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George Spencer-Brown in his 1969 " Laws of Form" (LoF) begins by first taking as given that "we cannot make an indication without drawing a distinction". This, therefore, presupposes the law of excluded middle. He then goes on to define two axioms, which describe how distinctions (a "boundary") and indications (a "call") work: We are talking about this, which involves information structured in four simple ways: Figure 3.4: Ideas and Simple Rules Lead to Systems Thinking Let us then agree to represent the class of individuals to which a particular name or description is applicable, by a single letter, as z. ... By a class is usually meant a collection of individuals, to each of which a particular name or description may be applied; but in this work the meaning of the term will be extended so as to include the case in which but a single individual exists, answering to the required name or description, as well as the cases denoted by the terms "nothing" and "universe," which as "classes" should be understood to comprise respectively 'no beings,' 'all beings.'" (Boole 1854:28) Now, whatever may be the extent of the field within which all the objects of our discourse are found, that field may properly be termed the universe of discourse. ... Furthermore, this universe of discourse is in the strictest sense the ultimate subject of the discourse." Let the predicate "function" f(x) be "x is a mammal", and the subject-domain (or universe of discourse) (cf Kleene 1967:84) be the category "bats": The formula ∀xf(x) yields the truth value "truth" (read: "For all instances x of objects 'bats', 'x is a mammal'" is a truth, i.e. "All bats are mammals"); But if the instances of x are drawn from a domain "winged creatures" then ∀xf(x) yields the truth value "false" (i.e. "For all instances x of 'winged creatures', 'x is a mammal'" has a truth value of "falsity"; "Flying insects are mammals" is false); However over the broad domain of discourse "all winged creatures" (e.g. "birds" + "flying insects" + "flying squirrels" + "bats") we can assert ∃xf(x) (read: "There exists at least one winged creature that is a mammal'"; it yields a truth value of "truth" because the objects x can come from the category "bats" and perhaps "flying squirrels" (depending on how we define "winged"). But the formula yields "falsity" when the domain of discourse is restricted to "flying insects" or "birds" or both "insects" and "birds".
In his next chapter ("On Our Knowledge of General Principles") Russell offers other principles that have this similar property: "which cannot be proved or disproved by experience, but are used in arguments which start from what is experienced." He asserts that these "have even greater evidence than the principle of induction ... the knowledge of them has the same degree of certainty as the knowledge of the existence of sense-data. They constitute the means of drawing inferences from what is given in sensation". [19] We all envy the natural thinkers of this world. They have the best ideas, make the smartest decisions, are open minded and never indecisive. The sequel to Bertrand Russell's 1903 "The Principles of Mathematics" became the three-volume work named Principia Mathematica (hereafter PM), written jointly with Alfred North Whitehead. Immediately after he and Whitehead published PM he wrote his 1912 "The Problems of Philosophy". His "Problems" reflects "the central ideas of Russell's logic". [13] The Principles of Mathematics (1903) [ edit ] Dasgupta, Surendranath (1991), A History of Indian Philosophy, Motilal Banarsidass, p.110, ISBN 81-208-0415-5 Hamilton opines that thought comes in two forms: "necessary" and "contingent" (Hamilton 1860:17). With regards the "necessary" form he defines its study as "logic": "Logic is the science of the necessary forms of thought" (Hamilton 1860:17). To define "necessary" he asserts that it implies the following four "qualities": [12] (1) "determined or necessitated by the nature of the thinking subject itself ... it is subjectively, not objectively, determined; (2) "original and not acquired; (3) "universal; that is, it cannot be that it necessitates on some occasions, and does not necessitate on others. (4) "it must be a law; for a law is that which applies to all cases without exception, and from which a deviation is ever, and everywhere, impossible, or, at least, unallowed. ... This last condition, likewise, enables us to give the most explicit enunciation of the object-matter of Logic, in saying that Logic is the science of the Laws of Thought as Thought, or the science of the Formal Laws of Thought, or the science of the Laws of the Form of Thought; for all these are merely various expressions of the same thing." Hamilton's 4th law: "Infer nothing without ground or reason" [ edit ]
Let it further be agreed, that by the combination xy shall be represented that class of things to which the names or descriptions represented by x and y are simultaneously, applicable. Thus, if x alone stands for "white things," and y for "sheep," let xy stand for 'white Sheep;'" (Boole 1854:28) The restriction is that the generalization "for all" applies only to the variables (objects x, y, z etc. drawn from the domain of discourse) and not to functions, in other words the calculus will permit ∀xf(x) ("for all creatures x, x is a bird") but not ∀f∀x(f(x)) [but if "equality" is added to the calculus it will permit ∀f:f(x); see below under Tarski]. Example:
Laws of thought". The Cambridge Dictionary of Philosophy. Robert Audi, Editor, Cambridge: Cambridge UP. p. 489. Aristotle, "The Categories", Harold P. Cooke (trans.), pp.1–109 in Aristotle, Vol. 1, Loeb Classical Library, William Heinemann, London, UK, 1938. His objections to Kant then leads Russell to accept the 'theory of ideas' of Plato, "in my opinion ... one of the most successful attempts hitherto made."; [33] he asserts that " ... we must examine our knowledge of universals ... where we shall find that [this consideration] solves the problem of a priori knowledge.". [33] Principia Mathematica (Part I: 1910 first edition, 1927 2nd edition) [ edit ] 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.That’s precisely why, despite the fact that the two orange circles in Figure 3.7 are exactly the same size, they appear to be different sizes. Your mind is distinguishing them in relation to what is near them. You can see that the other is often a whole system or set of things (e.g., all of the gray circles surrounding the orange circle). Also important to note is that while each circle is distinct from the others, there is also a distinction between the group of circles in A and those in B. At every level of scale we are making distinctions, boundaries between what something is and what it is not. In his introduction to Post 1921, van Heijenoort observes that both the "truth-table and the axiomatic approaches are clearly presented". [40] This matter of a proof of consistency both ways (by a model theory, by axiomatic proof theory) comes up in the more-congenial version of Post's consistency proof that can be found in Nagel and Newman 1958 in their chapter V "An Example of a Successful Absolute Proof of Consistency". In the main body of the text they use a model to achieve their consistency proof (they also state that the system is complete but do not offer a proof) (Nagel & Newman 1958:45–56). But their text promises the reader a proof that is axiomatic rather than relying on a model, and in the Appendix they deliver this proof based on the notions of a division of formulas into two classes K 1 and K 2 that are mutually exclusive and exhaustive (Nagel & Newman 1958:109–113). Intuitionistic logic', sometimes more generally called constructive logic, is a paracomplete symbolic logic that differs from classical logic by replacing the traditional concept of truth with the concept of constructive provability. The expression "laws of thought" gained added prominence through its use by Boole (1815–64) to denote theorems of his "algebra of logic"; in fact, he named his second logic book An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities (1854). Modern logicians, in almost unanimous disagreement with Boole, take this expression to be a misnomer; none of the above propositions classed under "laws of thought" are explicitly about thought per se, a mental phenomenon studied by psychology, nor do they involve explicit reference to a thinker or knower as would be the case in pragmatics or in epistemology. The distinction between psychology (as a study of mental phenomena) and logic (as a study of valid inference) is widely accepted. Given these definitions he now lists his laws with their justification plus examples (derived from Boole):
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