Logic – Venn & Peano
Super-Useful Syllogisms & Logical Diagramming
We’ve covered Mill, De Morgan, Boole and Carroll, British logicians who were trying to work with symbolic algebras for systematizing logic in the years before Venn, Peano, Frege and Russell founded formal logic on Boolean operations with various but somewhat developmentally consistent notation and practices. Formal logic did eventually settle into a form we teach in classrooms today, but even so various symbolic notation is used in textbooks and there are still widely differing interpretations of nearly unified practices, of ways of using the symbols in the same ways as everyone else.
What was a pastime for philosophers and mathematicians following Aristotle became the Boolean foundations for the modern electric world, with logic gates controlling increasingly complicated circuits of information. The forms of communication and debate that Gautama, Aristotle and Gongsun Long debated about are intelligible and meaningful to us, and we can use them to communicate, but our devices also use these forms as systematic co-operation, whether or not this is meaningful communication to the devices themselves or to us. Allan Turing’s followers claimed even light switches understand two things, on and off, though many wouldn’t agree. To Turing’s credit, he didn’t want to live in a world that treated human-like robots as mere things, fearing what that would lead to, which is a very utilitarian argument, worthy of Mill, that strives for the least pain we can project for the most.
In his first book, The System of Objects (1968) the French postmodern philosopher and sociologist Baudrillard says there are functional, nonfunctional, and metafunctional objects, useful, sub-useful and super-useful things that surround us, intertwine with our lives, and order them and us. A hammer is useful as we use it to hammer in nails, occasionally open pain cans or defend ourselves from biker gangs one fateful evening, so a hammer somewhat orders and structures our lives as a standard thing we use along with others. A beautiful painting is sub-useful, as is a glass of wine, because we use it for the immediate moment and enjoyment, with little if any thought to how it will be useful in the future or for this or that in particular.
Super-useful things, the metafunctional as Baudrillard says, are useful for much more than one thing, like writing, mathematics, government, science, religion, telephones, electronics, computers or the internet. These things, like smartphones that many carry around the world, are useful for all kinds of things, for enjoyment in the moment, like art and games, for planning out one’s life overall, like dating apps, spreadsheets, political websites, and for most things if we so use them, and so they centralize, accelerate, commodify, order and determine much of our lives, including how we feel and what our lives mean to us in ways we can’t count. Logic, including Aristotle’s Organon, was such a super-useful text, as are his four perfect forms of the syllogism at its core. Let’s consider the four forms again, before getting into Venn, as Venn diagrams are still the best visual way to teach Aristotle’s syllogisms, a counting-board that shows where this or that is or where this or that leads.
BARBARA, the Positive Universal Syllogism: If All A is B, and All B is C, then All A is C. If all things are possible to think if you Shut your eyes and try very hard, as the White Queen suggests to Alice, and if all impossible things are things indeed, even if they, unicorns and we are all quite mental, then Alice can think six or more impossible things before breakfast if she shuts her eyes, imagines, and tries very hard, as the White Queen implies but doesn’t say directly, meaning what she doesn’t say syllogistically. In Venn diagram form, if A is entirely B, and B is similarly C, then A must also be C.
CELARENT, the Negative Universal Syllogism: If All A is B, and No B is C, then No A is C. If All ways are mine, as the Red Queen says, and None of what’s mine is yours, as the Duchess moralizes, then none of these ways are yours, is what the Red Queen means but doesn’t say, which we understand and infer quite syllogistically from what is given in her words. As a Venn diagram, if A is entirely B, and no B is C, then no A can be C.
DARII, the Positive Particular Syllogism: If Some A is B, and All B is C, then Some A is C. If the White King says he sent almost all his horses along with his men, but not two of them who are needed in the game later, and if Alice has met all the thousands that were sent, 4,207 precisely who pass Alice on her way, then Alice has met some but not all of the horses, namely the Red and White Knights who stand between Alice and the final square where she becomes a queen. As a Venn diagram, if some A is B and all B is C then some A must be C.
FERIO, the Negative Particular Syllogism: If Some A is B, and No B is C, then Some A is not C. If all things are dreams, as Tweedle Dum and Tweedle Dee tell Alice, and some dreams are untrue or not ours alone, then all things are somewhat untrue, and somewhat aren’t ours alone, which is what Tweedle Dum, Dee and the Red King dreaming silently imply, but don’t say. As a Venn diagram, if some A is B and no B is C then some of A is C. As Aristotle says, if we have only some and no all or none, we can’t draw syllogistic judgements completely, leaving us with only a relative, somewhat satisfying conclusion, just as the Red King silently dreams and says nothing to Alice after she happily dances around hand in hand with both twin brothers.
