Paraconsistent Logic: Fundamentals and Applications

By Fouad Sabry

What Is Paraconsistent Logic

An attempt to create a logical framework that can deal with contradictions in a discriminating manner is an example of a paraconsistent logic. Alternately, paraconsistent logic is a subfield of logic that rejects the principle of explosion and is concerned with the study and development of "inconsistency-tolerant" systems of logic. This subfield focuses on the study and development of such systems.

How You Will Benefit

(I) Insights, and validations about the following topics:

Chapter 1: Paraconsistent Logic

Chapter 2: Disjunctive Syllogism

Chapter 3: Logical Connective

Chapter 4: Propositional Calculus

Chapter 5: Proof by Contradiction

Chapter 6: Contradiction

Chapter 7: Negation

Chapter 8: Dialetheism

Chapter 9: Principle of Explosion

Chapter 10: Philosophical Logic

(II) Answering the public top questions about paraconsistent logic.

(III) Real world examples for the usage of paraconsistent logic in many fields.

(IV) 17 appendices to explain, briefly, 266 emerging technologies in each industry to have 360-degree full understanding of paraconsistent logic' technologies.

Who This Book Is For

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of paraconsistent logic.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: Jun 30, 2023

Book preview

Chapter 1: Paraconsistent logic

A paraconsistent logic is an effort to develop a logical framework that can handle contradictions in a selective manner. As an alternative, paraconsistent logic is the branch of logic that studies and creates inconsistency-tolerant systems of logic that do not subscribe to the explosion principle.

Since at least 1910 (and possibly much earlier, as in the writings of Aristotle, for example), inconsistency-tolerant logics have been debated; this includes the dialetheistic school of thought.

Contradictions entail everything in classical logic (as well as intuitionistic logic and the majority of other logics). The formal formulation of this characteristic, often known as the principle of explosion or ex contradictione sequitur quodlibet (Latin meaning from a contradiction, anything follows), is

Which means: if P and its negation ¬P are both assumed to be true, the two claims P and (somearbitrary) A are then, One or more are true.

Therefore, Is P or A true?.

However, if we are aware that P or A are both true, and also that P is false (that ¬P is true) we can conclude that A, of which everything could be, is true.

Consequently, if a theory has just one contradiction, It is unimportant, that is, Every sentence is a theorem in it.

A paraconsistent logic opposes the explosion principle, which is its defining property. Therefore, unlike classical and other logics, paraconsistent logics can be used to formulate contradictory yet non-trivial theories.

Compared to classical logic, paraconsistent logics are propositionally weaker because they accept fewer valid propositional inferences. The key argument is that a paraconsistent logic cannot propositionally validate all that classical logic does, i.e., be a propositional extension of classical logic. Therefore, paraconsistent logic is in some ways more restrained or circumspect than classical logic. Paraconsistent languages, such as the hierarchy of metalanguages developed by Alfred Tarski et al., can be more expressive than their classical counterparts because of this conservatism. Natural language is abounding with directly or indirectly self-referential yet seemingly innocent expressions—all of which are prohibited from the Tarskian framework, according to Solomon Feferman. Paraconsistent logic can bypass this expressive constraint.

The belief that it ought to be feasible to reason with contradictory knowledge in a controlled and selective manner is a major driving force for paraconsistent logic. This is impossible due to the explosion principle, which calls for its abandonment. The trivial theory that treats each phrase as a theorem is the sole inconsistent theory in non-paraconsistent logics. It is feasible to recognize and reason with inconsistent theories using paraconsistent logic.

The philosophical school of dialetheism (most notably promoted by Graham Priest), which maintains that genuine contradictions occur in reality, such as groups of people holding opposite viewpoints on certain moral concerns, was also founded as a result of research into paraconsistent logic.

The three laws of Aristotle in classical logic, namely, the excluded middle (p or ¬p), non-contradiction ¬ (p ∧ ¬p) and identity (p iff p), are viewed as being the same, since the connectives have different definitions.

Moreover, Contradictions in a theory or body of knowledge and triviality (the notion that a theory includes all potential implications) are two concepts that are typically thought to be mutually exclusive, assuming that negation is possible.

