Law of thought


The laws of thought are fundamental axiomatic rules upon which rational discourse itself is often considered to be based. The formulation and clarification of such rules have a long tradition in the history of philosophy and logic. Generally they are taken as laws that guide and underlie everyone's thinking, thoughts, expressions, discussions, etc. However, such classical ideas are often questioned or rejected in more recent developments, such as intuitionistic logic, dialetheism and fuzzy logic.
According to the 1999 Cambridge Dictionary of Philosophy, 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. 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, the law of contradiction, and the law of excluded middle.
Sometimes, these three expressions are taken as propositions of formal ontology having the widest possible subject matter, propositions that apply to entities as such:, everything is itself; no thing having a given quality also has the negative of that quality ; every thing either has a given quality or has the negative of that quality. Equally common in older works is the use of these expressions for principles of metalogic about propositions: every proposition implies itself; no proposition is both true and false; every proposition is either true or false.
Beginning in the middle to late 1800s, these expressions have been used to denote propositions of Boolean Algebra about classes: every class includes itself; every class is such that its intersection with its own complement is the null class; every class is such that its union with its own complement is the universal class. More recently, the last two of the three expressions have been used in connection with the classical propositional logic and with the so-called protothetic or quantified propositional logic; in both cases the law of non-contradiction involves the negation of the conjunction of something with its own negation, ¬, and the law of excluded middle involves the disjunction of something with its own negation, A∨¬A. In the case of propositional logic, the "something" is a schematic letter serving as a place-holder, whereas in the case of protothetic logic the "something" is a genuine variable. The expressions "law of non-contradiction" and "law of excluded middle" are also used for semantic principles of model theory concerning sentences and interpretations: under no interpretation is a given sentence both true and false, under any interpretation, a given sentence is either true or false.
The expressions mentioned above all have been used in many other ways. Many other propositions have also been mentioned as laws of thought, including the dictum de omni et nullo attributed to Aristotle, the substitutivity of identicals attributed to Euclid, the so-called identity of indiscernibles attributed to Gottfried Wilhelm Leibniz, and other "logical truths".
The expression "laws of thought" gained added prominence through its use by Boole 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. 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 and logic is widely accepted.

The three traditional laws

History

offers a history of the three traditional laws that begins with Plato, proceeds through Aristotle, and ends with the schoolmen of the Middle Ages; in addition he offers a fourth law :

Three traditional laws: identity, non-contradiction, excluded middle

The following will state the three traditional "laws" in the words of Bertrand Russell :

The law of identity

The law of identity: 'Whatever is, is.'
For all a: a = a.
Regarding this law, Aristotle wrote:
More than two millennia later, George Boole alluded to the very same principle as did Aristotle when Boole made the following observation with respect to the nature of language and those principles that must inhere naturally within them:

The law of non-contradiction

The law of non-contradiction : 'Nothing can both be and not be.'
In other words: "two or more contradictory statements cannot both be true in the same sense at the same time": ¬.
In the words of Aristotle, that "one cannot say of something that it is and that it is not in the same respect and at the same time". As an illustration of this law, he wrote:

The law of excluded middle

The law of excluded middle: 'Everything must either be or not be.'
In accordance with the law of excluded middle or excluded third, for every proposition, either its positive or negative form is true: A∨¬A.
Regarding the law of excluded middle, Aristotle wrote:

Rationale

As the quotations from Hamilton above indicate, in particular the "law of identity" entry, the rationale for and expression of the "laws of thought" have been fertile ground for philosophic debate since Plato. Today the debate—about how we "come to know" the world of things and our thoughts—continues; for examples of rationales see the entries, below.

Plato

In one of Plato's Socratic dialogues, Socrates described three principles derived from introspection:

Indian logic

The law of non-contradiction is found in ancient Indian logic as a meta-rule in the Shrauta Sutras, the grammar of Pāṇini, and the Brahma Sutras attributed to Vyasa. It was later elaborated on by medieval commentators such as Madhvacharya.

Locke

claimed that the principles of identity and contradiction were general ideas and only occurred to people after considerable abstract, philosophical thought. He characterized the principle of identity as "Whatsoever is, is." He stated the principle of contradiction as "It is impossible for the same thing to be and not to be." To Locke, these were not innate or a priori principles.

