Linked Questions
41 questions linked to/from Implies ($\Rightarrow$) vs. Entails ($\models$) vs. Provable ($\vdash$)
17
votes
1
answer
13k
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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 ...
- 75
4
votes
2
answers
570
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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 ...
- 818
3
votes
2
answers
202
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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 ...
- 799
0
votes
1
answer
548
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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.
...
- 129
3
votes
0
answers
98
views
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 $\psi \Rightarrow \...
- 97
40
votes
1
answer
21k
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What's the difference between material implication and logical implication?
When I read the definitions of material and logical implications, they seem to me pretty much equivalent. Could someone give me an example illustrating the difference?
(BTW, I have no problem with ...
- 14.2k
16
votes
4
answers
3k
views
What's the difference between "$\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 ...
- 4,173
16
votes
1
answer
11k
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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.
- 273
8
votes
3
answers
416
views
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 ...
- 23k
9
votes
3
answers
2k
views
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 ...
- 3,326
4
votes
5
answers
2k
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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.&...
- 2,572
7
votes
2
answers
546
views
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 ...
- 312
3
votes
2
answers
1k
views
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 ...
- 4,094
4
votes
3
answers
2k
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Premise vs. Assumption
I have just asked about the difference between A,B and A∧B in A,B ⊢ M However, I have ...
- 1
5
votes
3
answers
500
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$\to$ 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 ...
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