These opinions could be philosophically contradicted, precisely because they don't recognize the difference between contradiction and other types of inconsistency.

On the other hand, once consistency and contradictions have been clearly defined, it is easy to derive triviality from the conflict between them. At the object language level, the concepts of consistency and inconsistency itself may be further internalized.

Tradeoffs are involved in paraconsistency. In particular, giving up the principle of explosion necessitates giving up at least one of the two principles listed below:

Both of these ideas have faced opposition.

One strategy is to maintain transitivity and disjunctive syllogism while rejecting disjunction introduction.

In this method, Natural deduction rules are valid, excluding the disjunction-introducing clause and the excluded middle; moreover, inference A⊢B does not necessarily mean entailment A⇒B.

Also, the double negation and associativity standard Boolean characteristics are valid, commutativity, distributivity, De Morgan, and conclusions about idempotence (for conjunction and disjunction).

Furthermore, inconsistency-robust proof of negation holds for entailment: (A⇒(B∧¬B))⊢¬A.

Disjunctive syllogism rejection is another strategy.

From the dialetheistic vantage point, Failure of the disjunctive syllogism is entirely logical.

This syllogism's thesis is that, if ¬ A, then A is excluded and B can be inferred from A ∨ B.

However, if A may hold as well as ¬A, then, the case for the inference is compromised.

Combining the two is still another strategy. There are often two distinct disjunctive connectives in relevant logic systems as well as linear logic. One permits the introduction of a disjunction, whereas another permits a disjunctive syllogism. Of course, this has the drawbacks associated with separate disjunctive connectives, such as complication in connecting them and confusion between them.

Additionally, the rule of proof of negation (below) is inconsistency non-robust in the sense that every proposition's negation can be demonstrated by a contradiction.

Strictly speaking, Because not every statement can be proven from a contradiction, having merely the rule above is paraconsistent.

However, if the rule double negation elimination ( \neg \neg A\vdash A ) is added as well, therefore any claim can be supported by a contradiction.

For intuitionistic logic, double negation elimination is invalid.

The LP system is a well-known example of paraconsistent logic (Logic of Paradox), first proposed by the Argentinian logician Florencio González Asenjo in 1966 and later popularized by Priest and others.

The binary relation V\, relates a formula to a truth value: V(A,1)\, means that A\, is true, and V(A,0)\, means that A\, is false.

There must be at least one truth value supplied to a formula, nonetheless, it is not necessary to only assign it one truth value.

The following list includes the semantic sentences for negation and disjunction:

V(\neg A,1)\Leftrightarrow V(A,0)V(\neg A,0)\Leftrightarrow V(A,1){\displaystyle V(A\lor B,1)\Leftrightarrow V(A,1){\text{ or }}V(B,1)}{\displaystyle V(A\lor B,0)\Leftrightarrow V(A,0){\text{ and }}V(B,0)}

(As is customary, the definitions of the other logical connectives are in terms of negation and disjunction.) Or, to express it in a less symbolic way:

If and only if A is untrue, then not A is true.

If and only if A is true, then not A is untrue.

If and only if A is true or B is true, then A or B is true.

If and only if both A and B are false, then A or B is false.

The definition of (semantic) logical consequence is then truth-preservation:

\Gamma \vDash A if and only if A\, is true whenever every element of \Gamma \, is true.

Now consider a valuation V\, such that V(A,1)\, and V(A,0)\, but it is not the case that V(B,1)\, .

It is simple to verify that this valuation serves as an example that contradicts both explosion and disjunctive syllogism.

However, Additionally, it serves as a refutation of modus ponens for the material conditional of LP.

Because of this, Expanding the system to incorporate a stronger conditional connective that is not defined in terms of negation and disjunction is typically recommended by LP proponents.

(LP and classical logic are identical except for the conclusions they accept as true.) The weaker form of paraconsistency known as first-degree entailment is produced when the condition that every formula be either true or untrue is relaxed (FDE).

Unlike LP, There are no logical truths in FDE.

There have been numerous paraconsistent logics proposed, and LP is just one of them. It is only used here as an example of how a paraconsistent logic can