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In a "math structures" class at the community college I'm attending (uses the book Discrete Math by Epp, and is basically a discrete math "light" edition), we've been covering some basic logic.

I've been reading some of the logic questions on here to get used to notation, etc. However, when I came across the question Visualizing Concepts in Mathematical Logic, I didn't understand what the $\vdash$ symbol means.

It's not in Discrete Math by Epp, nor is it in my mom's old logic book from when she went to college.

Wikipedia's Math Symbols page says it means "can be derived from" when used in a logic context. However, that doesn't make any sense in the above question, as there is nothing on the left of the $\vdash$.

So, what does $\vdash$ mean, especially in the context of the question linked above?

Community
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apnorton
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    I see you've accepted an answer. But before you go away, you should look at my answer, since it addresses some points that the others don't that are worth being aware of. – Michael Hardy Jan 17 '13 at 04:06
  • I too am using this textbook and I will bookmark this post for that chapter when I get there. I've used other Discrete Math books but this one in particular is good at forming the picture so I don't forget things. However, there are accounts, such as you have described, where they are a bit vague and I have to look elsewhere to find the answers I want. – bjd2385 Jan 22 '15 at 14:28

5 Answers5

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Let $S$ be a set of (logical) formulae and $\psi$ be a formula. Then $S \vdash \psi$ means that $\psi$ can be derived from the formulae in $S$. Intuitively, $S$ is a list of assumptions, and $S \vdash \psi$ if we can prove $\psi$ from the assumptions in $S$.

$\vdash \psi$ is shorthand for $\varnothing \vdash \psi$. That is, $\psi$ can be derived with no assumptions, so that in some sense, $\psi$ is 'true').

More precisely, systems of logic consist of certain axioms and rules of inference (one such rule being "from $\phi$ and $\phi \to \psi$ we can infer $\psi$"). What it means for $\psi$ to be 'derivable' from a set $S$ of formulae is that in a finite number of steps you can work with (i) the formulae in $S$, (ii) the axioms of your logical system, and (iii) the rules of inference, and end up with $\psi$.

In particular, if $\vdash \psi$ then $\psi$ can be derived solely from the axioms by using the rules of inference in your logical system.

