- #1

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Does anyone recognize the symbols ⊃, ≡, |, |–, –||–, |=, =||= ??

They are logical connectives, just different than the ones I know.

Help would be greatly appreciated as soon as possible.

Thank you.

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- Thread starter penitor
- Start date

- #1

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Does anyone recognize the symbols ⊃, ≡, |, |–, –||–, |=, =||= ??

They are logical connectives, just different than the ones I know.

Help would be greatly appreciated as soon as possible.

Thank you.

- #2

EnumaElish

Science Advisor

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⊃ is superset

≡ is identical or equivalent

I don't have a clue about the rest.

≡ is identical or equivalent

I don't have a clue about the rest.

- #3

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thanks man, anyone else have ideas about the rest?

- #4

AKG

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The single turnstyle "|-" has to do with syntax. It has nothing to do, essentially, with what your sentences mean. Just stipulate a set of rules for manipulating symbols, and if you have some sentences S, where each sentence is just a string of symbols that adhere to some rules as to what counts as a proper string of symbols, then if you manipulate these sentences according to the rules, you can get another string of symbols P, and you can say S |- P. The double turnstyle has to do with semantics. You choose some way of interpreting your strings of symbols. Some symbols have a standard interpretation, like the logical connectives. But if you have a sentence P & Q, then P and Q can be interpreted to mean just about any English sentence you want, but regardless of what you choose them to mean, if P & Q is true, then P will be true, so {P & Q} |= P. So, giving "P & Q" whatever meaning you want with the condition that P stands for some sentence and & stands for "and," but otherwise having total freedom to choose a meaning, whenever your interpretation makes P & Q true, it must make P true.

Note: the first two symbols are logical symbols, the rest (well I'm not sure about "|") are 'metalogical' symbols. So if p and q are sentences of your language, p -> q and p = q are also sentences of your language. p |- q and p |= q are not sentences of your language, they are metalogical sentences about sentences in your language.

I've never seen -||- or =||= before, but I can guess that if you have P -||- Q then it just means P |- Q and Q |- P. That is P is derivable from Q and vice versa. In this case, we say that Q and P are deductively equivalent. Similarly, if you have P =||= Q, then P |= Q and Q |= P, so you would say that P and Q are truth-functionally equiavlent or quantificationally equivalent, depending on the context.

Still not sure about the "|".

- #5

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penitor said:Does anyone recognize the symbols ⊃, ≡, |, |–, –||–, |=, =||= ??

⊃ is usually implication, so (P ⊃ Q) means "if P then Q".

≡ is usually syntactic equivalence.

|- is usually inference.

|= has a different meaning depending on the context. For example, A |= B could mean B is a logical consequence of A, or structure A satisfies B.

I am not sure about the others.

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- #6

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Thanks so much you guys, I am extremely grateful.

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- #8

honestrosewater

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"|" could be [URL [Broken] stroke[/url]. (Notice that it can be used to define negation, conjunction, disjunction, and implication, so you can use it as your only connective.) Sheffer's stroke would usually be one of AKG's *logical symbols*, along with the other connectives.

An extra note on the*meta* thing: There's an important difference between the object language (the language that you talk about) and the metalanguage (the language that you use to talk about the object language). All of the symbols that you use are symbols of the metalanguage. Additionally, some of them __denote__ symbols of the object language. However, they are not necessarily symbols of the object language -- the object language doesn't necessarily have a written form.

An extra note on the

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