Venn & Intersecting Circles
John Venn (1834 – 1923) was born near Hull on the east coast of England, lectured in Moral Sciences at Caius College, teaching philosophy, ethics and logic, wrote The Logic of Chance (1866), Symbolic Logic (1881) The Principles of Empirical or Inductive Logic (1889), and developed the Venn diagrams in 1880 as a method for teaching logic in intercollegiate programs to students of all majors, “the diagrammatical device of representing propositions by inclusive and exclusive circles.”
The Tree of Porphyry was used in medieval Europe to teach Aristotelian syllogisms, with an A branching into B and C, etc. Thus, it can teach the first two perfect universal forms, showing that D is B, and B is A, so D is A, and that B is A, B isn’t C, so C isn’t A. Leibniz was one of the first modern European logicians to study logical diagrams, but only a small portion of this was published and preserved during his lifetime.
The Swiss mathematician Leonhard Euler (1707 – 1783) had earlier used circles to diagram propositions such as All B is A and Some A is not B, such as boats (A) including sailing boats (B), but more than just sailing boats (Some A is not B), with one circle (B) inside another (A), but Euler diagrams did not show more than one proposition consistent or complementary with another, but rather aided visualizing each step at a time. The French logician and mathematician Joseph Diez Gergonne (1771 – 1859) studied how circular diagrams could show substructures of syllogisms, with two circles related in 5 ways, apart, overlapping, identical, one in the other or this inverted.
Venn realized the circles can intersect, and the basic syllogisms and forms of logic can be represented with a small number of circular diagrams. For example, taking two intersecting circles, we have, like the Catuskoti of Nagarjuna, four regions, A, B, A and B, and neither A nor B, so if we want to say that all B is A, but some A is not B, we can cross out or fill in the B region, leaving us no B that is not A, but some A that is not B. For three classes, and all the ways they can intersect, Venn placed three circles sharing part in the middle. The German mathematician Mich had created a similar three circles just before (1871), and Venn drew on his work, but Mich didn’t see he could use all the intersecting spaces, merely some of them, ignoring the other unused sections.
Venn carefully surveyed all the work of those before him more than anyone, and based his on the algebra of classes of George Boole. Venn’s absence of a universal class was a bit of an embarrassment for other logicians. Venn says the limits of discourse are “whatever I chose to consider them,” and may cover a sheet of paper, or contained in a larger drawn circle, or go off indefinitely. Carroll said he was astonished that Venn allowed the fourth subset the rest of the Infinite plane to range about in and Carroll’s logical diagram method closes the area that “Mr. Venn” allows “liberal sway” but now is “dismayed” to find itself cabined and confined in a limited cell like the other classes, as opposed to being the infinite sea stretching off into the distance.
Venn wrote that Boole’s work is not so much reducing logic to mathematics, but enumerating and symbolizing general logical processes, actualities, and probabilities such as dichotomy and subdivision with symbols, and the circular diagrams can serve the same purpose. In binary, which is how Boolean algebra is done and the basis of computing systems, there are 0s or 1s for each class, so for three circles A, B and C, 000 means none of them, 111 means all of them, and 101 means A, not B, and C. In the 15th century, these were known as the Borromean rings that symbolize the Christian Trinity, found in the crest of the Borromeo family of Northern Italy.
Lewis Carroll was Venn’s double at Oxford, where he tried to represent the universal set as a closed area of rectangles rather than an open area with circles, writing in his Symbolic Logic that his diagrams resemble Venn’s, and he also marks the parts as occupied or empty, but with a closed area that places Venn’s infinite space in a simple compartment like all the other possible spaces. Carroll originally presented his diagrams in 1887 in The Game of Logic, over 20 years after Venn published his. Venn gave up at 4 categories in the original idea, but Carroll adapted his to take 8 or more, given the areas are broken up into non-continuous parts, such that A and B areas occur in different places apart. Much later, the mathematician Edwards figured out that drawing the circles as cogwheels allows many more to fit together simply.