Clive Newstead
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  • What do you mean by "formula"? – alancalvitti Jan 16 '13 at 23:27
  • @alancalvitti: If you like, think if 'formula' as another word for 'statement'. – Clive Newstead Jan 16 '13 at 23:28
  • I don't know what formula is, nor a statement, nor, really, what "derivable" means, in an universal sense. Brouwer: "a mathematical system of entities can never remain reliable as a guide along our perceptions, when it is indefinitely extended beyond the perceptions which it made understandable" – alancalvitti Jan 16 '13 at 23:31
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    @alancalvitti: I don't really get what you're saying. I'm trying to explain syntactic entailment to someone with little grounding in formal logic, so I don't want to get hung up on technical definitions or philosophical points. – Clive Newstead Jan 16 '13 at 23:32
  • What do you mean by "syntactic"? I've been programming computers for 20 years - you can use technical definitions. I just want to make sure there actual bits at the end, and not an endless fugue from one term to another. – alancalvitti Jan 16 '13 at 23:36
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    @alancalvitti: I was referring to the question-asker, not yourself $-$ I thought you were asking these questions on their behalf (because of shortcomings in my answer). The comment box isn't really a place for these questions though. If you want to learn about these things, there are plenty of good resources, websites and textbooks on the subject, or you could ask a new question here. – Clive Newstead Jan 16 '13 at 23:40
  • You wrote "syntactic entailment" whereas gvv wrote "syntactic implication". Which is correct? Or can either one be syntactically derived from the other? Or perhaps semantically? – alancalvitti Jan 16 '13 at 23:43
  • @alancalvitti: I'll indulge your questions one last time but if you have any more then you really should post it as a new question rather than a comment. Either gvv's 'syntactic implication' is another way of saying 'syntactic entailment', in which case they're the same; or else it refers to logical implication (of the type $\phi \to \psi$), in which case you may be interested in the deduction theorem. – Clive Newstead Jan 16 '13 at 23:49
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    @alancalvitti: both are correct and (in standard systems) synonymous, but at least in my experience, entailment is used more often for the analogous semantic notion (usually denoted $\vDash$), while implication and provability are more common for this syntactic notion. – Peter LeFanu Lumsdaine Jan 16 '13 at 23:51
  • @PeterLeFanuLumsdaine, gvv just edited his answer. Your comment seems at odds with his revised answer ie, which syntactic v. semantic is associated with which symbol? – alancalvitti Jan 17 '13 at 00:00
  • @CliveNewstead, I'm not trying but bust your b*lls. These comments serve as a reminder that there's no universally accepted definition of syntax, semantics, implication, entailment. – alancalvitti Jan 17 '13 at 00:01
  • I'm not expert, but I agree with Peter in that, so for as I have seen, entailment is used more for the semantic notion. I changed my answer to be more consistent and less confusing if anyone looks at this question. – ggg Jan 17 '13 at 00:06
  • @CliveNewstead, from Lawvere & Rosebrugh, SFM p.198: "$B$ implies $D$ or 'if $B$ then $D$' is to be distinguished from $B \vdash D$, which is a statement about statements usually written down only when we mean to assert that $D$ in fact follows from $B$, whereas $B \Rightarrow D$ is a statement, which, like other statements, may be important to consider even when it is only partly true. The rule of inference characterizing $\Rightarrow$ is not only in terms of $\vdash$ but also involves $\wedge$" (their italics). Is this assertion consistent with the ongoing discussion? – alancalvitti Jan 17 '13 at 00:15
  • I wasn't aware of these conventions regarding the use of 'entailment' and so on; when I learnt this stuff (last year) it was in this language and with this notation that I learnt it, but the lecturer is known for his quirks (e.g. he insists that structures can be empty). @alancalvitti: I know you're not trying to break my b*lls but I don't think this discussion is suitable for this website, at least not in the comments section of an unrelated question. – Clive Newstead Jan 17 '13 at 00:15
  • @alancalvitti: (Yes, it is consistent with it.) – Clive Newstead Jan 17 '13 at 00:17
  • @Clive: I have a more serious question. How can a statement be "derived from no assumptions?" I thought we always need at least some axioms to start off with, to have something from which to prove things (well, and to assume that the laws of logic are valid, but I guess that's always implied). Or do you mean statements like, "By definition, 1=1"? – BlueRaja - Danny Pflughoeft Jan 17 '13 at 00:21
  • @BlueRaja: See the final paragraph in my answer. – Clive Newstead Jan 17 '13 at 00:25
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    Derivations depend on the logical calculus you are using. For example, if you're using an axiomatic calculus, you can derive formulas by just following the axioms, thus you have derived something without any assumptions. However, if you are using another calculus, for example Natural Deduction Calculus, ⊢ A iff you can derive A without any open assumptions. – ggg Jan 17 '13 at 00:26
  • The only thing here comes as that systems of logic can consist of just rules of inference without any axioms at all. – Doug Spoonwood Jan 17 '13 at 01:36
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⊢ means "can be derived from" or "proves", and denotes syntactic entailment. For example, let G be a set of sentences in logic, and A be any sentence in logic. G ⊢ A (read: G proves A) iff A can be derived using only the sentences in G as assumptions. Thus, if for a certain A we have ⊢ A, then A can be derived without any open assumptions.