Leibniz

formulated two additional principles, either or both of which may sometimes be counted as a law of thought:
In Leibniz's thought, as well as generally in the approach of rationalism, the latter two principles are regarded as clear and incontestable axioms. They were widely recognized in European thought of the 17th, 18th, and 19th centuries, although they were subject to greater debate in the 19th century. As turned out to be the case with the law of continuity, these two laws involve matters which, in contemporary terms, are subject to much debate and analysis. Leibniz's principles were particularly influential in German thought. In France, the Port-Royal Logic was less swayed by them. Hegel quarrelled with the identity of indiscernibles in his Science of Logic.

Schopenhauer

Four laws

"The primary laws of thought, or the conditions of the thinkable, are four: - 1. The law of identity . 2. The law of contradiction. 3. The law of exclusion; or excluded middle. 4. The law of sufficient reason."
Arthur Schopenhauer discussed the laws of thought and tried to demonstrate that they are the basis of reason. He listed them in the following way in his On the Fourfold Root of the Principle of Sufficient Reason, §33:
  1. A subject is equal to the sum of its predicates, or a = a.
  2. No predicate can be simultaneously attributed and denied to a subject, or a ≠ ~a.
  3. Of every two contradictorily opposite predicates one must belong to every subject.
  4. Truth is the reference of a judgment to something outside it as its sufficient reason or ground.
Also:
To show that they are the foundation of reason, he gave the following explanation:
Schopenhauer's four laws can be schematically presented in the following manner:
  1. A is A.
  2. A is not not-A.
  3. X is either A or not-A.
  4. If A then B.

    Two laws

Later, in 1844, Schopenhauer claimed that the four laws of thought could be reduced to two. In the ninth chapter of the second volume of The World as Will and Representation, he wrote:

Boole (1854): From his "laws of the mind" Boole derives Aristotle's "Law of contradiction"

The title of George Boole's 1854 treatise on logic, An Investigation on the Laws of Thought, indicates an alternate path. The laws are now incorporated into an algebraic representation of his "laws of the mind", honed over the years into modern Boolean algebra.

Rationale: How the "laws of the mind" are to be distinguished

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":
Contrasted with this are what he calls "laws of the mind": Boole asserts these are known in their first instance, without need of repetition:

Boole's signs and their laws

Boole begins with the notion of "signs" representing "classes", "operations" and "identity":
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:
He then defines what the string of symbols e.g. xy means :
Given these definitions he now lists his laws with their justification plus examples :
Logical OR: Boole defines the "collecting of parts into a whole or separate a whole into its parts". Here the connective "and" is used disjunctively, as is "or"; he presents a commutative law and a distributive law for the notion of "collecting". The notion of separating a part from the whole he symbolizes with the "-" operation; he defines a commutative and distributive law for this notion:
Lastly is a notion of "identity" symbolized by "=". This allows for two axioms: : equals added to equals results in equals, : equals subtracted from equals results in equals.
Nothing "0" and Universe "1": He observes that the only two numbers that satisfy xx = x are 0 and 1. He then observes that 0 represents "Nothing" while "1" represents the "Universe".
The logical NOT: Boole defines the contrary as follows :
The notion of a particular as opposed to a universal: To represent the notion of "some men", Boole writes the small letter "v" before the predicate-symbol "vx" some men.
Exclusive- and inclusive-OR: Boole does not use these modern names, but he defines these as follows x + y and x + y, respectively; these agree with the formulas derived by means of the modern Boolean algebra.

Boole derives the law of contradiction

Armed with his "system" he derives the "principle of contradiction" starting with his law of identity: x2 = x. He subtracts x from both sides, yielding x2 - x = 0. He then factors out the x: x = 0. For example, if x = "men" then 1 - x represents NOT-men. So we have an example of the "Law of Contradiction":

Boole defines the notion "domain (universe) of discourse"

This notion is found throughout Boole's "Laws of Thought" e.g. 1854:28, where the symbol "1" is used to represent "Universe" and "0" to represent "Nothing", and in far more detail later :
In his chapter "The Predicate Calculus" Kleene observes that the specification of the "domain" of discourse is "not a trivial assumption, since it is not always clearly satisfied in ordinary discourse... in mathematics likewise, logic can become pretty slippery when no D has been specified explicitly or implicitly, or the specification of a D is too vague''.

