Wikipedia:Reference desk/Archives/Mathematics/2018 February 27

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February 27

Material conditional

I must have asked this question before: Is $p\to q\equiv \neg (p\land \neg q)$ "intuitively" defined, or can be proven by logic? (not truth-tables)

I found these 2 proofs here and here – are they valid? I do not quite understand them. יהודה שמחה ולדמן (talk) 00:18, 27 February 2018 (UTC)

Depending on which system of logic you're talking about, p→q might be something you define via a truth table, or as an abbreviation for an expression such as ~p∨q or ~(p∧~q), or → might be taken as a fundamental operation whose behavior is determined by axioms and rules of inference. The problem with something like Proof Wiki is that you can't necessarily pull a single proof out of context; a proof of a given statement depends on other statements which in turn depend on others. The proofs you linked to seem to be using a particular flavor of a subgenre of logic systems called natural deduction, and I'm guessing that the proofs are valid withing the context of one the sources listed as references. Specifically, the linked proofs seem to be taking the → symbol as fundamental with behavior determined by the Rule of Implication and Modus Ponens, but even within natural deduction it might be a defined symbol where the Rule of Implication and Modus Ponens are proved as theorems. So the short answer to the question is that it depends on which system of logic you're talking about. The good news is that different systems of propositional logic are equivalent in the sense that they prove the same statements, at least for classical systems. (In intuitionist logic e.g., the linked proofs are not valid.) But there can be a lot of variation in how you get there; you could build a system of logic built just from the ↓ (nor) operation, or a system using just truth tables with no normal proofs at all. --RDBury (talk) 10:36, 27 February 2018 (UTC)

The thing is, I understand pretty well why the $\land ,\lor$ functions give their answers; it is called "by definition".

But it seems to me that $\to$ is not "Well-defined". And now you tell me that Modus Ponens can be proven in some systems... does this ever end?

So I wonder: Is it $p\to q:=\neg (p\land \neg q)$ , or is it $\equiv$ ? יהודה שמחה ולדמן (talk) 11:25, 27 February 2018 (UTC)

As an example of a similar situation, you could define an even number as one that can be written as x+x, or you could define an even number as one that is divisible by 2. They are two different definitions that turn out to be equivalent. Euclid used the first definition but most modern authors use the second. But it's like you're asking which is 'the' definition of even; the answer is it depends which book you're reading, but in the end it doesn't really matter because the two definitions give the same results. It's the same with logic; five books on logic will have five different sets of axioms, rules of inference, and definitions of the operations. The book here defines → in terms of ~ and ∨, another book might define it a different way, and another might define ∨ in terms of ~ and →. Again, it doesn't matter in the end because the same set of statements can be proved. --RDBury (talk) 12:43, 27 February 2018 (UTC)

OK, so what are the axioms and rules of inference for mathematical logic? יהודה שמחה ולדמן (talk) 13:41, 27 February 2018 (UTC)

As RDBury said above, that depends on which system of logic you have in mind. The axioms of propositional logic are different from the axioms of intuitionist logic, for example. Gandalf61 (talk) 16:06, 27 February 2018 (UTC)

I finally found a proof on youtube, but I keep getting a posting-error. יהודה שמחה ולדמן (talk) 00:29, 2 March 2018 (UTC)

Meter to French foot and vice versa

Our Units of measurement in France before the French Revolution says that in 1799 one meter was set to 443.296 French lines, or 3 feet 11.296 lines. That corresponds to ⁠9,000/27,706⁠ m. A note follows: This can be shown by noting that 27706 x 16 = 443296 and that 9 x 16 = 144, the number of lignes in a pied. I feel that it must be entirely correct, but I simply can't get it. How does it work? Could you elaborate, step by step, how we get from 1 m = 443.296 lines to 1 ft (144 lines) = ⁠9,000/27,706⁠ m? --Lüboslóv Yęzýkin (talk) 17:49, 27 February 2018 (UTC)

It's simple arithmetic. If 1 foot is 144 lines and 1 meter is 443.296 lines, then 1 foot is 144/443.296 meter. Multiply both numerator and denominator by 1,000 to get rid of the decimals: 144,000/443,296. Now divide both by the common factor of 16 and you have 9,000/27,706. That is what the footnote is clumsily trying to say. I have no idea why the remaining common factor of 2 was not eliminated to write the fraction in lowest terms as 4,500/13,853. --69.159.62.113 (talk) 00:44, 28 February 2018 (UTC)

• One more point. One of the motivations for adopting metric in France was that each region had its own definitions of different units. So speaking of the "French foot" is wrong; it's only the "Paris foot" or the "French royal foot" or some such. In the article, look back a few paragraphs from where it mentions 9,000/27,706. --69.159.62.113 (talk) 00:47, 28 February 2018 (UTC)