Lewis Carroll had Venn’s works, though Venn’s diagrams were developed after both Alice books were completed, so Carroll could not have used Venn diagrams to plan or plot out either of Alice’s adventures, but he was struggling to create a square-shaped counting-board method that visually presented syllogisms and Boolean logical operations of inclusion and exclusion before and after Venn’s diagrams became the standard for teaching syllogisms and Boolean operations much as Boolean operations later became the standard for electronic circuits. It is arguable that Wonderland and through the Looking Glass are themselves visual presentations of logical forms, and in more ways than one, with the first book much like a trial in a court and the second book much like a game of chess, and with the second book mirroring and canceling out the other, much as fractions and Boolean algebra rely on elimination of terms on both sides of the bar to narrow things down to the conclusion and solution of the problem.
Peano & Set Theory
Another powerful and useful addition to formal logic that serves as a visual and mathematical bridge between the work of Boole, Carroll and Venn and the work of Russell and Wittgenstein is Set Theory, founded by the Italian logician Giuseppe Peano (1858 – 1932) and extended by the work of the German logician Frege (1848 – 1925). We will cover Frege, Russell, early Wittgenstein and the basics of formal proofs next week, but before that we will consider the arithmetic theory of Peano, who was unfortunately never stationed in the Italian army at Fort Issimo, so we can’t make that terrible musical theory joke, nor should we, and complete one last assignment on Truth Tables and the rules of replacement.
Peano was born on a farm in Spinetta of Northern Italy, and then taught at the University of Turin, so if Peano had grown large enough, he could be a Gran Torino. He tried to popularize a universal utilitarian language he created out of simplified Latin without inflections called Latino sine flexione, and gave a lecture about it in 1908 that explained the language as the lecture turned from Italian into his simplified Latin with each step. He wrote a textbook on calculus, created a space-filling curve which prefigured the modern fractal, and at one point was so frustrated with publishing delays that he bought his own printing press to print his own works and distribute them.
At the Second International Congress of Mathematics in 1900 Peano met Bertrand Russell, who left the congress and went home to London to study Peano’s logical notation. Russell completed more than three hundred pages of his Principles of Mathematics before discovering Frege’s work, which he combined with Aristotle, Leibniz, Kant, Boole, Carroll and Peano to form the basis of what became Logical Positivism, the close ancestor of Analytic Philosophy. Unfortunately, the next year in 1901 Russell discovered a paradox in Frege’s work, inspired by a logical problem about a barber by Carroll we will discuss next week.
In basic Set Theory, union is a set of A and B together, intersection is all that is both A and B, the center slice of the Venn diagram with two circles overlapping, set difference between A and B is whatever A isn’t also B, like an exclusive OR but that works one way, also called the compliment of B in A, symmetric difference are the parts of A and B they do not share, and the Cartesian product, as if Descartes was a drug kingpin, is all possible ordered pairs of sets A and B. Wittgenstein’s truth tables use a Cartesian product in the arranging of cases, as the sets of TT, TF, FT and FF are all possible combinations of the truth values for all four possible cases.
Peano’s arithmetical system assumes 5 postulates: 1) Zero is a number, 2) Anything that follows a number, its successor, is itself a number, 3) No two numbers have the same successor, something that follows something similar in type, 4) Zero doesn’t follow any number, is not a successor, and 5) If Zero has property P, and every number with P has a successor that follows it that also has P, then all numbers have P. Unfortunately for Peano, Godel’s incompleteness theorems threw a large wrench into his plans for a complete and unified set that grounds all of mathematics coherently together. In Logicomix, a graphic novel I recommend about the life of Russell and formal logic, after hearing Godel’s theorems, Peano is pictured as pacing in circles, cigarette puffing away, saying It’s impossible! over and over to himself.
Rules of Replacement
In the last truth table assignment, we demonstrated De Morgan’s theorems, such that ~ (A ^ B) and (~A v ~B) are equivalent and ~ (A v B) and (~A ^ ~B) are equivalent. We can use the biconditional connective to see whether or not two expressions are equivalent if we connect two expressions with a biconditional symbol, evaluate this new biconditional expression with the truth table method, and find that it is true in all possible cases. Likewise, if we find that the biconditional is not true in all possible cases, we know the two expressions aren’t equivalent. In formal logic, there are several basic equivalences known as the rules of replacement that are used to make substitutions in proofs other than De Morgan’s two theorems, including Commutativity, the Material Conditional, and Modus Tollens. We are now going to show that we can prove each of these using truth tables.