Hamilton">Sir William Hamilton, 9th Baronet">Hamilton (1837–38 lectures on Logic, published 1860): a 4th "Law of Reason and Consequent"

As noted above, Hamilton specifies four laws—the three traditional plus the fourth "Law of Reason and Consequent"—as follows:

Rationale: "Logic is the science of the Laws of Thought as Thought"

Hamilton opines that thought comes in two forms: "necessary" and "contingent". With regards the "necessary" form he defines its study as "logic": “Logic is the science of the necessary forms of thought”. To define "necessary" he asserts that it implies the following four “qualities”:

Hamilton's 4th law: "Infer nothing without ground or reason"

Here's Hamilton's fourth law from his LECT. V. LOGIC. 60-61:

Welton

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:

Russell (1903–1927)

The sequel to Bertrand Russell's 1903 "The Principles of Mathematics" became the three volume work named Principia Mathematica, 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".

''The Principles of Mathematics'' (1903)

In his 1903 "Principles" Russell defines Symbolic or Formal Logic as "the study of the various general types of deduction". He asserts that "Symbolic Logic is essentially concerned with inference in general" and with a footnote indicates that he does not distinguish between inference and deduction; moreover he considers induction "to be either disguised deduction or a mere method of making plausible guesses". This opinion will change by 1912, when he deems his "principle of induction" to be par with the various "logical principles" that include the "Laws of Thought".
In his Part I "The Indefinables of Mathematics" Chapter II "Symbolic Logic" Part A "The Propositional Calculus" Russell reduces deduction to 2 "indefinables" and 10 axioms:
From these he claims to be able to derive the law of excluded middle and the law of contradiction but does not exhibit his derivations. Subsequently, he and Whitehead honed these "primitive principles" and axioms into the nine found in PM, and here Russell actually exhibits these two derivations at ❋1.71 and ❋3.24, respectively.

''The Problems of Philosophy'' (1912)

By 1912 Russell in his "Problems" pays close attention to "induction" as well as "deduction", both of which represent just two examples of "self-evident logical principles" that include the "Laws of Thought."
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 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."
He then collects all the cases of the induction principle into a "general" law of induction which he expresses as follows:
He makes an argument that this induction principle can neither be disproved or proved by experience, the failure of disproof occurring because the law deals with probability of success rather than certainty; the failure of proof occurring because of unexamined cases that are yet to be experienced, i.e. they will occur in the future. "Thus we must either accept the inductive principle on the ground of its intrinsic evidence, or forgo all justification of our expectations about the future".
In his next chapter 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".
Inference principle: Russell then offers an example that he calls a "logical" principle. Twice previously he has asserted this principle, first as the 4th axiom in his 1903 and then as his first "primitive proposition" of PM: "❋1.1 Anything implied by a true elementary proposition is true". Now he repeats it in his 1912 in a refined form: "Thus our principle states that if this implies that, and this is true, then that is true. In other words, 'anything implied by a true proposition is true', or 'whatever follows from a true proposition is true'. This principle he places great stress upon, stating that "this principle is really involved -- at least, concrete instances of it are involved -- in all demonstrations".
He does not call his inference principle modus ponens, but his formal, symbolic expression of it in PM is that of modus ponens; modern logic calls this a "rule" as opposed to a "law". In the quotation that follows, the symbol "⊦" is the "assertion-sign" ; “⊦" 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". Embedded in this notion of "implication" are two "primitive ideas", "the Contradictory Function" and "the Logical Sum or Disjunction" ; these appear as "primitive propositions" ❋1.7 and ❋1.71 in PM. With these two "primitive propositions" Russell defines "p ⊃ q" to have the formal logical equivalence "NOT-p OR q" symbolized by "~p ⋁ q":
In other words, in a long "string" of inferences, after each inference we can detach the "consequent" “⊦q” from the symbol string “⊦p, ⊦” and not carry these symbols forward in an ever-lengthening string of symbols.

