formal, and that the phrase *the man who is Superman*, or *the King of France in the year 3000* expresses something formal that only incidentally involves actual men or objects. Frege argued that phrases such as *The King of France is bald* make sense to us, but clearly refer to no one, as there is no King of France today. Russell complained that this meant saying *Mont Blanc is 4,000 kilometers high* would thus tell us nothing at all about the actual *Mont Blanc*. Russell’s *On Denoting* (1905), which he thought was his *finest philosophical essay* and one of his most important and shortest works, sought to solve this problem. Russell’s essay opens with the following passage:

*By a `*denoting phrase*‘ I mean a phrase such as any one of the following: a man, some man, any man, every man, all men, the present King of England, the presenting King of France, the center of mass of the solar system at the first instant of the twentieth century, the revolution of the earth round the sun, the revolution of the sun round the earth. Thus a phrase is denoting solely in virtue of its form. We may distinguish three cases: (1) A phrase may be denoting, and yet not denote anything; e.g., `*the present King of France*‘. (2) A phrase may denote one definite object; e.g., `*the present King of England*‘ denotes a certain man. (3) A phrase may denote ambiguously; e.g. `*a man’* denotes not many men, but an ambiguous man. The interpretation of such phrases is a matter of considerably difficulty; indeed, it is very hard to frame any theory not susceptible of formal refutation. All the difficulties with which I am acquainted are met, so far as I can discover, by the theory which I am about to explain.*

Russell argues that we speak certainly and scientifically about things we never meet or know personally, such as the center of mass of the solar system, but still speak as if they are definite things that are shared and real in our world, such that we do not need to sense them to know that we are exclusively referring to them together, meaning the same thing with our words. We do not even see into others minds, but simply behave with them as if we are in agreement as we speak and listen to each other. Russell says that propositions are either always, sometimes, or never true, and if we say *All men are mortal*, we assert it as if it is always true, but when we say we met a man, and we have not met each man, we are saying we met an x, x is a man, and what we say is sometimes true, at least once, in the meeting of the particular man.

What happens, Russell asks, when we use the word *the*? Wittgenstein later says in his *Philosophical Investigations* that when we say the phrase, *The word THE*, we understand the word two ways without thinking or choosing either one such that we may not notice the first *the* refers to the second, but the second refers to nothing, and is rather an example of a word, which here does not yet say anything about why or how the word is being used as an example. Similarly, we use OR inclusively and exclusively, without a thought to it, and we similarly say *a man* without thinking much as to confusions that could arise unless those confusions happen, as they can if we are misunderstood.

If we say *the father of Charles II*, who would be the beheaded Charles the First, we are excluding everyone but one possible man. Russell says that we must formally add, and does not mention how this is done, *X was the father of Charles II, AND if y is the father of Charles II, then y is identical with x*. Russell says he is not *at present giving reasons*, but *merely stating the theory*, which *seem irreducible* to him, but it is unclear how if x can refer to more than one individual why y cannot as well, leaving several individuals the father, and no one other than that set of individuals. Russell argues *The present King of France* is a meaningful statement that refers to no one and nothing, just as the statement *a round square* does not refer to any particular existing thing, but *The present King of England* refers to one and no one else, which seems an indirect way of reminding the French that Charles II restored the British monarchy while Napoleon did not. Perhaps Russell is still sore about the Battle of Waterloo.

Russell says Frege is wrong to completely separate meaning and denotation, as if the content in a phrase indicating it applies to nothing, one thing, some things, or everything is independent of the rest of what is logical and certain, and reasons that if Frege is right, then *The present King of France is bald* is either true or false according to the *principle of bivalence*, also known as *the excluded middle*, and *The present King of France is not bald* should be false if the first statement is true, and true if the first is false, but the first and second statements refer to no individual at all, and so neither is true. Russell mocks the Hegelians, saying that they love a synthesis in the middle, and would probably conclude that the present King of France wears a wig, and so he is both bald and not.

