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Arthur Schopenhauer, The World as Will and Representation, Volume 2, Dover Publications, Mineola, New York, 1966, ISBN 0-486-21762-0 Aristotle, "On Interpretation", Harold P. Cooke (trans.), pp.111–179 in Aristotle, Vol. 1, Loeb Classical Library, William Heinemann, London, UK, 1938. Boole begins his chapter I "Nature and design of this Work" with a discussion of what characteristic distinguishes, generally, "laws of the mind" from "laws of nature": He does not call his inference principle modus ponens, but his formal, symbolic expression of it in PM (2nd edition 1927) is that of modus ponens; modern logic calls this a "rule" as opposed to a "law". [23] In the quotation that follows, the symbol "⊦" is the "assertion-sign" (cf PM:92); "⊦" means "it is true that", therefore "⊦p" where "p" is "the sun is rising" means "it is true that the sun is rising", alternately "The statement 'The sun is rising' is true". The "implication" symbol "⊃" is commonly read "if p then q", or "p implies q" (cf PM:7). Embedded in this notion of "implication" are two "primitive ideas", "the Contradictory Function" (symbolized by NOT, "~") and "the Logical Sum or Disjunction" (symbolized by OR, "⋁"); these appear as "primitive propositions" ❋1.7 and ❋1.71 in PM (PM:97). With these two "primitive propositions" Russell defines "p ⊃ q" to have the formal logical equivalence "NOT-p OR q" symbolized by "~p ⋁ q": 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]
All of the above "systems of logic" are considered to be "classical" meaning propositions and predicate expressions are two-valued, with either the truth value "truth" or "falsity" but not both(Kleene 1967:8 and 83). While intuitionistic logic falls into the "classical" category, it objects to extending the "for all" operator to the Law of Excluded Middle; it allows instances of the "Law", but not its generalization to an infinite domain of discourse. Lastly is a notion of "identity" symbolized by "=". This allows for two axioms: (axiom 1): equals added to equals results in equals, (axiom 2): equals subtracted from equals results in equals. When we take a CAS perspective on systems thinking, we ask ourselves: what are its simple underlying rules? The simple rules are based on distinctions (D), systems (S), relationships (R), and perspectives (P). That is, each bit of information can distinguish itself from other bits, each bit can contain other bits or be part of a larger bit, each bit can relate to other bits of information, and each bit of information can be looked at from the perspective of another bit of information and can also be a perspective on any other bit. DSRP Rules Occur Simultaneously Logical OR: Boole defines the "collecting of parts into a whole or separate a whole into its parts" (Boole 1854:32). Here the connective "and" is used disjunctively, as is "or"; he presents a commutative law (3) and a distributive law (4) for the notion of "collecting". The notion of separating a part from the whole he symbolizes with the "-" operation; he defines a commutative (5) and distributive law (6) for this notion:
p" and "⊦(p ⊃ q)" " have occurred, then "⊦q" will occur if it is desired to put it on record. The process of the inference cannot be reduced to symbols. Its sole record is the occurrence of "⊦q". ... An inference is the dropping of a true premiss; it is the dissolution of an implication". [24] The images here are provided by a software program our research lab developed. The software, called Plectica, treats every bit of information (text or image) as a rule-following card. Each card follows DSRP rules, which can be executed one at a time or simultaneously. Each card has a pop-up (see Figure 3.2) that appears when you hover over them. The user decides which rule(s) to execute for that card. These little rule-following cards allow us to visualize systems thinking. So instead of this, which involves fish or birds or ants, etc. Figure 3.3: Fish and Simple Rules Lead to Superorganism
In a nutshell: given that "x has every property that y has", we can write "x = y", and this formula will have a truth value of "truth" or "falsity". Tarski states this Leibniz's law as follows: As noted above, Hamilton specifies four laws—the three traditional plus the fourth "Law of Reason and Consequent"—as follows:
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The law of identity: 'Whatever is, is.' (2) The law of contradiction: 'Nothing can both be and not be.' (3) The law of excluded middle: 'Everything must either be or not be.'" [25] The "simple" type of implication, aka material implication, is the logical connective commonly symbolized by → or ⊃, e.g. p ⊃ q. As a connective it yields the truth value of "falsity" only when the truth value of statement p is "truth" when the truth value of statement q is "falsity"; in 1903 Russell is claiming that "A definition of implication is quite impossible" (Russell 1903:14). He will overcome this problem in PM with the simple definition of (p ⊃ q) = def (NOT-p OR q).