Note that ⊢ is different than ⊨, which stands for semantic entailment.

ggg
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  • You wrote "syntactic implication" whereas Newstead wrote "syntactic entailment". Which is correct? Or can either one be syntactically derived from the other? Or perhaps semantically? – alancalvitti Jan 16 '13 at 23:43
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    Honestly, I don't think we meant anything different. I think syntactic entailment/consequence would be better, and will edit my answer now. This could help clarify though: http://en.wikipedia.org/wiki/Logical_consequence – ggg Jan 16 '13 at 23:48
  • I want you to acknowledge that you edited your answer (first line) from "syntactic implication" to "syntactic entailment" and (last line) from "syntactic implication" to "semantic entailment" – alancalvitti Jan 16 '13 at 23:59
  • I didn't edit my last line. – ggg Jan 17 '13 at 00:07
  • no problem, I would partially revise my previous comment but it timed out. – alancalvitti Jan 17 '13 at 00:16
  • @alancalvitti There is no separate syntactic or semantic derivation. Derivation is semantic, and not just a juggling of symbols and syntax: the logic symbols have meaning. semantic implication versus syntactic implication is an abuse of these terms. A "syntactic" implication is one that holds for all possible interpretations, whereas a "semantic" one is one that holds for some interpretations of interest, not necessarily all. When the conclusions follow from the premises in all "real world" situations, that is useful even if the conclusions do not absolutely follow. – Kaz Jan 17 '13 at 02:40
  • @Kaz, can you please give an example and a counterexample from each (syntactic vs semantic as you characterized them?) – alancalvitti Jan 17 '13 at 03:43
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It's called a 'turnstile'. See here: http://en.wikipedia.org/wiki/Turnstile_(symbol)

adam W
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Erik G.
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We should attend to a distinction between "$\vdash$" and "$\models$". The notation $A\vdash B$ means $B$ can be deduced from $A$ in some reasonable system of deduction, and "reasonable" should mean at the very least

  • There is an algorithm for deciding which deductions are valid ("effectiveness"); and
  • If $B$ can be deduced from $A$ then $B$ is true in every structure in which $A$ is true (soundness).

One may also have

  • If $B$ is true in every structure in which $A$ is true, then $A\vdash B$ (completeness).

(The word "completeness" here should not be confused with the "completeness" referred to in Gödel's incompleteness theorem; that is different.)

The notation $A\models B$ means simply that $B$ is true in every structure in which $A$ is true.

Michael Hardy
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  • How would one know if $A\models B$ in case there is an open-ended or unbounded list of models? Wouldn't there have to be an algorithm for deciding (presumably based on some reasonable system of deduction) when $A$ and $B$ are true (or false) in all model instances? – alancalvitti Jan 18 '13 at 17:51
  • Also, when you write "at the very least..there is an algorithm for deciding which deductions are valid: ('effectiveness')" does this mean that $A\vdash B$ only applies to constructive math systems, typically characterized, eg by McLarty as having: no Choice, no Excluded Middle, exhibit specific instances of solutions? – alancalvitti Jan 18 '13 at 17:55
  • "$B$ can be deduced from $A$" is that logically equivalent to $A$ implies $B$?" – alancalvitti Jan 18 '13 at 19:01
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    Also note the potential (or actual) consistency issues between the answers and comments to this question versus here: http://math.stackexchange.com/questions/238872/difference-between-implies-and-therefore/238880#238880 – alancalvitti Jan 18 '13 at 19:05
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I usually read $\vdash$ as "entails". You can also use "proves".

Code-Guru
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  • Is your interpretation consistent with Michael Hardy's? He wrote "$A \vdash B$ means $B$ can be deduced from $A$". Is that logically equivalent to "$A$ proves $B$"? – alancalvitti Jan 18 '13 at 19:02
  • "$A$ proves $B$ uses the word "proves" loosely. I believe it conveys the same meaning and is less wordy than "$B$ can be deduced from $A$" which is more technically accurate. Of course, "$A$ entails $B$" is also technically accurate, but uses a word which is unfamiliar to many. – Code-Guru Jan 18 '13 at 23:58
  • What about the above versus "$A$ implies $B$"? – alancalvitti Jan 19 '13 at 00:00