In Russell’s *The Problems of Philosophy* (1912), he begins asking if knowledge can be so certain that no one can doubt it. We *assume* many things are true all day long. Just as Descartes says that he seems to be sitting by the fire but it could be a deceiving demon or mad scientist, a *man of great industry*, Russell says that he seems to be sitting at a desk writing on white sheets of paper with the sun shining outside through the window, that he believes that the sun is a hot ball of gas which will continue to rise as the earth circles it for many, many years to come, and that others who walk into his room see and believe in a world similar to his, and yet all of this *can* be doubted, as Descartes said in the opening of his *Meditations* at the dawn of modern European philosophy.

Russell says that we can judge with our eyes that a particular thing is a smooth, brown, rectangular, wooden table, we can use words to describe it and others agree with this description, but as soon as we try to be more precise than this we have problems. An actual table is not uniformly brown, perfectly smooth or absolutely rectangular, and others can view the table in different lights and from different points of view. While we and others use common sense to simply say the table is brown, it is not simply brown to the painter or to the philosopher. Appearance and reality are not simply the same, and while the painter looks beyond common sense to study the ways things actually appear, the philosopher looks beyond common sense to study the ways things *actually* are.

Russell gives the name *sense-data* to the particular sensations we have of colors, textures and shapes, which he says are *immediately known in sensation*, even though ancient Greek skeptics that Descartes discussed said no sensation happens immediately, without any mediation. Russell says we know the table *by way of the sense-data*, as if the data is the thing and its medium, so the data itself is the immediate medium through which we sense and know things such as the table, the mind as reality itself. While Russell equates idealism with a denial of the independent, external world, he says philosophy at least shows us what strange possibilities for thought lurk in typically unexamined tables.

Like Descartes, Russell says that we must try to find *some more or less fixed point from which to start*, and that even if we doubt the independent physical existence of the table we do not doubt the existence of the *sense-data* as an immediate appearance, what Hume would say is a sense impression that can lead us to habitually assume ideas. Russell says Descartes invented the valuable method of systematic doubt, but that *I think therefore I am* assumes a stable self which is not itself immediately given. It is the *sense-data* which are certain, of the self and other things equally, even if they turn out to be dreams and hallucinations that do not correspond to actual existing things, and so we *can* critically investigate and gain objective knowledge.

Common sense tells us that tables exist when we leave the room and the sense-data is no longer immediate, and that we all see the same table even as we see it from different points of view. It seems absurd to think that tables disappear when not seen or that there are as many tables as there are viewers, but philosophy should not fear absurdities. There seem to be independent, public objects observable to many in common, but this also seems so in a dream when it is not actually real. We cannot prove the independent existence of our world and the things in it, but we naturally assume this instinctive belief and there is no good reason for rejecting it.

Russell admits that this argument is weak, but argues that we cannot know anything if we throw out all instinctive belief, and we can use philosophy to separate our stronger instinctive beliefs from weaker instinctive beliefs, and beliefs that seem to be true but are not, to leave us with a harmonious system of clarified, isolated and identified beliefs without contradictions and clashes between them. Russell argues, much like Gautama and Aristotle in ancient times, that if nothing contradicts our beliefs, including our beliefs with themselves, we have the best reasons possible to believe the total whole is coherently certain and true.

Russell argues that if we want to know anything other than the obvious and immediate, we need to gather things in our conceptions into *general principles*. We can perceive lightning following thunder again and again, but we must draw a general principle from this if we want to know the relationship between the two. Russell admits that this creates a problem, the *Problem of Induction*: *If we only know general principles by induction, then how can we know anything for certain*? Russell argues, we must found our understandings in the principle of the uniformity of nature, the object of all science. He admits, however, that we are never far from the position of the chicken who is sure that nature is uniform and she will continue be fed each day, even the day when her neck will be wrung and she is served as food herself.

