When you have to prove that two formulas are logically equivalent, you make a truth table and check whether their their truth table columns are identical. And when you have to prove that a set of formulas logically entails a formula, you are doing the same thing. If I am correct, what is the difference?
Difference between logical equivalence and semantic entailment?
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logic
propositional-calculus
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0Logical equivalence and **mutual** semantic entailment are the same. $\phi \vDash \psi$ **iff** $\phi \to \psi$ is a *tautology*; thus, ($\phi \vDash \psi$ and $\psi \vDash\ \phi$) **iff** $\phi \leftrightarrow \psi$ is a *tautology*. – 2017-01-21
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To show two formulas, $\varphi$ and $\psi$, are logically equivalent you need to construct a formal proof in the language of propositional logic of $\varphi \leftrightarrow \psi$. To show that $\varphi \vDash \psi$ prove that whenever $\varphi$ is assigned the truth value, then $\psi$ also has the truth value, which as Mauro states is more related to $\varphi \to \psi$. You can do this with truth tables. A priori, constructing a truth table is neither here nor there when it comes to logical (or syntactic) entailment. A witness to logical entailment is a formal proof, and a truth table is not a formal proof.
Now, what we do have is the soundness and completeness theorems which say that $\varphi$ logically entails $\psi$ if and only if $\varphi$ semantically entails $\psi$. These theorems, particularly the completeness theorem, imply that if you can show semantic entailment using truth tables then there exists a formal proof that will show logical entailment. In fact, the completeness theorem is true constructively which is to say that not only does such a formal proof exist in that case, but a constructive proof of completeness is an algorithm that will produce it. The above theorems allow us to conflate logical and semantic entailment for propositional logic, but they are still different things. If I ask you for a formal proof of $\varphi \to \psi$, constructing a truth table and then saying "by completeness" would be like if when I asked you for the inverse of a matrix you checked that the determinant is non-zero and gave me a program for inverting matrices with non-zero determinant. What I expected was an actual matrix! Of course, if I merely asked if the matrix was invertible, then the above would suffice. Luckily, this latter question is indeed what we are usually asking.
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0"A witness to logical entailment is a formal proof, and a truth table is not a formal proof. " Waaaa..? I was under the impression that a truth table was a formal proof. Are not all logical "things" (for lack of a better word) that can be described by a truth table defined by them? So writing out th truth table would be a proof? – 2017-01-22
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0A formal proof is a collection of applications of formal rules. There are various ways of presenting these. You've probably seen [Hilbert-style proofs](https://en.wikipedia.org/wiki/Hilbert_system) before. I'll use a [natural deduction style](https://en.wikipedia.org/wiki/Natural_deduction) with sequents for a bit more compactness and human-friendliness. Here's a proof of $P\land Q\to Q\land P$:$$\frac{\frac{P\land Q\ \vdash P\land Q}{P\land Q\ \vdash Q}\quad\frac{P\land Q\ \vdash P\land Q}{P\land Q\ \vdash P}}{\frac{P\land Q\ \vdash Q\land P}{\vdash P\land Q\ \to\ Q \land P}}$$A much more ... – 2017-01-22
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0... compact representation exploits the [Curry-Howard Correspondence](https://en.wikipedia.org/wiki/Curry-Howard_correspondence) to represent the above proof as a program, namely $\lambda x:P\land Q.(\pi_2(x),\pi_1(x))$. Either way, neither of these look anything like truth tables. Instead, the completeness theorem states that if semantic entailment holds, which we can show with truth tables, then we can make a proof like one of the above. Semantics is about interpreting syntax and truth tables are but one interpretation of the syntax of propositional logic. – 2017-01-22
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The most important difference between equivalence and entailment is that equivalence expresses a 'two-way' (symmetrical, if you want) relationship of logical implication, whereas entailment expresses this only 'one-way'.
Example:
$P \land Q$ is equivalent to $Q \land P$, since they imply each other
$P$ entails $P \lor Q$ (But $P \lor Q$ does not imply $P$)
Also, because entailment is one way, it can also be used to say that a set of sentences entails some othe sentence, which is another difference, since you would never say that some set of sentences is equivalent to some other single sentence.
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0Why is it one way though? – 2017-01-22
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0@user400188 to take the $P$ and $P \lor Q$ case: if $P$ is true, then $P\lor Q$ will also have to be true. But going the other way: just because $P\lor Q$ is true does not mean that $P$ is true. So, the entailment only goes one way in this case. – 2017-01-22
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0its a strange sort of operation to my eyes. Information is gained from left to right. Should this thing be read backwards (right to left) instead? – 2017-01-22