The three traditional "laws" of thought: Russell goes on to assert other principles, of which the above logical principle is "only one". He asserts that "some of these must be granted before any argument or proof becomes possible. When some of them have been granted, others can be proved." Of these various "laws" he asserts that "for no very good reason, three of these principles have been singled out by tradition under the name of 'Laws of Thought'. And these he lists as follows:



Rationale: Russell opines that "the name 'laws of thought' is... misleading, for what is important is not the fact that we think in accordance with these laws, but the fact that things behave in accordance with them; in other words, the fact that when we think in accordance with them we think truly." But he rates this a "large question" and expands it in two following chapters where he begins with an investigation of the notion of "a priori" knowledge, and ultimately arrives at his acceptance of the Platonic "world of universals". In his investigation he comes back now and then to the three traditional laws of thought, singling out the law of contradiction in particular: "The conclusion that the law of contradiction is a law of thought is nevertheless erroneous... , the law of contradiction is about things, and not merely about thoughts... a fact concerning the things in the world."
His argument begins with the statement that the three traditional laws of thought are "samples of self-evident principles". For Russell the matter of "self-evident" merely introduces the larger question of how we derive our knowledge of the world. He cites the "historic controversy... between the two schools called respectively 'empiricists' and 'rationalists' ". Russell asserts that the rationalists "maintained that, in addition to what we know by experience, there are certain 'innate ideas' and 'innate principles', which we know independently of experience"; to eliminate the possibility of babies having innate knowledge of the "laws of thought", Russell renames this sort of knowledge a priori. And while Russell agrees with the empiricists that "Nothing can be known to exist except by the help of experience,", he also agrees with the rationalists that some knowledge is a priori, specifically "the propositions of logic and pure mathematics, as well as the fundamental propositions of ethics".
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:
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."; he asserts that "... we must examine our knowledge of universals... where we shall find that solves the problem of a priori knowledge.".

''Principia Mathematica'' (Part I: 1910 first edition, 1927 2nd edition)

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 that is sufficient for deductive reasoning by means of the propositional 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. What is missing in PM's treatment is a formal rule of substitution; in his 1921 PhD thesis Emil Post fixes this deficiency.
Russell sums up these principles with "This completes the list of primitive propositions required for the theory of deduction as applied to elementary propositions".
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 =def ~. ).

Ladd-Franklin (1914): "principle of exclusion" and the "principle of exhaustion"

At about the same time that Russell and Whitehead were finishing the last volume of their Principia Mathematica, and the publishing of Russell's "The Problems of Philosophy" at least two logicians were asserting that two "laws" of contradiction" and "excluded middle" are necessary to specify "contradictories"; Ladd-Franklin renamed these the principles of exclusion and exhaustion. The following appears as a footnote on page 23 of Couturat 1914:
In other words, the creation of "contradictories" represents a dichotomy, i.e. the "splitting" of a universe of discourse into two classes that have the following two properties: they are mutually exclusive and exhaustive. In other words, no one thing can simultaneously be a member of both classes, but every single thing must be a member of one class or the other.

Post (1921): The propositional calculus is consistent and complete

As part of his PhD thesis "Introduction to a general theory of elementary propositions" Emil Post proved "the system of elementary propositions of Principia " i.e. its "propositional calculus" 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 it is impossible to derive both a formula S and its contradictory ~S . 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.
Given PM's tiny set of "primitive propositions" and the proof of their consistency, Post then proves that this system is complete, meaning every possible truth table can be generated in the "system":

A minimum set of axioms? The matter of their independence

Then there is the matter of "independence" of the axioms. In his commentary before Post 1921, van Heijenoort states that Paul Bernays solved the matter in 1918 -- the formula ❋1.5 Associative Principle: p ⋁ ⊃ q ⋁ can be proved with the other four. As to what system of "primitive-propositions" is the minimum, van Heijenoort states that the matter was "investigated by Zylinski, Post himself, and Wernick " but van Heijenoort does not answer the question.

Model theory versus proof theory: Post's proof

Kleene observes that "logic" can be "founded" in two ways, first as a "model theory", or second by a formal "proof" or "axiomatic theory"; "the two formulations, that of model theory and that of proof theory, give equivalent results". This foundational choice, and their equivalence also applies to predicate logic.
In his introduction to Post 1921, van Heijenoort observes that both the "truth-table and the axiomatic approaches are clearly presented". This matter of a proof of consistency both ways 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 . 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 K1 and K2 that are mutually exclusive and exhaustive

Gödel (1930): The first-order predicate calculus is complete

The "first-order predicate calculus" is the "system of logic" that adds to the propositional logic the notion of "subject-predicate" i.e. the subject x is drawn from a domain of discourse and the predicate is a logical function f: x as subject and f as predicate. Although Gödel's proof involves the same notion of "completeness" as does the proof of Post, Gödel's proof is far more difficult; what follows is a discussion of the axiom set.