Unfortunately for Russell, he was writing the piece before the work of Einstein, when Newton’s laws of nature were the dominant model of science and physics. Einstein’s work did much to swing scientists away from the use of the word *law* and toward use of the word *theory* to describe proposed conceptions of the regularities of nature, humanity and the universe. In spite of this, we can frequently hear scientists and scholars refer to the *Laws of Nature*, following Newton, who shared with Plato and Aristotle a teleological view of the universe, such that our world behaves according to coherent, mathematical laws that were intelligently designed, not by the ideal mind or the gathering of experience with Russell. Descartes argued much like Russell, but for the objectivity of reason, logic, mathematics, science, but also the authority of the Catholic Church and truth of its theological doctrines. Russell, an open atheist, argues that Descartes’ arguments hold for science, but not for religion.

In *The Empiricist Answer to Skepticism*, Russell calls out Hegelians and skeptics by name and says we cannot get the sort of independent certainties we want to call facts or laws by their methods. Russell says that we can *whittle away* at our theories to get to the *pure data*, the facts beneath interpretation, much as Frege thought logic worked beyond Boole. Russell has faith in a *minimal theory*, and though he concedes to the Hegelians and skeptics that there is always some uncertainty in truth, some room for error, he argues that some data has *independent credibility*, seemingly known *in itself*, as the Kantian *thing-in-itself*. Russell argues that some things must be more than opinion, as opinions can contradict each other but the true fact never contradicts itself or other things, and that we should give the *greatest weight* to that which is most regular and most certain, and in this way form our principles and propositions.

Russell wrote, “*the business of philosophy, as I conceive of it, is essentially that of logical analysis, followed by logical synthesis*.” We should take things completely apart, into their basic components to *atomize* them into the fundamental pieces, and then build up the thing into the combinations of the parts that show how it works in all of its possibilities. This follows the entire project of Hegel, but in total rejection of contradiction, which Hegel sees in every step. For Russell, analysis is less *definition* than *reduction*. Logical reconstruction substitutes known, certain entities for unknown, unsure ones, until the whole is analyzed, and then synthesizes out of basics that can be sensed in the world, a position Russell called *Logical Atomism*.

Unfortunately, there is a problem of whether or not we *can* atomize things entirely that can be called *Wittgenstein’s broom*. Russell’s protege argued in his later work that if we call for a broom, we are calling for the broom handle and brush, but not *separately*, so we are calling for all of the atoms in the broom, but not that way, or for any atomic purpose. In the Philosophical Investigations, Wittgenstein suggests that the pieces in a chess game could be atomized and categorized differently as the game goes on with interests and positions shifting, such that sometimes a valuable piece would be best, and other times a piece that can jump over others becomes far more valuable than the piece with the higher score. Unless we can compartmentalize contexts, this could be a fatal flaw to Russell’s Logical Atomism, even if the shifting situations can, moment by moment, be atomized in particular passing contexts.

**Basic Proofs In Sentential Logic**

Now that we have demonstrated the rules of replacement and additional derived rules, we can explore doing proofs in Sentential Logic (*SL*) without truth tables. Why should we abandon truth tables for another method? In the last chapter, we saw that truth tables can become quite complicated when more than two statements are being evaluated. What if we have more than two or three statements, and we want to derive conclusions? We could use truth tables, but we would need sixteen, thirty two, or exponentially more possible cases of truth values, which would be needlessly confusing. By learning a simplified method of proof that relies on rules already demonstrated by truth tables, we can eliminate using T and F altogether.

Let us say that we know several expressions are given as true, such as (A > B) and (B > C), and we want to determine whether or not we can arrive at a particular conclusion, such as (A > C). We could prove that in all possible cases the Hypothetical Syllogism is true, but we could simply assume a single case, in which (A > B) and (B > C) are known to be true, and attempt to derive (A > C) from this single case, using the rules of replacement and derived rules we also know to be true. Consider that if I know (A > B), and (B > C), and (C > D), and (D > E), and (E > F), there should be a way of concluding that (A > F) without determining each truth value for each expression for sixty four possible cases. This way, we can take a set of several complex expressions, such as A, ((~A > (B ^ D)), (D v E), F, (G > H) etc., and draw further conclusions from any complicated set without determining all possible truth values for each expression.

