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Peirce's law in logic is named after the philosopher and logician Charles Sanders Peirce. It was taken as an axiom in his first axiomatisation of propositional logic. The axiom can be used as an alternative to the excluded middle. Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy among philosophers. ...
A philosopher is a person who thinks deeply regarding people, society, the world, and/or the universe. ...
A logician is a philosopher, mathematician, or other whose topic of scholarly study is logic. ...
Charles Sanders Peirce Charles Sanders Peirce (September 10, 1839 – April 19, 1914) was an American logician, philosopher, scientist, and mathematician. ...
In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ...
A propositional calculus is a formal, deduction system, or proof theory for reasoning with propositional formulas as symbolic logic. ...
The law of excluded middle (tertium non datur in Latin) states that for any proposition P, it is true that (P or ~P). ...
In propositional calculus, Peirce's law says that ((P→Q)→P)→P. Written out, this says that P must be true if you can show that P implying Q forces P to be true. In mathematical logic the propositional calculus or sentential calculus is a formal deduction system whose atomic formulas are propositional variables. ...
When someone sincerely agrees with an assertion, they are claiming that it is the truth. ...
Peirce's law does not hold in intuitionistic logic or intermediate logics. Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ...
Intermediate logics are intermediate between intuitionistic logic and classical logic in the sense that they contain theorems that are not provable in intuitionistic logic, without giving rise to the whole of classical logic. ...
Under the Curry-Howard isomorphism, Peirce's law is the type of continuation operators. The Curry-Howard correspondence is the close relationship between computer programs and mathematical proofs; the correspondence is also known as the Curry-Howard isomorphism, or the formulae-as-types correspondence. ...
In computing, a continuation is a representation of the execution state of a program (for example, the register contents and call stack) at a certain point in time. ...
History
Here is Peirce's own statement of the law: - A fifth icon is required for the principle of excluded middle and other propositions connected with it. One of the simplest formulae of this kind is:
- This is hardly axiomatical. That it is true appears as follows. It can only be false by the final consequent x being false while its antecedent (x —< y) —< x is true. If this is true, either its consequent, x, is true, when the the whole formula would be true, or its antecedent x —< y is false. But in the last case the antecedent of x —< y, that is x, must be true. (Peirce, CP 3.384).
Peirce goes on to point out an immediate application of the law: The law of excluded middle (tertium non datur in Latin) states that for any proposition P, it is true that (P or ~P). ...
- From the formula just given, we at once get:
- where the a is used in such a sense that (x —< y) —< a means that from (x —< y) every proposition follows. With that understanding, the formula states the principle of excluded middle, that from the falsity of the denial of x follows the truth of x. (Peirce, CP 3.384).
Proof of Peirce's law Showing Peirce's Law applies does not mean that P→Q is true, we have that P is true but only (P→Q)→P, not P→(P→Q) (see affirming the consequent). Look up Proof on Wiktionary, the free dictionary The word proof can mean: Shit and wanker originally, a test assessing the validity or quality of something. ...
Affirming the consequent is a logical fallacy in the form of a hypothetical proposition. ...
To prove:
(1) ((P→Q)→P)→P
One way to proof Peirce's Law is to prove the contrapositive: In predicate logic, the contrapositive (or transposition) of the statement p implies q is not-q implies not-p. ...
(2) ¬P→¬((P→Q)→P)
Prove:
(3) :¬P→(P→Q) is a valid formula. Or more precisly a vacuous truth. Informally, a logical statement is vacuously true if it is true but doesnt say anything; examples are statements of the form everything with property A also has property B, where there is nothing with property A. It is tempting to dismiss this concept as vacuous or silly. ...
(2) is equivalent to ¬P→((P→Q) ∧ ¬P)
This is equivalent to
P ∨ ((P→Q) ∧ ¬P)
which can be transformed into
(P ∨ (P→Q)) ∧ (P ∨¬P).
Again, this is equivalent to
(4) (¬P→(P→Q)) ∧ true.
Using (3) in (4) results in true.
=> (1) <=> (2) <=> true
Q.E.D. Q. E. D. is an abbreviation of the Latin phrase quod erat demonstrandum (literally, which was to be demonstrated). This is a translation of the Greek (hóper édei deĩxai) which was used by many early mathematicians including Euclid and Archimedes. ...
Peirce's law, explained for non-mathematicians Peirce's law can be very confusing, especially if you're new to logic, so let's make it a little more concrete with an example.
Unfortunately for us, we have been drafted into the navy. Life at sea is no fun, but at least the lunch line serves pears. We like pears, so we're glad that they're served often. The navy has a very odd custom. Sometimes when we go through a line, every pear is glued to a kiwi. We call these "glue lines". The custom says that every glue line must have pears. Customs often have the force of laws, and this custom is so old that it's as strong as the law of gravity. Today, we come upon a line which has no pears. How sad for us. On the other hand, a funny thing happens when there are none of something: nearly any claim about it is true. For instance, all of the pears in this line are purple with orange polka dots. They are also all three feet long and taste like steak. And they are all very sweet pears. More to the point, each of the pears in this line is glued to a kiwi, so this must be a glue line. But every glue line has pears, which means that this line had pears after all. Odd that we didn't notice them earlier, isn't it? Well, now that we see some, they aren't purple with orange polka dots after all, nor are they three feet long or taste like steak. They are, however, very sweet. Hmm... my pear has a kiwi glued to it. Did you notice if this line was a glue line? No? Me neither.
Let's put this back into our logic context now. Okay, so P means that the line has pears and Q means that it has kiwi. (Well, can you come up with a common fruit starting with Q? Quinces are not very good raw.) Now, let's say that P→Q represents a glue line, since it's certainly true in a glue line. That means (P→Q)→P says that glue lines always have pears. Now, if we find a line with no pears (i.e. ¬P), then that line is a glue line (¬P→(P→Q)), but glue lines always have pears ((P→Q)→P), so not having pears means that the line has pears (¬P→P), which doesn't make sense (contradiction). That means that we can't find a line without pears, or rather that all lines have pears (P). If we put that all together, we see that the fact that glue lines always have pears means that all lines have pears, but not necessarially that all lines are glue lines (((P→Q)→P)→P, but not →(P→Q)). Binomial name Cydonia oblonga Mill. ...
Broadly speaking, a contradiction is an incompatibility between two or more statements, ideas, or actions. ...
References - CE n, m = Writings of Charles S. Peirce: A Chronological Edition, vol. n, page m.
- CP n.m = Collected Papers of Charles Sanders Peirce, vol. n, paragraph m.
Books - Peirce, C.S., Collected Papers of Charles Sanders Peirce, Vols. 1-6, Charles Hartshorne and Paul Weiss (eds.), Vols. 7-8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931-1935, 1958.
Articles Charles Sanders Peirce Charles Sanders Santiago Peirce (pronounced purse), (September 10, 1839, Cambridge, Massachusetts – April 19, 1914, Milford, Pennsylvania) was an American polymath. ...
Charles Hartshorne (June 5, 1897 – October 9, 2000) was a prominent philosopher who concentrated primarily on the philosophy of religion and metaphysics. ...
Paul Weiss is a leading nanoscientist at the Pennsylvania State University. ...
- Peirce, C.S., "On the Algebra of Logic: A Contribution to the Philosophy of Notation", American Journal of Mathematics, 7.2, 180-202 (1885). Reprinted, CP 3.359-403 and CE 5, 162-190.
Charles Sanders Peirce Charles Sanders Santiago Peirce (pronounced purse), (September 10, 1839, Cambridge, Massachusetts – April 19, 1914, Milford, Pennsylvania) was an American polymath. ...
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