Rules of Thinking, The: A Personal Code To Think Yourself Smarter, Wiser And Happier

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Starting from these eight tautologies and a tacit use of the "rule" of substitution, PM then derives over a hundred different formulas, among which are the Law of Excluded Middle ❋1.71, and the Law of Contradiction ❋3.24 (this latter requiring a definition of logical AND symbolized by the modern ⋀: (p ⋀ q) = def ~(~p ⋁ ~q). ( PM uses the "dot" symbol ▪ for logical AND)).

Figure 3.10 visually depicts the MECE and NONG requirement filled (A) and not filled (B and C). MECE and NONG mean the same thing, so you can use whichever one makes more sense to you. No overlaps and mutually exclusive mean that the distinctions you make in a system are not overlapping. They are in fact distinct. No gaps and collectively exhaustive mean that the system of distinctions you have assembled to describe the problem is sufficient and complete and that everything that needs to be considered has been. MECE and NONG establish what is inside and outside the boundary of any system. What is important to understand is that while MECE and NONG help you to consider the system of distinctions that you are using, all of the items inside your system of distinctions is also a thing-other distinction. Distinctions are occurring at different scales with regard to the smallest ideas and things as well as the largest ideas and things. So you can see that in (A) there are distinctions occurring wherever there are red lines, which includes not only the distinctions between the parts inside the system, but also the distinction between what lies inside and outside of the system (not-S). there is one main objection which seems fatal to any attempt to deal with the problem of a priori knowledge by his method. The thing to be accounted for is our certainty that the facts must always conform to logic and arithmetic. ... Thus Kant's solution unduly limits the scope of a priori propositions, in addition to failing in the attempt at explaining their certainty". [32] This article is about axiomatic rules due to various logicians and philosophers. For Boole's book on logic, see The Laws of Thought.He then collects all the cases (instances) of the induction principle (e.g. case 1: A 1 = "the rising sun", B 1 = "the eastern sky"; case 2: A 2 = "the setting sun", B 2 = "the western sky"; case 3: etc.) into a "general" law of induction which he expresses as follows: Also remember that we interact with the world indirectly through our mental models, not directly. When one considers this fact, the thing (real world) and the idea (mental model) that represent it are effectively the same. So in our descriptions we will use the term idea and thing interchangeably. The mental model of a thing is determined by both its information and its structure—the simple rules it follows (DSRP).

Theaetetus, by Plato". The University of Adelaide Library. November 10, 2012. Archived from the original on 16 January 2014 . Retrieved 14 January 2014. Armed with his "system" he derives the "principle of [non]contradiction" starting with his law of identity: x 2 = x. He subtracts x from both sides (his axiom 2), yielding x 2 − x = 0. He then factors out the x: x(x − 1) = 0. For example, if x = "men" then 1 − x represents NOT-men. So we have an example of the "Law of Contradiction": Boole then clarifies what a "literal symbol" e.g. x, y, z,... represents—a name applied to a collection of instances into "classes". For example, "bird" represents the entire class of feathered winged warm-blooded creatures. For his purposes he extends the notion of class to represent membership of "one", or "nothing", or "the universe" i.e. totality of all individuals:The latter asserts that the logical sum (i.e. ⋁, OR) of a simple proposition p and a predicate ∀xf(x) implies the logical sum of each separately. But PM derives both of these from six primitive propositions of ❋9, which in the second edition of PM is discarded and replaced with four new "Pp" (primitive principles) of ❋8 (see in particular ❋8.2, and Hilbert derives the first from his "logical ε-axiom" in his 1927 and does not mention the second. How Hilbert and Gödel came to adopt these two as axioms is unclear. Unfortunately, Russell's "Problems" does not offer an example of a "minimum set" of principles that would apply to human reasoning, both inductive and deductive. But PM does at least provide an example set (but not the minimum; see Post below) that is sufficient for deductive reasoning by means of the propositional calculus (as opposed to reasoning by means of the more-complicated predicate calculus)—a total of 8 principles at the start of "Part I: Mathematical Logic". Each of the formulas:❋1.2 to:❋1.6 is a tautology (true no matter what the truth-value of p, q, r ... is). What is missing in PM’s treatment is a formal rule of substitution; [34] in his 1921 PhD thesis Emil Post fixes this deficiency (see Post below). In what follows the formulas are written in a more modern format than that used in PM; the names are given in PM).