A deductive proof always starts with one or more premises, with a bar underneath to show where the given and assumed premises end. Then, beneath the bar, each line is a conclusion derived from the previous premises and conclusions using a rule, which must be justified in standard notation.

There are three types of rules for doing deductive proofs in SL. First, is **reiteration**, repeating something that has already been determined. This is even simpler than our first truth table we used to demonstrate negation. We are going to assume that a single simple premise, A, is true, and from this prove that A, our premise, is true. Notice that we give each line of proof, including premises and each conclusion, a number to the left of the proof such that we can refer to it later, and we give each conclusion we derive from the premises a justification. In this case, we reiterated our only premise, A, and symbolized this justification as R1, a reiteration of line 1. When my mother, aunt and uncle were very young and going on car trips, they would sing, “We’re here, because we’re here, because we’re here, because we’re here…” to pass the time and annoy their parents. This is also an example, a purposefully annoying one, of reiteration.

Other than reiteration, all additional rules involve either **introduction** or **elimination**. For each of these, based on one or several lines one introduces or eliminates a connective, rule of replacement or derived rule, each of the elements we demonstrated in previous chapters with truth tables. Here, to begin, is the rule for **conjunction introduction**, adding two lines together to introduce an AND. Notice that with two premises, A and B, we can determine A ^ B. Our justification is that we can introduce (I) an AND (^) given lines 1 and 2. You can introduce an AND from lines that are derived from the premises. Let us say that our premises are A, B and C, and we want to derive (A ^ B) ^ C as our conclusion.

Notice that we first introduced an AND, then reiterated line 3 (C), and then introduced another AND to derive our desired conclusion. Some might say that line 5 is redundant, as we could have introduced an AND given lines 3 and 4 rather than reiterate line 3 as line 5, but we do this to show each step of our work carefully. It is better to use line 23, 24 and 25 to derive line 26 than it is to use line 2, line 16, and line 25 to derive line 26. Reiteration may seem redundant, but it keeps our work simple. This makes deductive proof easier than truth tables, which sometimes require glancing between columns that look similar and can lead to mistakes.

If we know A ^ B, we can also eliminate the conjunction, leaving us with either A or B alone. Here is the rule for both types of **conjunction elimination. **If we started from the premise (A ^ B) ^ C, and we wanted to derive A for our conclusion, we would have to first eliminate C, leaving A ^ B, and then eliminate B, leaving A. While this may seem obvious and redundant, keep in mind that we will be dealing with many variables and complex expressions, and will need to carefully show our work at each step.

**Disjunction introduction** also seems silly in its simple form, but in complex proofs it can be quite useful. If we know A, we know that A v B, or A v C, or A v (~F > G), or A v anything else. If I know that my name is Roberto, then I know that my name is Roberto OR I am a purple chicken. While this would mean that if we proved my name is not Roberto, it would prove I am a purple chicken, if we start knowing my name is Roberto, and introduce a disjunction, we will not be able to prove my name is not Roberto without contradicting ourselves.

**Disjunction elimination** works like the Material Conditional. If we know A v B, and we know ~A, then we know B. Similarly, if we know A v B, and we know ~B, we know A. Here is the rule for disjunction elimination, of both types.

**Conditional introduction** involves adding another hypothetical level to a proof, another vertical and horizontal line to the right of the original vertical line of proof. As we have seen, any line that can be derived to the right of the original vertical line is true given that the premises are true. This means that our proof is already hypothetical. It may be that the premises are wrong, but if we work by the rules, step by step, our proof is valid even if the premises are wrong. Just like we supposed that there could be a hypothetical universe in which puppies are evil, we can add any number of hypothetical proofs within hypothetical proofs to show what would be the case if we start from any set of premises.

