Cameron Miranda
The law of excluded middle is a fundamental principle in logic that asserts that every statement of the form 'P or not-P' is true. In other words, it excludes the middle ground where a statement can be both true and false. This law, also known as the principle of double negation, has been a topic of debate among philosophers and logicians, with some stressing its semantic nature and others its ontological aspect. While it is widely used in mathematics and classical logic, there are also mathematicians and intuitionists who choose to reject it. The law of excluded middle is untrue in many-valued logic, and its rejection has deep consequences, rendering proofs by contradiction invalid. However, it remains a valuable tool for many, and understanding when to apply it is crucial in fields such as mathematics and philosophy.
| Characteristics | Values |
|---|---|
| Type | Semantic, Ontological, or both |
| Usage | Used by mathematicians, logicians, and engineers in their daily work |
| Validity | Valid in classical logic, not in many-valued logic |
| Truth Values | Bivalent logic, not applicable in cases where principle of bivalence fails |
| Examples | "He’s in the room or he’s not in the room" |
| Other Names | Principle of double negation, Principle of non-contradiction |
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What You'll Learn
- The Law of Excluded Middle is untrue in many-valued logic
- The Law of Excluded Middle is a semantic principle
- The Law of Excluded Middle is an ontological principle
- The Law of Excluded Middle is not the same as the Principle of Bivalence
- The Law of Excluded Middle is equivalent to the law of double negation elimination
The Law of Excluded Middle is untrue in many-valued logic
The Law of Excluded Middle (or its statement) is a fundamental principle in logic and philosophy. It states that for every proposition, either the proposition or its negation is true. This principle is also known as the law or principle of the excluded third, in Latin "principium tertii exclusi" or "tertium non datur".
However, this law is untrue in many-valued logic, which is a propositional calculus with more than two truth values, such as ternary logic. In many-valued logic, there is an indeterminate value, and fuzzy logic allows for degrees of truth. Fuzzy logic is useful in situations where things aren't black and white, such as controlling the temperature in a shower or recognizing handwritten characters. It allows for statements to be "mostly true" or "somewhat false".
The Law of Excluded Middle is a cornerstone of logic, providing a foundation for propositional and predicate logic. It helps us evaluate and construct arguments by forcing a truth value, which is essential for logical inferences. For example, if we know that the statement "All cats are mammals" is true, then according to the Law of Excluded Middle, "Not all cats are mammals" must be false.
While the Law of Excluded Middle is widely recognized in classical logic, intuitionistic logic challenges its universal validity. In intuitionistic logic, double negation elimination doesn't work because proving that something isn't false doesn't mean it's true. This reflects the intuitionist belief that mathematical objects and truths must be constructed rather than simply inferred.
Some modern logic systems replace the Law of Excluded Middle with the concept of negation as failure, where a proposition is either true or unable to be proved true. This principle is used as a foundation for autoepistemic logic and is widely used in logic programming.
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The Law of Excluded Middle is a semantic principle
The Law of Excluded Middle (LEM) is a principle in logic that states that for any proposition, either that proposition is true or its negation is. In other words, it excludes the middle ground between two contradictory propositions, asserting that one must be true and the other false. This is often expressed in propositional terms using the symbol 'p', such as in the statement "p v not p".
LEM can be seen as a semantic principle, as any statement of it is true by virtue of the meanings of the words expressing it. For example, consider the statement, "Boris Johnson is either mortal or he's not mortal." This statement is true based on the semantics of the words used, regardless of whether anyone has expressed it or not.
However, LEM is also considered "true of the world itself" or "true of thought itself". This means that even if a statement is never uttered or expressed, the LEM still applies. For instance, even without the concept of mortality, Boris Johnson's state of being, whether mortal or not, would still conform to the LEM.
While LEM is a widely applied principle, it is not without its problems. One issue arises when considering statements that are unprovable now but may be provable in the future. In such cases, the LEM may apply even when the principle of bivalence, which states that a proposition is either true or false, fails. This was demonstrated in Aristotle's discussion of future contingents, where he denied the LEM, stating that a proposition can be neither true nor false.
Another problem arises when considering many-valued logic, where there are more than two truth values. In such cases, the LEM is untrue, as it does not account for indeterminate or vague values. For example, consider a man who is half in and half out of a room. The LEM would state that he is either in the room or not, without accounting for the partial state.
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The Law of Excluded Middle is an ontological principle
The Law of Excluded Middle (or LEM) is a principle in logic that states that for every proposition, either the proposition or its negation is true. In other words, there is no middle ground or third possibility between two contradictory statements; one must be true, and the other must be false. This is often expressed as "A is either B or not-B", where the statement "A is neither B nor not-B" is excluded by logic.
