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I know semantic consequence to mean that all statements on the left can all be true (are satisfiable) if the right side is true. If the right side is false, than the statements on the left cannot all be true.

There are a few statements giving me trouble.

The first: $$ [\{\Gamma, \phi \} \vDash \psi] \ \ iff \ \ [\{\Gamma\} \vDash (\phi \rightarrow \psi)] $$ If I start with the left of the iff, the statements all make sense.

The problem is when I start with the right side of the iff and $\Gamma$ is true, $\phi$ is false, and $\psi$ is true. That is a legitimate statement, but it proves the whole statement wrong.

The second: $$ [\{\bot\} \vDash \psi] $$

$\psi$ could be true despite the left side being false. I thought this was impossible.

The third:

$$ If \ [\{\Delta, \lnot \phi\} \vDash \bot]\ then \ [\{\Delta\} \vDash \phi] $$

If $\Delta$ is unsatifiable and $\phi$ is true, the if portion is true and the then portion is wrong.

Surely I am misunderstanding something if I keep running into this issue.

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I know semantic consequence to mean that all statements on the left can all be true (are satisfiable) if the right side is true.

No, that's not what it means. It's exactly the other way round: The right-hand side is true if all statements on the left-hand side are true. I.o.w., the definition of semantic consequence is that under any given interpretation, either the RHS is true or at least one statements on the LHS is false. It is not required that the LHS is true if the RHS is!
Perhaps it is easier to see it from the negative: The only thing that must not happen is for all the statements on the LHS to be true but the RHS false simulateneously. If, under some interpretation, the RHS is true but the LHS not, that's fine. This means in particular that if the LHS can never simulateneously be true (= is unsatisfiable), then there can be no such counter interpretation, and the consequence holds vacusously.
(Also see the note on (un)satisfiability in the last paragraph; your usage here suggests a misunderstanding of what it means.)


$$ [\{\Gamma, \phi \} \vDash \psi] \ \ iff \ \ [\{\Gamma\} \vDash (\phi \rightarrow \psi)] $$ If I start with the left of the iff, the statements all make sense.

The problem is when I start with the right side of the iff and $\Gamma$ is true, $\phi$ is false, and $\psi$ is true. That is a legitimate statement, but it proves the whole statement wrong.

You are misreading the structure of the statement. You are looking at one concrete assignment of truth values and try to make out from that one interpretation whether the semantic consequences on the left and right hold. But that's not what it says: The statement translates to

[Under all interpretations, either one of the statements in $\Gamma, \phi$ is false or $\psi$ is true]
iff
[Under all interpretations, either one of the statements in $\Gamma$ is false or $\phi \to \psi$ is true].

That is, we first need to look at all interpretations to determine whether the semantic consequences hold, and then evaluate the "if and only if". Looking at just one case where $\Gamma$ is true, $\phi$ false and $\psi$ true does not allow us to make a conclusion about whether either side of the "iff" holds.


The second: $$ [\{\bot\} \vDash \psi] $$

$\psi$ could be true despite the left side being false. I thought this was impossible.

See above: It's the other way round; it is only required that it's not possible for the RHS to be false despite the LHS being true. And this can never be the case if the LHS can not become true under any interpretation in the first place, which is the case for $\bot$, so the consequence holds vacuously.


$$ If \ [\{\Delta, \lnot \phi\} \vDash \bot]\ then \ [\{\Delta\} \vDash \phi] $$

If $\Delta$ is unsatifiable and $\phi$ is true, the if portion is true and the then portion is wrong.

You can stop reading after "If $\Delta$ is unsatisfiable": Then neither of the LHS's can ever become true, so both consequences hold vacuously, and the "if then" is satisfied.


And just to clarify the terminology: "$\Delta$ satisfiable/unsatisfiable" means that it is possible/impossible for all of its statements to become simultaneously true under any interpretation whatsoever, that is, $\Delta$ is not contradictory/contradictory. If it is just the case under one particular interpretation that all/not all of the statements in $\Delta$ are true, then we don't say that $\Delta$ is satisfiable/unsatisfiable, but just true/false. The same goes for single formulas: $\phi$ is true/false in a particular interpretation, and satisfiable/unsatisfiable if there is at least one/no interpretation under which it is true.

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  • $\begingroup$ Thank you for clearing that up! I shouldn't have flipped the definition. This will teach me to be more careful reading. $\endgroup$
    – John Glen
    Commented Aug 19, 2020 at 14:36
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A model of $\Gamma$ in which $\phi$ is false says nothing about the statement $\{\Gamma,\phi\}\vDash\psi$: that statement just says that $\psi$ is true in every model of $\Gamma$ and $\phi$, which is indeed the case if $\phi\to\psi$ is true in every model of $\Gamma$.

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