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# what is the law of the excluded middle

## What is the law of excluded middle in logic?

The law in classical logic stating that one of the two statements A or not A is true. The law of the excluded middle is expressed in mathematical logic by the formula \$A\lor eg A\$, where \$\lor\$ denotes disjunction and \$ eg\$ denotes negation.

## What is the basis for the excluded middle?

The basis for the principle of the excluded middle is found in the notion of contradictory opposites (see opposition). Things that are opposed as affirmation and negation are such that it is always necessary that one should be true but the other false ( Cat. 13b 1 – 3).

## Where is excluded middle third from the Oxford English Dictionary?

OED attests ‘excluded middle, third’ (in the entry for excluded, adj., …. b.) from 1849, in William Thomson’s An Outline of the Necessary Laws of Thought. Although OED does not give a quote, merely the citation, the relevant text is this:

## What is the meaning of Lex exclusi medii?

The principle of the middle being excluded, (lex exclusi medii.) Either a given judgement must be true of any subject, or its contradictory; there is no middle course. I did not find any earlier work containing the phrase lex exclusi medii. Some earlier works contain the phrase excluded third.

Contradictory opposition is between being and nonbeing expressed in affirmative and negative statements; it is between being and nonbeing absolutely, and not within a genus. Either of the two opposites may be true or false, but not both true or both false at the same time. Future Contingents. When two enunciations are in contradictory opposition, …

Rather, he says that neither contradictory is determinately true or false now.

## What is the basis for the principle of the excluded middle?

Contradictory Opposition. The basis for the principle of the excluded middle is found in the notion of contradictory opposites (see opposition).

## What is Aristotle’s explanation of the proposition?

The proposition is made clear from the definitions of the true and the false, for it is false to say of what is that it is not, or of what is not that it is; and it is true to say of what is that it is, and of what is not that it is not. If anyone says something is, he either says something true or something false.

## Which philosophers held the truth of every proposition?

Because of their strict determinism, Stoics held the determinate truth or falsity of every proposition, eliminating the possibility of future contingents. epicurus, on the other hand, is reported by Cicero ( De fato 21) to have denied that every proposition is true or false.

## When two enunciations are in contradictory opposition, is it necessary that one be true and the other false?

Future Contingents. When two enunciations are in contradictory opposition, is it necessary that one be true and the other false? This question has concerned logicians and philosophers since the time of Aristotle ( Interp. 18a 28 – 19b 4). His answer is that propositions about the past or the present must be true or false; likewise, for any universal proposition and its contradictory one must be true and the other false; but for a singular proposition about the future, the case is different. For propositions about the past or the present there is a state of affairs against which the truth or falsity of a proposition can be measured, and this is true regardless of whether the propositions are about necessary or contingent matter. But for singular propositions about the future, there is no state of affairs that can be enunciated truly or falsely. Although singular propositions in necessary or impossible matter do have a determinate truth or falsity, future singular propositions in contingent matter do not.

## Is "event E" true or false?

A statement such as "Event E will occur on such and such a day" is a timeless statement and is true or false now. But since the truth or falsity of that proposition cannot be calculated on the basis of propositions about present events, we cannot know whether the proposition is true until the point of time has passed.

## What is the principle of non-contradiction?

The earliest known formulation is in Aristotle’s discussion of the principle of non-contradiction, first proposed in On Interpretation, where he says that of two contradictory propositions (i.e. where one proposition is the negation of the other) one must be true, and the other false. He also states it as a principle in the Metaphysics book 3, saying that it is necessary in every case to affirm or deny, and that it is impossible that there should be anything between the two parts of a contradiction.

## What is the truth value of a proposition?

Truth-values. The “truth-value” of a proposition is truth if it is true and falsehood if it is false*

• This phrase is due to Frege]…the truth-value of “p ∨ q” is truth if the truth-value of either p or q is truth, and is falsehood otherwise … that of “~ p” is the opposite of that of p…” (p. 7-8)

But Aristotle also writes, “since it is impossible that contradictories should be at the same time true of the same thing, obviously contraries also cannot belong at the same time to the same thing” (Book IV, CH 6, p. 531). He then proposes that “there cannot be an intermediate between contradictories, but of one subject we must either affirm or deny any one predicate” (Book IV, CH 7, p. 531). In the context of Aristotle’s traditional logic, this is a remarkably precise statement of the law of excluded middle, P ∨ ¬ P.

## What is Aristotle’s assertion that it will not be possible to be and not to be the same thing?

Aristotle’s assertion that “it will not be possible to be and not to be the same thing”, which would be written in propositional logic as ¬ ( P ∧ ¬ P ), is a statement modern logicians could call the law of excluded middle ( P ∨ ¬ P ), as distribution of the negation of Aristotle’s assertion makes them equivalent, regardless that the former claims that no statement is both true and false, while the latter requires that any statement is either true or false.

## What is the name of the thing that is immediately known in sensation?

Let us give the name of “sense-data” to the things that are immediately known in sensation: such things as colours, sounds, smells, hardnesses, roughnesses, and so on. We shall give the name “sensation ” to the experience of being immediately aware of these things… The colour itself is a sense-datum, not a sensation. (p. 12)

## When did the rancorous debate end?

The rancorous debate continued through the early 1900s into the 1920s; in 1927 Brouwer complained about “polemicizing against it [intuitionism] in sneering tones” (Brouwer in van Heijenoort, p. 492). But the debate was fertile: it resulted in Principia Mathematica (1910–1913), and that work gave a precise definition to the law of excluded middle, and all this provided an intellectual setting and the tools necessary for the mathematicians of the early 20th century:

## Which group restricted mathematics to the study of concrete operations on finite or potentially (but not actually) infinite structures?

And finally constructivists … restricted mathematics to the study of concrete operations on finite or potentially (but not actually) infinite structures; completed infinite totalities … were rejected, as were indirect proof based on the Law of Excluded Middle. Most radical among the constructivists were the intuitionists, led by the erstwhile topologist L. E. J. Brouwer (Dawson p. 49)