Commutativity is when an expression is the equivalent of itself when the things on either side are reversed. We can show with truth tables that AND and OR are commutative, such that AND still comes out TFFF and OR comes out TTTF whether or not we exchange A and B on either sides. This means that saying I have an apple and a banana is the same thing as saying I have a banana and an apple and saying I have an apple or a banana is the same thing as saying I have a banana or an apple with regard to truth as far as modern formal logic finds it. Of course, logically the phrase People are people is tautologically meaningless, telling us nothing, but it is something worth saying often in life, and similarly it does matter which things are included first or last or offered first or last sometimes, and a decent attorney would know both the formalities and circumstances.
We can also see by constructing a truth table that IF-THEN is not commutative. This means that saying If I have an apple then I have a banana is not the same as saying If I have a banana then I have an apple, which is exactly what Alice is taught by the Tea Party, the Hatter saying I see what I eat isn’t the same thing as I eat what I see. The Hare follows with another example, I like what I get isn’t the same as I get what I like. The Dormouse offers another, I breathe when I sleep isn’t the same as I sleep when I breathe, but the Hatter again interrupts and says that the two are the same for the Dormouse, as the Dormouse is always both breathing and sleeping.
The Material Conditional shows us relationships between OR and IF-THEN. The three parts are first, (A > B) <> (~A v B), second (A v B) <> (~A > B), and third (A v B) <> (~B > A). The first part means that saying, If I have an apple then I have a banana is the same thing as saying Either I don’t have an apple or I have a banana. The second and third parts mean that saying Either I have an apple or I have a banana is the same thing as saying If I do not have an apple then I have a banana, and is also the same thing as saying If I do not have a banana then I have an apple. This can be confusing, as it seems to say that OR is used exclusively when we know that it is used inclusively, but when we construct the truth tables we can see that an inclusive OR does mean that if one side is false then the other side must be true, both ways.
Modus Tollens, Latin for The Way of Denial, due North in Egypt, is a dual reversal of the conditional connective, both inverse and converse, both sides negated and switched, such that If A then B becomes If not B, then not A. We have just seen that conditionals are not commutative. A > B is not the same as B > A, but it turns out that if you add a NOT to both sides, it is. While the name comes from medieval European logicians, who gave it the Latin name, it was known by the ancient Greek stoics and can be found in the difference between the positive and negative proofs of the Indian Nyaya school of Gautama. The example of the positive proof is, Whatever is created is impermanent, like a cup, and sound is created, therefore sound is impermanent. The example of the negative proof switches and negates created and impermanent, and says, Whatever is permanent is not created, like the soul and sound is created, therefore sound is impermanent. In ancient India, the Nyaya, an orthodox Hindu school, argued that the soul or self was permanent and uncreated.
In addition to the rules of replacement, there are two standard types of replacement useful for proofs, the Hypothetical Syllogism and the Dilemma, known as derived rules. For these two, we will need to construct truth tables that involve three statements, symbolized by A, B and C. This will mean we will need eight cases of truth values rather than four. The Hypothetical Syllogism is nothing more than the Barbara form of Aristotle. When we say All As are Bs this is the same as saying If it is an A, then it is a B, or A > B. This means that the expression, If A then B and if B then C, then if A then C is equivalent to Barbara. Thus we can substitute If A then C for the longer expression if we can chain each element legitimately together.
The Dilemma, also known as Damned if you do, damned if you don’t shows us that if we know that (A > C), (B > C) and (A v B) are true, then we also know that C must be true. If we know that both A and B lead to C, and that A or B must be true, whether or not the OR is inclusive and we can have both A and B, then either way C will be true. Let’s say that I like to eat both apples and bananas with coconut, but I do not like eating apples and bananas together. Whenever I go to the market, I make sure to buy either an apple or a banana (A v B). When I buy an apple I also buy a coconut (A > C), and when I buy a banana I also buy a coconut (B > C). If my friend wants me to buy either an apple or a banana, but is afraid of coconuts, she is faced with the dilemma that, either way, whichever fruit I choose to buy, I am certainly going to buy a coconut.
To make a truth table out of this, we first determine the truth values for (A > C) and (B > C), then determine the truth values for ((A > C) ^ (B > C)), then determine the truth values for (((A > C) ^ (B > C)) ^ (A v B)), and only then, using those truth values and the truth values for C, can we determine the final truth values for the entire expression, ((((A > C) ^ (B > C)) ^ (A v B)) > C).
Eighth Assignment: On a piece of paper construct a truth table that proves the conditional or biconditional expressions for Commutativity, the Material Conditional, Modus Tollens, the Hypothetical Syllogism and the Dilemma. Send me the four truth values for each biconditional expression.