The LEM can be seen as either a semantic or an ontological principle, or both. As a semantic principle, the LEM is true by virtue of the meanings of the words used to express it. In other words, the statement "Boris Johnson is either mortal or not mortal" is true because of the definitions of the words "mortal" and "not". This is also known as the semantical principle of bivalence, which states that every proposition is either true or false.
However, some philosophers and logicians argue that the LEM is also "true of the world itself" or "true of thought itself". This ontological interpretation of the LEM suggests that certain statements are true regardless of whether they are expressed or even conceived by human thought. For example, even if no one had ever considered the mortality of Boris Johnson, it would still be the case that he is either mortal or not mortal. This interpretation of the LEM as an ontological principle highlights its role in understanding the nature of reality and thought, rather than simply the semantics of propositions.
The LEM has been a topic of debate among philosophers and logicians, with some arguing for its applicability in certain cases and others pointing out its limitations. For instance, Aristotle discussed the principle of non-contradiction, which aligns with the LEM, but he also denied the LEM in the case of future contingents in his work "On Interpretation". Additionally, the LEM is untrue in many-valued logic, where there are more than two truth values, and in cases where statements are unprovable at the present time but may be provable in the future.
Despite these debates and limitations, the LEM has been influential in the development of logic and mathematics. It has provided a foundation for important works such as Principia Mathematica by Russell and Whitehead, and it continues to be used by mathematicians, logicians, and engineers in their daily work.
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The Law of Excluded Middle is not the same as the Principle of Bivalence
The Law of Excluded Middle (or Principle of Excluded Middle) is a logical truth, and one of the three laws of thought. It states that for any proposition, either that proposition is true, or its negation is. It is important to note that the LEM can be seen as a semantic or ontological principle, or both. It is semantic in the sense that any statement of it is true by virtue of the meanings of the words which express it.
The Principle of Bivalence, on the other hand, is a semantical principle, and a law of classical two-valued logic. It states that every proposition is either true or false, and has only a semantical formulation. It is not a logical truth, because it has the same logical form as some falsehoods. For example, the principle that "every number is either odd or prime" is a falsehood, as the number 2 is neither odd nor prime.
The difference between the two principles is important because there are logics that validate the Law of Excluded Middle, but not the Principle of Bivalence. The Principle of Bivalence always implies the Law of Excluded Middle, but the converse is not always true. A commonly cited counterexample is that the Law of Excluded Middle may apply when the Principle of Bivalence fails, for example, when a statement is unprovable now but provable in the future.
To illustrate the Law of Excluded Middle, consider the statement "He is in the room". According to the LEM, this statement is true, or its negation ("He is not in the room") is true. The LEM does not stipulate how much of the person needs to be in the room, only that he is either in the room or he isn't. This does not prevent the additional statement that the person is "half in and half out of the room" from also being true.
In summary, the Law of Excluded Middle and the Principle of Bivalence are distinct principles. The former is a logical truth that applies to propositions, while the latter is a semantical principle of classical logic that applies to declarative sentences. The Law of Excluded Middle states that for any proposition, either that proposition or its negation is true, while the Principle of Bivalence states that every proposition has exactly one truth value, either true or false.
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The Law of Excluded Middle is equivalent to the law of double negation elimination
The Law of Excluded Middle (LEM) is a principle in logic that states that a statement is either true or false. In other words, it excludes the middle ground between contradictory propositions, asserting that one must be true and the other false. This is often expressed as 'P or not-P', where P represents a statement.
The LEM can be seen as a semantic principle, where its truth is derived from the meanings of the words used. It can also be viewed as ontological, meaning it is "true of the world itself". For example, consider the statement, "Boris Johnson is either mortal or he's not mortal." Regardless of whether this statement is expressed or not, it remains true by virtue of its semantics and its ontological status.
LEM is also known as the "principle of double negation". This is derived from the LEM formula, where substituting ~p for p yields ~p ∨ ~(~p), which simplifies to p → ~(~p). This demonstrates that the LEM removes the "middle" of the inclusive-or used in its formulation.
The Law of Double Negation Elimination states that 'not-not-φ implies φ' or '¬¬φ → φ'. This law allows for the use of Proof by Contradiction, where assuming ¬φ and deriving a contradiction proves φ. For example, if a set of equations has a solution in real numbers, assuming the solution is irrational (¬φ) and deriving a contradiction proves that the solution is not-not-rational (¬¬φ).
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Frequently asked questions
The Law of Excluded Middle is the assertion that every statement of the form ‘P or not-P’ is true. In other words, a statement is either true or false, and there is no "middle ground".
The Law of Excluded Middle can be applied when a statement is presented in a way that it can be true or false. For example, "Boris Johnson is either mortal or not mortal".
No, the Law of Excluded Middle is untrue in many-valued logic, where there are more than two truth values. It also does not apply when the principle of bivalence fails, as shown in some counterexamples. Additionally, some mathematicians, known as intuitionists, choose not to include the Law of Excluded Middle.