A true hypothesis in an implication may be dropped, and the consequent asserted. This is a principle incapable of formal symbolic statement ..." (Russell 1903:16) 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] We all envy the natural thinkers of this world. They have the best ideas, make the smartest decisions, are open minded and never indecisive. Given these definitions he now lists his laws with their justification plus examples (derived from Boole): By 1912 Russell in his "Problems" pays close attention to "induction" (inductive reasoning) as well as "deduction" (inference), both of which represent just two examples of "self-evident logical principles" that include the "Laws of Thought." [4]According to the 1999 Cambridge Dictionary of Philosophy, [1] laws of thought are laws by which or in accordance with which valid thought proceeds, or that justify valid inference, or to which all valid deduction is reducible. Laws of thought are rules that apply without exception to any subject matter of thought, etc.; sometimes they are said to be the object of logic [ further explanation needed]. The term, rarely used in exactly the same sense by different authors, has long been associated with three equally ambiguous expressions: the law of identity (ID), the law of contradiction (or non-contradiction; NC), and the law of excluded middle (EM).
I now go on to the fourth law. " Par. XVII. Law of Sufficient Reason, or of Reason and Consequent: "XVII. The thinking of an object, as actually characterized by positive or by negative attributes, is not left to the caprice of Understanding – the faculty of thought; but that faculty must be necessitated to this or that determinate act of thinking by a knowledge of something different from, and independent of; the process of thinking itself. This condition of our understanding is expressed by the law, as it is called, of Sufficient Reason ( principium Rationis Sufficientis); but it is more properly denominated the law of Reason and Consequent ( principium Rationis et Consecutionis). That knowledge by which the mind is necessitated to affirm or posit something else, is called the logical reason ground, or antecedent; that something else which the mind is necessitated to affirm or posit, is called the logical consequent; and the relation between the reason and consequent, is called the logical connection or consequence. This law is expressed in the formula – Infer nothing without a ground or reason. 1 Relations between Reason and Consequent: The relations between Reason and Consequent, when comprehended in a pure thought, are the following: 1. When a reason is explicitly or implicitly given, then there must ¶ exist a consequent; and, vice versa, when a consequent is given, there must also exist a reason. 1 See Schulze, Logik, §19, and Krug, Logik, §20, – ED. 2. Where there is no reason there can be no consequent; and, vice versa, where there is no consequent (either implicitly or explicitly) there can be no reason. That is, the concepts of reason and of consequent, as reciprocally relative, involve and suppose each other. The logical significance of this law: The logical significance of the law of Reason and Consequent lies in this, – That in virtue of it, thought is constituted into a series of acts all indissolubly connected; each necessarily inferring the other. Thus it is that the distinction and opposition of possible, actual and necessary matter, which has been introduced into Logic, is a doctrine wholly extraneous to this science. Welton [ edit ] What makes something a distinction is not so simple a question. Every thing or idea has an other. But in many cases the other is either implicit or absent in your thinking. If a thing or idea exists, then an other exists, even if it's not clear what the other is. Induction principle: Russell devotes a chapter to his "induction principle". He describes it as coming in two parts: firstly, as a repeated collection of evidence (with no failures of association known) and therefore increasing probability that whenever A happens B follows; secondly, in a fresh instance when indeed A happens, B will indeed follow: i.e. "a sufficient number of cases of association will make the probability of a fresh association nearly a certainty, and will make it approach certainty without limit." [15] All men (x) except Asiatics (y)" is represented by x − y. "All states (x) except monarchical states (y)" is represented by x − ycf Boole 1854:226 ARISTOTELIAN LOGIC. CHAPTER XV. [CHAP. XV. THE ARISTOTELIAN LOGIC AND ITS MODERN EXTENSIONS, EXAMINED BY THE METHOD OF THIS TREATISE In his Part I "The Indefinables of Mathematics" Chapter II "Symbolic Logic" Part A "The Propositional Calculus" Russell reduces deduction ("propositional calculus") to 2 "indefinables" and 10 axioms:
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