Notice that we start with the single premise, A. We then suppose, hypothetically, that B, introducing another vertical and horizontal line to indicate that we are on another hypothetical plane of existence within our proof. We do not know that B is true, given A, our original premise, but we are seeing what we could determine is true if we suppose that B. Because we know our original premise A, we can reiterate it. We can then introduce a conditional, B > A. Notice that we are not proving ~A, nor are we proving B. We are proving, given the premise A v B, then IF ~A, then B. Notice also that our justification uses a dash between line numbers, indicating a continuous range, rather than numbers separated by commas.

**Conditional elimination** does not require introducing another hypothetical level to our proof. If we know A, and we know A > B, then we know B. Here is the rule for conditional elimination. Let us say that we know A, B and (A ^ B) > C, and we want to derive C. First we must use conjunction introduction to combine A and B to get A ^ B, and then use conditional elimination to arrive at C:

We can use conditional introduction and elimination to prove the Hypothetical Syllogism. Given both of the premises A > B and B > C, we can hypothetically suppose A, derive B through conditional elimination because of A > B (line 1) and A (line 3), then similarly derive C through conditional elimination because of B > C (line 2) and B (line 4). We can then conclude, by conditional introduction, that A > C, given lines 3 through 5:

**Biconditional introduction** combines two conditionals together, which requires us to first have presumed or derived both conditionals, and then make two separate hypothetical suppositions. Notice that in the justification for line 7, the biconditional introduction and our conclusion, there are two sets of numbers, each containing a dash and separated by a comma. While it seems redundant to create two separate hypothetical suppositions when we could simply justify the biconditional introduction with the premises, lines 1 and 2, remember that proofs can get complicated and can involve many expressions and steps. **Biconditional elimination** works just like conditional elimination, except we can not only derive B given A and A <> B, but we can also derive A given B, as it works like a conditional both ways.

When we studied connectives and truth tables we covered negation first, as introducing it is simple and easy, but with proofs we cover negation last, as **Negation introduction**, known as *reductio ad absurdum* to Aristotle, but not in Latin, *reduction to absurdity* to us who speak English, involves a hypothetical supposition like we use in both conditional and biconditional introduction. Given that contradictions are impossible and necessarily false in formal logic, if we arrive at a contradiction such that an expression and its negation are both true, then we know that at least one of the premises must be false or the rules improperly applied. This means that if we make a hypothetical supposition with a single premise and follow the rules properly, we can prove that the premise is necessarily false. Let us say that we know both A and B > ~A. From these premises, we can use negation introduction to prove ~B. Notice the use of repetition in line 5. Typically, negation introduction and elimination involve repetition, bringing in something previously presumed or derived to contradict something derived in a hypothetical supposition. With negation introduction, we can demonstrate both the impossibility of contradiction (admittedly, already presupposed) without any premises. Negation elimination is essentially the same as negation introduction, through one starts by supposing a negated expression and through negation eliminates the negation.

We can use negation introduction to demonstrate Modus Tollens. To do this, we must assume A > B, then hypothetically suppose ~B, then, within our hypothetical supposition, hypothetically suppose A. This means that we are creating a possible universe within a possible universe, a dream within a dream (much like the movie Inception). In this supposition within a supposition, we can derive B, as we know A > B, our original premise, which we introduce into the supposition within our supposition. Then, with repetition, we introduce ~B from our first supposition into our second supposition within a supposition. Remember, we have proved neither ~B or A, we have merely supposed ~B and then supposed A within our supposing ~B. Now we have a contradiction within our second supposition within a supposition, so we can introduce a negation and have proved that, within our first supposition, ~A. Finally, we can conclude that, given our A > B, we can deduce ~B > ~A.

**Using Rules of Replacement and Derived Rules**

Now that we have covered repetition, as well as introduction and elimination for each of the connectives, we can have a new appreciation for the rules of replacement and derived rules. We can use them in deductive proofs to save ourselves a great deal of work. For instance, rather than go through the motions of our last proof, we can simply use **Modus Tollens** as a rule, symbolized as MT in our justification, and have a two step proof. This becomes even more useful when we allow ourselves to use rules of replacement and derived rules to transform expressions within expressions.

