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Per Ebbinghaus' Mathematical Logic, or any other standard mathematical logic books:

  • Is $\models$ called (logical) consequence relation between formulas?

  • Is $\unicode{x27DA}$ called (logical) equivalence relation between formulas?

  • Is $\vdash$ called derivable relation between formulas?

  • Is $\unicode{x27DB}$ (a symmetric relation between formulas, defined in terms of $\vdash$) called derivable equivalence? (Does Ebbinghaus' book ever use it?)

  • Is $\to$ called (material) implication (a connective between formulas, to create a compound formula)? Is $⇒$ the same as $\to$?

  • Is $\leftrightarrow$ (a connective between formulas, to create a compound formula, defined in terms of $\to$) called (material) equivalence? Is $⇔$ the same as $\leftrightarrow$?

  • What does $\equiv$ mean in logic? ($\unicode{x27DA}$, $\unicode{x27DB}$, $\leftrightarrow$, $=$, or something else?). ($\equiv$ is called "equality" symbol in II.3.1 in Ebbinghaus' book and used to create a formula from two terms. Does it have a different meaning $\unicode{x27DA}$ in this Wikipedia page ?)

Thanks.

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    $\begingroup$ Note that the second symbol isn't right - it should be a backwards "$\models$" followed by a standard "$\models$" (so "$\models$ in both directions"). Unfortunately it seems MathJax doesn't support the relevant LaTeX code, which is "\Dashv" (see page $52$ here). There's also the symmetric version of "$\vdash$," which again MathJax doesn't seem to like unfortunately. $\endgroup$ – Noah Schweber Sep 7 '20 at 0:34
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    $\begingroup$ Does en.wikipedia.org/wiki/List_of_logic_symbols help answer any of this? $\endgroup$ – Barry Cipra Sep 7 '20 at 0:37
  • $\begingroup$ See math.meta.stackexchange.com/questions/31987/… on how to typeset =||=/-||-. $\endgroup$ – lemontree Sep 7 '20 at 1:31
  • $\begingroup$ If satisfied with one of the answer below, please accept it $\endgroup$ – Mauro ALLEGRANZA Sep 9 '20 at 14:18
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  • Is $\models$ called (logical) consequence relation between formulas?

Yes. Or (logical) inference or (logical) entailment or semantic consequence/inference/entailment.

  • Is $\unicode{x27DA}$ called (logical) equivalence relation between formulas?

Yes.

  • Is $\vdash$ called derivable relation between formulas?

It is called derivability.

  • Is $\unicode{x27DB}$ (a symmetric relation between formulas, defined in terms of $\vdash$) called derivable equivalence? (Does Ebbinghaus' book ever use it?)

It is called interderivability. Don't know the entire Ebbinghaus book by heart, but you don't see it as often as logical equivalence or unidirectional derivability.

  • Is $\to$ called (material) implication (a connective between formulas, to create a compound formula)?

Yes, or (material) conditional. Sometimes (esp. in older texts) you also see

⊃

being used for material implication.

  • Is $⇒$ the same as $\to$?

Sometimes yes; sometimes $⇒$ means logical consequence; sometimes $⇒$ means a meta-linguistic (= mathematical English) "if ... then".

  • Is $\leftrightarrow$ (a connective between formulas, to create a compound formula, defined in terms of $\to$) called (material) equivalence?

It is ususally called biimplication or biconditional, perhaps sometimes (material) equivalence.

  • Is $⇔$ the same as $\leftrightarrow$?

Analogous to above: Sometimes yes; sometimes it means logical equivalence; sometimes sometimes a meta-linguistic "if and only if".

  • What does $\equiv$ mean in logic? ($\unicode{x27DA}$, $\unicode{x27DB}$, $\leftrightarrow$, $=$, or something else?).

Usually it means logical equivalence, sometimes biimplication and sometimes syntactic identity (= literal sameness of formulas).

$=$

usually means term equality in FOL; sometimes it is used for logical equivalence and sometimes for syntactic identity.

$\bumpeq$

is sometimes seen for syntactic identity.

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  • $\begingroup$ Thanks. What is $≡$ in en.wikipedia.org/wiki/… ? By en.wikipedia.org/wiki/List_of_logic_symbols, it looks like material equivalence? Or does it mean logic equivalence? $\endgroup$ – Tim Sep 7 '20 at 1:34
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    $\begingroup$ If you are unsure about different language levels, this can not be comprehensively answered in a comment and is best asked as a new question. $\endgroup$ – lemontree Sep 7 '20 at 1:43
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    $\begingroup$ Theoretically yes, but it doesn't make much sense to throw an object language formula (such as a chain of material equivalences) on one line within a text, because an object-linguistic formula is not an English statement, whereas a meta-linguistic statement (such as a logical equivalence) is, so the context makes it clear that logical equivalence is probably the intended meaning. $\endgroup$ – lemontree Sep 7 '20 at 1:50
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    $\begingroup$ Sorry, but you'll have to figure out yourself; I can''t tell you what you want to know. $\endgroup$ – lemontree Sep 7 '20 at 2:51
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    $\begingroup$ I check the site regularly and will answer questions whenever I have an interest, time, and an answer; in this case I don't. $\endgroup$ – lemontree Sep 9 '20 at 13:52
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An example using all of the symbols:

The second symbol ($\unicode{x27DA}$) is stating that 2 formula are each a logical consequence of each other i.e.

$A \unicode{x27DA} B$ states that:

$A \vDash B $ and $ B \vDash A$
Therefore, $ A \equiv B $ ($\equiv$ meaning logical equivalence).

They are logically equivalent because $ A \leftrightarrow B $ is a Tautology.
$ (A \rightarrow B) \wedge (B \rightarrow A) $ would also be a Tautology.

$A \unicode{x27DA} B$ would also mean that $ A \unicode{x27DB} B $ (assuming a 'complete' inference system) i.e.

$ A \vdash B $ and $ B \vdash A $

Meaning that B can be derived from A and that A can be derived from B.

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Original question: "What does $\leftrightarrow$ mean?"

It is called a bi-conditional relation. If given statements $P$ and $Q$, then $P\leftrightarrow Q$ means that $$(P\rightarrow Q )\wedge (Q\rightarrow P)$$

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    $\begingroup$ I misread the original question when I said "I think Tim is talking about "$P$ can be used to prove $Q$ and vice versa". You need something like the deduction theorem to get from that to a biconditional." You have answered part of the original question. $\endgroup$ – Mark S. Sep 7 '20 at 0:30
  • $\begingroup$ @MarkS. Its all good. There are a lot of questions in this post; I only answered the one that I knew. It seems like you would be more suited to answer the rest of them though. $\endgroup$ – C Squared Sep 7 '20 at 0:33
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    $\begingroup$ You should add parentheses or $ Q \wedge Q $ will be the first operation. $\endgroup$ – 0implies0 Sep 7 '20 at 3:42

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