Gottfried Leibniz formulated two additional principles, either or both of which may sometimes be counted as a law of thought: Boole, George, An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, Macmillan, 1854. Reprinted with corrections, Dover Publications, New York, NY, 1958.

Aristotle, "On Interpretation", Harold P. Cooke (trans.), pp.111–179 in Aristotle, Vol. 1, Loeb Classical Library, William Heinemann, London, UK, 1938.Alfred Tarski in his 1946 (2nd edition) "Introduction to Logic and to the Methodology of the Deductive Sciences" cites a number of what he deems "universal laws" of the sentential calculus, three "rules" of inference, and one fundamental law of identity (from which he derives four more laws). The traditional "laws of thought" are included in his long listing of "laws" and "rules". His treatment is, as the title of his book suggests, limited to the "Methodology of the Deductive Sciences". 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:

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)). This first half of this axiom – "the maxim of all" will appear as the first of two additional axioms in Gödel's axiom set. The "dictum of Aristotle" ( dictum de omni et nullo) is sometimes called "the maxim of all and none" but is really two "maxims" that assert: "What is true of all (members of the domain) is true of some (members of the domain)", and "What is not true of all (members of the domain) is true of none (of the members of the domain)".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": 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. Rationale: In his introduction (2nd edition) he observes that what began with an application of logic to mathematics has been widened to "the whole of human knowledge": This axiom also appears in the modern axiom set offered by Kleene (Kleene 1967:387), as his "∀-schema", one of two axioms (he calls them "postulates") required for the predicate calculus; the other being the "∃-schema" f(y) ⊃ ∃xf(x) that reasons from the particular f(y) to the existence of at least one subject x that satisfies the predicate f(x); both of these requires adherence to a defined domain (universe) of discourse. This question of how such a priori knowledge can exist directs Russell to an investigation into the philosophy of Immanuel Kant, which after careful consideration he rejects as follows:

A deeper understanding and greater application of systems thinking requires that we identify the patterns that connect all of the varied systems thinking methods. If all of these "big tent" methods and approaches (i.e., the MFS universe) are types of systems thinking, what cognitive patterns are universal to them all? In other words, the creation of "contradictories" represents a dichotomy, i.e. the "splitting" of a universe of discourse into two classes (collections) that have the following two properties: they are (i) mutually exclusive and (ii) (collectively) exhaustive. [35] In other words, no one thing (drawn from the universe of discourse) can simultaneously be a member of both classes (law of non-contradiction), but [and] every single thing (in the universe of discourse) must be a member of one class or the other (law of excluded middle). As part of his PhD thesis "Introduction to a general theory of elementary propositions" Emil Post proved "the system of elementary propositions of Principia [PM]" i.e. its "propositional calculus" [36] described by PM's first 8 "primitive propositions" to be consistent. The definition of "consistent" is this: that by means of the deductive "system" at hand (its stated axioms, laws, rules) it is impossible to derive (display) both a formula S and its contradictory ~S (i.e. its logical negation) (Nagel and Newman 1958:50). To demonstrate this formally, Post had to add a primitive proposition to the 8 primitive propositions of PM, a "rule" that specified the notion of "substitution" that was missing in the original PM of 1910. [37] Arthur Schopenhauer, The World as Will and Representation, Volume 2, Dover Publications, Mineola, New York, 1966, ISBN 0-486-21762-0 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.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: Another example is that for some people the concept of SpongeBob may contain within it the degradation of the intellect and the decay of the fabric of society, whereas for others, it’s just a funny character who is part of a kid's show. Any idea or thing that we might represent with words—dog, socialism, run, it, SpongeBob, or any other of the over one million words in the English language—defines not only what something is, but what it is not.

It may be a question whether that formula of reasoning, which is called the dictum of Aristotle, de Omni et nullo, expresses a primary law of human reasoning or not; but it is no question that it expresses a general truth in Logic" (1854:4) To review, DSRP rules operate on information simultaneously and a single bit of information can be a distinction, system, relationship, and/or perspective. Imagine, for example, a systemic diagram representing some set of ideas or a network as in the thought bubble in Figure 3.5. First, notice that each bit of information in the network is distinct from other bits (cards). Note that when a relationship is distinguished, a card exists in the center of the line indicating not only that there is a relationship, but also explicating what it is. Some of the relationships (lines) have not yet become distinctions (i.e., they are currently undefined). Some of the cards are also whole systems because they contain parts, whereas other cards are not yet whole systems (perhaps because we haven't explored them yet). And some but not all of the cards are acting as perspectives, viewing or experiencing the system in different ways from each other. DISTINCTIONS RULE: Any Idea or Thing Can Be Distinguished from the Other Ideas or Things It Is with In the 19th century, the Aristotelian laws of thoughts, as well as sometimes the Leibnizian laws of thought, were standard material in logic textbooks, and J. Welton described them in this way: 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":That is to say, if we wish to prove that something of which we have no direct experience exists, we must have among our premises the existence of one or more things of which we have direct experience"; Russell 1912, 1967:75 All men (x) except Asiatics (y)" is represented by x − y. "All states (x) except monarchical states (y)" is represented by x − y Full Book Name: The Rules of Thinking: A Personal Code to Think Yourself Smarter, Wiser and Happier