"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.

# [Math] Semantic Consequence Definition

logicpropositional-calculus

## Best Answer

$\vDash$ means: logical consequence.

The general definition of it is:

In the context of propositional logic, this means that:

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

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 $p$.

$\Gamma \vdash_{\mathcal S} \varphi$, instead, means that $\varphi$ is

derivable(in theproof system$\mathcal S$) from the set ofassumptions$\Gamma$.