**DeMorgan’s Theorem** allows us to exchange ~(A v B) with ~A ^ ~B, as well as exchange ~(A ^ B) with ~A v ~B. The **Material Conditional** allows us to exchange A > B with ~ A v B, as well as exchange A v B with ~A > B.

The **Hypothetical Syllogism** allows us to derive A > C given we have presumed or derived A > B and B > C.

Derived rules do not allow us to transform expressions within expressions. They do, however, allow us to transform expressions composed of expressions. If we know that something leads to something, such as (C v D), and we know that something leads to something else, we can use the rule to cut out the middle link.

The **Dilemma** allows us to derive C given we have A v B, A > C and B > C. Notice that in the justification for line 4, there are three numbers, separated by commas. Similar to the Hypothetical Syllogism, we can transform expressions of expressions with the Dilemma. The lines used for justification also do not have to be in any particular order.

For example, if we know that both (Z ^ B) and ~D lead to C, and we have a disjunction between those two , then we know that C is an inevitable conclusion from the premises.

Let us conclude by examining several long proofs, showing how to solve each one, trying to figure out which kinds of introduction, elimination and rules of replacement are used in each step in deriving the final conclusion in the last line from the premises in the first lines above the horizontal bar.

In our first example, we know three complicated premises involving A, B, C, D and E, which would make this a nightmare-level-length of a truth table with sixteen cases, requiring sixteen Ts and a lot of work to get there before proving the final, simple conclusion E. To prove it easily, we can start in line 4 with conjunctive elimination, line 2, which gives us A from A ^ C, the sort of simple step we would not make in conversation. Then, using conditional elimination and line 1, we know ~B. Using conditional elimination and line 3, because we know ~B, we know (D ^ E) must be true on the other side of the OR, and again, because we know a conjunction, we can take away one side with conjunction elimination and the previous line 6, leaving us with E.

In our second example, longer than the first and quite complicated, we only have two premises, but we will go full *Inception*, and put a proof within a proof within a proof, a conditional within a conditional, within a proof, which is structured exactly like a conditional, determining what we can conclude IF the premises are true. We start by hypothetically supposing, introducing a conditional within our proof, that ~(B v C). We use Modus Tollens to turn it inside out, and the OR into an AND. We then, within our proof within a proof, suppose hypothetically that A, which leads with conditional elimination and line 1 to B ^ ~C. We use conjunctive elimination to give us B, but this leads to a contradiction within our dream within a dream, both B and ~B, which is what we were TRYING TO DO ALL ALONG!

Now we know ~A, because we tried A hypothetically, and it produced a contradiction. Given premise 2 and conditional elimination, we know C ^ ~B, which means we know C, but this leads to a second, *even more thrilling* contradiction, as we know ~C from line 4. That means that both our dream within a dream and the dream that dreamed the second lead to contradiction, which you might think means the first contradiction wasn’t actually a contradiction, but that’s not how this works. If an option contains a contradiction, it is impossible in its place in the proof, regardless whether it turns out true in the proof overall. Finally, because we found a contradiction pursuing line 3 hypothetically, we know B v C, and are left with an open question, as we are in much of life, even after formal logical proofs.

While it is nice to conclude something, we have, essentially, argued that if we know *If I have an apple, I have a banana and I don’t have a coconut,* and we know *If I don’t have an apple, I have a coconut and I don’t have a banana,* we have proved that I have a banana or I have a coconut, whether or not I have an apple, and we would only prove that if we were asked to prove that, as these premises could lead to quite a few conclusions in different directions, depending on what you are seeking in the proof.

**Ninth Assignment: Now that we have worked our way through the forms of logic we learn in the class, moving past formal logic with the final two lectures, I want you to write a two page reflection on what you think about logic and human thought given what you have learned, what you like and what you don’t like about the history, practice and forms. What does the study of logic do, and what could the study of logic and thought lead to?**

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