Linked Questions

17 votes
1 answer

what's the difference between Syntactic consequence ⊢ and Semantic consequence ⊨ [duplicate]

Can you help me to differentiate the Syntactic consequence $\vdash$ and Semantic consequence $\vDash$ ? I think $A \vdash B$ means, "$A$ proves $B$" and $A \vDash B$ means , if $A$ is true, then ...
martian03's user avatar
4 votes
2 answers

How can you derive a formula without premises? [duplicate]

What is the difference between $\vdash A $ and $\models A$? I'm not asking about the general difference between syntactic entailment and semantic entailment. I specifically don't understand the ...
Colin Hicks's user avatar
3 votes
2 answers

Semantical and syntactical derivation notation [duplicate]

What do the following two notations mean: $$A \models$$ $$A \vdash$$ The first one would be saying that A is a contradiction? The second one that A is not a theorem? That doesn't sound right to ...
SlowerPhoton's user avatar
0 votes
1 answer

Difference between implies and "turnstile" symbols (→ and ⊢) [duplicate]

According to Wikipedia's list of logic symbols: A → B means A → B is false when A is true and B is false but true otherwise. ...
Elliott's user avatar
  • 129
3 votes
0 answers

Where's the difference between $\vDash$ and $\Rightarrow$? [duplicate]

I'm sorry if that question has already been asked. Sometimes in propositional logic when proving "$\phi \rightarrow \psi$ is a tautology" using truth tables people write $\psi \Rightarrow \...
Knogger's user avatar
  • 97
40 votes
1 answer

What's the difference between material implication and logical implication?

When I read the definitions of material and logical implications, they seem to me pretty much equivalent. Could someone give me an example illustrating the difference? (BTW, I have no problem with ...
kjo's user avatar
  • 14.2k
16 votes
4 answers

What's the difference between "$\to$" (implication) and "$\vdash$" (therefore)?

In Wikipedia, here in the last axiom of the Natural deduction system, it says "From [accepting $p$ allows a proof of $q$], infer $(p \to q)$." Isn't that a tautology? In the big table "Basic and ...
xzhu's user avatar
  • 4,173
16 votes
1 answer

When do we use entailment vs implication?

I have seen both used and I have been unable to find by searching. They seem to have a similar meaning from what I have seen, however I have only seen entailment in relation to models.
stmfunk's user avatar
  • 273
8 votes
3 answers

Using $p\supset q$ instead of $p\implies q$

I saw that a use for the notation $p\supset q$ instead of $p\implies q$ that got me a bit confused. One occurrences is in this Wikipedia link. It seems to me opposite than what it should be, let me ...
Belgi's user avatar
  • 23k
9 votes
3 answers

In logic, do the $\Longrightarrow$ and $\rightarrow$ signify different things?

In logic, do the $\Longrightarrow$ and $\rightarrow$ signify different things? Are there contexts where one is more appropriate than the other? I had believed that the $\Longrightarrow$ was for ...
Hal's user avatar
  • 3,326
4 votes
5 answers

Not Necessarily Imply

Consider the statement $(1)$     If I'm in Richmond, then I'm in Virginia. This statement is an implication $p \implies q$, where $p$ is "I'm in Richmond" and $q$ is "I'm in Virginia.&...
chharvey's user avatar
  • 2,572
7 votes
2 answers

Are mathematical definitions logical equivalences or material equivalences?

Lets consider representing logical equivalence using the symbol $\equiv$ and material equivalence with the symbol $\longleftrightarrow$. I know that the formulas $P$ and $Q$ are logically equivalent ...
fire-bee's user avatar
  • 312
3 votes
2 answers

Using "implies" to refer to material conditional

Is it acceptable to translate the binary connective "$\let\ f\rightarrow$" into English with "implies"? I'm unsure because "implies" for me immediately brings to mind logical implication, but I've ...
CuriousKid7's user avatar
  • 4,094
4 votes
3 answers

Premise vs. Assumption

I have just asked about the difference between A,B and A∧B in A,B ⊢ M However, I have ...
Val's user avatar
  • 1
5 votes
3 answers

$\to$ vs. $\vdash$ in logic

I am really lost trying to understand the difference between the logical connective "implies", $\to$, and the metalogical symbol (or maybe it's also a connective?) $\vdash$. (This is all focusing on ...
user525966's user avatar
  • 5,531

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