# Logical Consequence In [Logical](Logic.md) [Deduction](Deduction.md), a **logical consequence** is also known as a **logical entailment**. Given two statements $x$, $y$, we write $x \vdash y$ if $x$ entails $y$. For example, if $\begin{aligned} &x =\text{All men are mortal}\\ &y =\text{Socrates is a man}\\ &z =\text{Socrates is mortal} \end{aligned}$ we can write $(x \wedge y) \vdash z$, which can be read “$x$ and $y$, therefore $z$”. Another example is: $ \begin{aligned} &S 1 \rightarrow \text{White}\\ &\neg \text{White}\\ &\therefore \neg S 1 \end{aligned} $ which can be rewritten using the turnstile symbol $\vdash$ as: $\begin{aligned} &x =\text{S1 is white}\\ &y =\text{That bird is white}\\ &z =\text{That bird is S1}, \end{aligned}$ where $(x \wedge \lnot y) \vdash (\lnot z)$ This can be read “$x$ and not $y$, therefore not $z$”. But what does “entailment” really mean? I like to think of it as an operation which allows us to “**move truth around**”, from one sentence over to the next. If $x$ “entails” $y$, what we’re saying is, if we’re willing to assume sentences on the left of the turnstile are true (which we will represent with the color green). $ \begin{aligned} \textcolor{green}{(x \wedge y)}\vdash z, \end{aligned} $ then we automatically know sentences on the right of the turnstile are true, $\textcolor{green}{(x \wedge y)\vdash z}$ In this way, we have moved the property is true from $(x \wedge y)$ over to $z$. Synonyms or near-synonyms for “logical entailment” you’re likely to encounter are: “**logical consequence**”, “derived”, “deduced”, “follows from”, “logically valid”, “necessarily true”, “necessary truth”, “thus”, “logical conclusion”, “therefore”, “hence”, etc. (And when Popper uses the word “logical”, this is what he has in mind - similarly with Deutsch and “derive”.) Now consider an example where we don’t have logical entailment: $\begin{aligned} x =&\text{All men are mortal} \\ y =&\text{Socrates is a man} \\ z =~&\text{Water is comprised of two parts hydrogen}\\ ~&\text{ and one part oxygen.} \end{aligned}$ True as $z$ may be, it isn’t entailed by $(x \wedge y)$, so assuming the truth value of $x$ and $y$ won’t give us any additional insight into the truth value of $z$ - we have to get that from elsewhere: $\textcolor{green}{(x \wedge y)} \nvdash z$ This can lead us nicely into [There Is No Problem Of Induction](There%20Is%20No%20Problem%20Of%20Induction.md). --- Date: 20250403 Links to: Tags: References: * [The Problem of Induction and Machine Learning \| Vaden Masrani](https://vmasrani.github.io/blog/2021/problem-of-induction/) * [Logical consequence - Wikipedia](https://en.wikipedia.org/wiki/Logical_consequence)