[Math] Semantic Consequence Definition

Question

"What is the difference between
⊨
(semantic consequence) and
⊢
(syntactic consequence)?" was a question that has been posted, but I am wanting a more specific answer. For example, this video explains what a syntactic consequence is. After watching this video, it is obvious that we say
p
⊢q
when p->q is a tautology where p and q are given propositions forming the tautology. What is an easy way to explain what a semantic consequence is? I have been obsessed looking at this question for awhile. Any help would be greatly appreciated.

Answer 1

$\vDash$ means: logical consequence.

The general definition of it is:

A sentence $\varphi$ is said to be a logical consequence of a set of sentences $\Gamma$ (in symbols: $\Gamma \vDash \varphi$) if and only if there is no model $\mathcal {I}$ in which all members of $\Gamma$ are true and $\varphi$ is false.

In the context of propositional logic, this means that:

for every truth assignment (or valuation) $v$, i.e. for every function that assign a truth-value (T or F) to every sentential letter occurring in the formulas in $\Gamma$ or $\varphi$, if $v$ satisfy every formulas in $\Gamma$, then it satisfy also $\varphi$.

Trivial example (where $\Gamma$ has only one formula):

$\{ p \land q \} \vDash p$.

A truth assignment $v$ satisfy $p \land q$ only if $v(p)=v(q)=$ T.

Thus, every truth assignment $v$ that satisfy every formulas in $\Gamma$, i.e. that satisfy $p \land q$, satisfy also $p$.

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$\Gamma \vdash_{\mathcal S} \varphi$, instead, means that $\varphi$ is derivable (in the proof system $\mathcal S$) from the set of assumptions $\Gamma$.