Completeness

in his 1930 doctoral dissertation "The completeness of the axioms of the functional calculus of logic" proved that in this "calculus" that every valid formula is "either refutable or satisfiable" or what amounts to the same thing: every valid formula is provable and therefore the logic is complete. Here is Gödel's definition of whether or not the "restricted functional calculus" is "complete":

The first-order predicate calculus

This particular predicate calculus is "restricted to the first order". To the propositional calculus it adds two special symbols that symbolize the generalizations "for all" and "there exists " that extend over the domain of discourse. The calculus requires only the first notion "for all", but typically includes both: the notion "for all x" or "for every x" is symbolized in the literature as variously as, ∀x, ∏x etc., and the notion of "there exists " variously symbolized as Ex, ∃x.
The restriction is that the generalization "for all" applies only to the variables and not to functions, in other words the calculus will permit ∀xf but not ∀f∀x . Example:
Kleene remarks that "the predicate calculus fully accomplishes what has been conceived to be the role of logic".

A new axiom: Aristotle's dictum – "the maxim of all and none"

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" is sometimes called "the maxim of all and none" but is really two "maxims" that assert: "What is true of all is true of some ", and "What is not true of all is true of none ".
The "dictum" appears in Boole 1854 a couple places:
But later he seems to argue against it:
But the first half of this "dictum" is taken up by Russell and Whitehead in PM, and by Hilbert in his version of the "first order predicate logic"; his includes a principle that Hilbert calls "Aristotle's dictum"
This axiom also appears in the modern axiom set offered by Kleene, as his "∀-schema", one of two axioms required for the predicate calculus; the other being the "∃-schema" f ⊃ ∃xf that reasons from the particular f to the existence of at least one subject x that satisfies the predicate f; both of these requires adherence to a defined domain of discourse.

Gödel's restricted predicate calculus

To supplement the four axioms of the propositional calculus, Gödel 1930 adds the dictum de omni as the first of two additional axioms. Both this "dictum" and the second axiom, he claims in a footnote, derive from Principia Mathematica. Indeed, PM includes both as
The latter asserts that the logical sum of a simple proposition p and a predicate ∀xf 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" of ❋8 applicable to predicates.

Tarski (1946): Leibniz's Law

in his 1946 "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. 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".
Rationale: In his introduction he observes that what began with an application of logic to mathematics has been widened to "the whole of human knowledge":

Law of identity (Leibniz's Law, equality)

To add the notion of "equality" to the "propositional calculus" Tarski symbolizes what he calls "Leibniz's law" with the symbol "=". This extends the domain of discourse and the types of functions to numbers and mathematical formulas.
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:
He then derives some other "laws" from this law:
Principia Mathematica defines the notion of equality as follows ; note that the generalization "for all" extends over predicate-functions f:
Hilbert 1927:467 adds only two axioms of equality, the first is x = x, the second is → → f); the "for all f" is missing. Gödel 1930 defines equality similarly to PM :❋13.01. Kleene 1967 adopts the two from Hilbert 1927 plus two more.

Contemporary developments

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. 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.

Intuitionistic logic

'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 generalized law of the excluded middle is not part of the execution of intuitionistic logic, but neither is it negated. Intuitionistic logic merely forbids the use of the operation as part of what it defines as a "constructive proof", which is not the same as demonstrating it invalid.

Paraconsistent Logic

'Paraconsistent logic' refers to so-called contradiction-tolerant logical systems in which a contradiction does not necessarily result in trivialism. In other words, the principle of explosion is not valid in such logics. Some argue that the law of non-contradiction is denied by dialetheic logic. They are motivated by certain paradoxes which seem to imply a limit of the law of non-contradiction, namely the Liar Paradox. In order to avoid a trivial logical system and still allow certain contradictions to be true, dialetheists will employ a paraconsistent logic of some kind.

Three-valued Logic

TBD cf Three-valued logic
try this
A Ternary Arithmetic and Logic - Semantic Scholar
http://www.iaeng.org/publication/WCE2010/WCE2010_pp193-196.pdf

Modal propositional calculi

: These "calculi" include the symbols ⎕A, meaning "A is necessary" and ◊A meaning "A is possible". Kleene states that:

Fuzzy logic

'Fuzzy logic' is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact.