# The Mathematical Analysis of Logic: Being an Essay Towards a Calculus of Deductive Reasoning By George Boole

I thought this book provided a brief but thorough explanation of the connection between the usage of logic within the thought process and its use in mathematical reasoning. Kindle Edition, Printed Access Code, Paperback super Kindle Edition, Printed Access Code, Paperback I really enjoyed this read. It's contents are expressive and artistic for a mathematician author, which is what attracted me to it. Obviously it takes some study and reflection to follow his train of thought but the man was a genius! He combined scientific thought and artistic reasoning to fill a void that not only gave us the groundwork for computer science but also allowed for advances in industrialism. Kindle Edition, Printed Access Code, Paperback Je ne l'ai pas encore lu, mais lire les travaux de Boole ne sera sûrement pas décevant. Kindle Edition, Printed Access Code, Paperback Concise and straightforward, using simple, easy to grasp mathematical principles, this powerful masterwork shows how to SWIFTLY sort out TRUTH and destroy FALLACY.Unlike the Socratic `circle of debate' or the `Organon' defined in Aristotle's `Prior and posterior analytics' which utilises a complex NEGATIVE method of HYPOTHESIS ELIMINATION using `cross examination' to search for commonly held truths that shape OPINION but often FAILS to sort out truth from fallacy (see my example further down), George Boole uses seven LOGICAL CONSTRUCTS and his unique BOOLEAN ALGEBRA to express LOGICAL PROPOSITIONS as a MATHEMATICAL EQUATION to establish or deny TRUTH with ABSOLUTE CERTAINTY (see example further down),But the method goes much deeper than that because Boole's BOOLEAN ALGEBRA forms the backbone of SYMBOLIC LOGIC as used in all DIGITAL COMPUTERS, so if you are a COMPUTER PROGRAMMER seeking to replace all of your nested `if then else' nonsense with powerful mathematical formulae that swiftly evaluate your propositions, or a DIGITAL HARDWARE DESIGNER wishing to swiftly design robust logic circuits with the minimal amount of `gates', then this book is your CANON nothing else comes close.To sum up if you or your children wish to have a ROBUST method to evaluate the VALIDITY of an argument, or you want a powerful method to design robust software and reliable hardware then this publication is an ESSENTIAL part of your arsenal.A little tip: when you read the book don't try and `dip into it' it will be confusing because each construct builds on the ones that precede it. As the king said to the white rabbit in `Alice in Wonderland' Begin at the beginning and go on till you come to the end: then stopBefore I go on to show the comparisons of Aristotle and Boole, here are a few everyday postulations that are SHOT TO PIECES using Boole's method.BUY ONE GET ONE FREE is a FALLACY! You CANNOT purchase ONE you are COMPELLED to purchase TWO! It's a fallacy that lines the pockets of the supermarkets and causes MILLIONS of TONS of food to be THROWN AWAY on a DAILY basis food that would feed those in WANT and DESTITUTION.HALF PRICE is a FALLACY. You CANNOT have HALF a price any than you can have `half a hole' a hole is a hole is a hole a price is a price is a PRICE.In Boole's masterwork, to aid persons coming from the CLASSICAL school of ANALYTICAL LOGIC with comprehending his BOOLEAN ALGEBRA, George Boole refers to the ORGANON a LOCI developed by mediaeval Muslim scholars in the fourteenth century as an aid to remember the `valid moods' when carrying out a LOGISM (`analysis') which leads to SYlogism analysis of the SOUNDNESS of the logic).Although Boole is about to DECIMATE the whole thing at a stroke (as did Lewis Carroll in HIS masterworks `Symbolic Logic' and `Game of Logic' (BOTH books now available in `Mathematical recreations of Lewis Carroll' another `must read' publication), by way of explanation, so that you may get your head around it all when it pops up, there are FOUR logical constructs, these being:ALL `A' are `B' (All new cakes are nice) represented in by the VOWEL `A'NO `A' are `B' (No new cakes are nice) represented by the vowel `E'SOME `A' are `B' (Some new cakes are nice) represented by the vowel `I'SOME `A' are NOT `B' (Some new cakes are NOT nice) represented by the vowel `O'Which yields 256 LOGICAL SUPPOSITIONS (`suppose that ?) used in CLASSICAL LOGISM to DEBATE a set of PREMISES for example AAA; ABA; BEI; BIE; IEB; EIB; IEA you get the idea.Of the 250 `possibilities', Aristotle (wrongly) considered that only NINETEEN of the 250 permutations were VALID and that these must be evaluated IN A SET ORDER.Aristotle was MISTAKEN there are some FORTY ODD valid premises and these can be evaluated in ANY order one wishes to choose. Lewis Carroll was the first person to realise this as you will discover for yourself if you purchase the book `Mathematical recreations of Lewis Carroll' (available from ) and play his `Game of logic' (which is just fabulous); whilst Boole was the first person to express logic MATHEMATICALLY.The nineteen LOGICAL MOODS use the VOWELS in a series of LOCI words to jog the MEMORY of the person evaluating a proposition of the ORDER OF EVALUATION.For example, the word `Barbara' stimulates for the order `b A rb A r A'; `calarent' yields `c A l A r E nt; Darrii yields `d A rr I I; `Ferioque' yields f E r I O qu E and so on (snore) ALL of which is REDUNDANT and NONE of which is required to master logic. One wonders if the Organon was developed by the church to keep the common man OUT.I have intentionally explained all of this to you because Boole refers to the Organon when he explains WHY it is NOT needed when using his MATHEMATICAL method (Mr Carroll you were SO close to making this HUGE leap in mathematics a leap that led to the invention of the machine that changed the world forever the COMPUTER)So since you are showing keen interest as promised before giving an illustration of the Boolean method, here is a little discourse on the FLAWS within the Aristotelian Method.In Lewis Carroll's masterworks `Alice in Wonderland' and `Alice through the looking glass (and what she found there)', Alice encounters many trials and tribulations to quickly teach children many valuable lessons about life in a humorous and memorable way.In `Alice in Wonderland' Alice finds herself trapped in a room surrounded by LARGE locked doors doors that she is BARRED from entering some for her own good such as the Duchesses house (pig and pepper) which teaches Alice to `look before you leap' and some because she is not PERMITTED to open them (because she is a woman) so Alice cannot make any progress (glass ceilings) which include the `gentlemen's club' where resides the `mad hatter' and his crony the March Hare who teach Alice to realise the importance of speaking LOGICALLY.Alice learns not to `wallow in self pity' or she will drown in her own sorrows (The pool of tears); before being CONNED out of her wealth (sweeties) by a bunch of VERY strange creatures which include a ROPER (someone who `ropes you in'); a SHILL (someone who allegedly `wins' the `con'); and a BOUNCER (someone who will beat the living daylights out of you if you reveal the con for what it is) all of whom `walk away' after they have `pulled the con' leaving Alice bewildered and ALONE.In `Alice's evidence' (chapter 12), Alice finds herself in a COURTROOM where a knave (The knave of Hearts) is accused of stealing some tarts.The knave is being FRAMED for a crime he did NOT commit, and Carroll uses the SOCRATIC TECHNIQUE in a very humorous way to show how USELESS it is, so as to explain to children why `criminals' often escape justice (the Socratic method of `cross examination' is taught to Barristers to REFUTE EVIDENCE so as to `win' cases) and to be on their guard for statements which are FALLACY.To set the scene: Alice is the size of a `playing card' but as Alice gains and confidence she begins to `grow' (in confidence and stature).KING: Rule Forty two ALL PERSONS MORE THAN A MILE HIGH TO LEAVE THE COURT.Everybody looks at Alice.ALICE: I'M not a mile high!KING: Yes you are!QUEEN: Nearly two miles high!ALICE: That's not a regular rule: you invented it just now.KING: It's the oldest rule in the book.ALICE: Then it ought to be Number One.WHITE RABBIT: There's evidence to come yet, please your Majesty it seems to be a letter written by the prisoner.JURYMAN: Is it in the prisoner's handwriting?WHITE RABBIT: No!KING: He must have imitated somebody else's hand!KNAVE: Please your Majesty, I didn't write it, and they can't prove that I did: there's no name signed at the end.KING: If you didn't SIGN it that only makes the matter worse. You MUST have meant some mischief, or else you'd have signed your name like an honest man.QUEEN: That PROVES his guilt!ALICE: IT PROVES NOTHING OF THE SORT!KING: Read it out!WHITE RABBIT: Where shall I begin, please your Majesty?KING: Begin at the beginning and go on till you come to the end: then stop.AND SO dear reader, when you stop laughing and you have composed yourself to read further, here is a TINY insight into the POWER of Boolean algebra.We are not measuring QUANTITY or MAGNITUDE as one NORMALLY does in mathematics; we are measuring VALIDITY hence:All considerations in ANY given CLASS (e.g. birds) must be reduced to ONE SINGULAR consideration in that class (a bird) represented by the numeric value `1' (one) and NO OTHER VALUE.The RESULT of performing the APPROPRIATE LOGICAL OPERATION will result in either a `1' (representing `TRUE') or a `0' (ZERO representing FALSE)There are no than TWO `inputs' to ANY proposition (the results of ONE proposition is fed into SUBSEQUENT propositions to create complex scenarios all fully expanded in the masterwork).ALL inputs MUST be tested using FOUR `STATES' these being:A = FALSE (0) B = FALSE (0) = A = 0 B = 0A = FALSE (0) B = TRUE (1) = A = 0 B = 1A = TRUE (1) B = FALSE (0) = A = 1 B = 0A = TRUE (1) B = TRUE (1) = A = 0 B = 0The constructs:Every LOGICAL CONSTRUCT has: A PROPOSITION (in the form of a question) AN AXIOM (see below) A MATHEMATICAL OPERATOR A LAW A BOOLEAN TEST (see above) A `TRUTH TABLE' (derived from the mathematical operator) A RULE (derived from the truth table)We shall now examine the logical construct `AND'AXIOM: This is a construct where you must have BOTH items in a given class for the proposition to be TRUE.To illustrate:ALL birds have FEATHERSALL ducks have feathersTherefore ALL ducks are BIRDS (as opposed to all birds are ducks)Lets examine the AND gate using the PROPOSITION that one needs a VALID bank card AND a valid PIN to obtain money from a cash point.Here we are dealing with a SINGULAR `bank card' AND a singular `PIN' (Personal Identification Number) so the LOGICAL operator is ANDThe AXIOM is: BOTH items MUST be in a given class for the proposition to be TRUE.The MATHEMATICAL OPERATOR is `MULTIPY'The `AND' LAW is BOTH statements must be `TRUE'The RULE is `any' FALSE = FALSEThe Boolean test isFALSE FALSE = 0 0FALSE TRUE = 0 1TRUE FALSE = 1 0TRUE TRUE = 1 1Let A = Valid bank cardLet B = Valid PINThe truth table is:A X B = C0 X 0 = 00 X 1 = 01 X 0 = 01 X 1 = 1This gives us:0 X 0 = 0 (false = you cannot obtain money from a cash point because you do not have a bank card or a PIN)0 X 1 = 0 (false = you cannot obtain money from a cash point because you do not have a bank card even though you have a PIN)1 X 0 = 0 (false = you cannot obtain money from a cash point because although you have a bank card, you do not have the PIN)1 X 1 = 1 (true = you CAN obtain money from a cash point because you possess a VALID card AND a VALID PIN)QEDBack to ducksALL birds have FEATHERS (One BIRD represents ALL birds AND one FEATHER represents ALL feathers)ALL ducks have feathers (ONE duck represents ALL ducks)Therefore ALL ducks are BIRDS1 Duck X 1 Feather = 1 X 1 = 1 = TRUE ALL DUCKS are BIRDSIs a dog a bird? NO because a dog does not possess FEATHERS.1 DOG X 0 FEATHERS = 0 = FALSEThis return TWO results: A DOG is NOT a BIRD; and a BIRD is NOT a DOG.Have fun and come away ENLIGHTENED! Kindle Edition, Printed Access Code, Paperback

The Mathematical Analysis of Logic: Being an Essay Towards a Calculus of Deductive Reasoning: Boole, George: 9781537082837: Books The Mathematical Analysis of Logic: Being an Essay Towards a Calculus of Deductive Reasoning