7.1: Logical Entailment
- Page ID
- 223895
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)If I was to tell you that either I was going to be promoted or get a big bonus because I just landed a huge account for the company I work for, then you would think a few things:
- If he doesn’t get promoted, then at least he’ll get compensated with a bonus
- If he doesn’t get a bonus, then at least he’ll get compensated with a promotion
- His company won’t both not promote him and not give him a bonus.
All of these claims follow from the original claim. They “follow” in the sense that if the original claim is in fact true, then this conclusion must be true. There’s some sense in which they “mean the same thing”: they describe the same world or claim the same thing to be true about the world. A lot of logic consequence is similar: it’s a relationship of “following” or “entailment” between statements which mean essentially the same thing.
Other logical consequences are cases where one statement entails another statement (if the first is true, the second must be true), but not because they essentially mean the same thing. Instead, because the first statement is making a “stronger” claim than the other. Here’s an example:
Franklin is a student at Butte College.
Entails the following claim:
Either Franklin is a student at Butte College or Franklin is a student at De Anza College, I’m not sure which.
This first statement entails this second statement because the second one is an “either...or...” statement and is therefore much weaker. If I told you “Either Franklin is a student at Butte College or Franklin is a student at De Anza College”, then you’d know that one of two worlds must be the real world: either a world in which Franklin is a student at Butte or a world in which Franklin is a student at De Anza. Alternatively, if I told you “Franklin is a student at Butte College,” then you’d know which of those two worlds was the real world: the one in which Franklin was a student at Butte College. It is therefore a stronger or more determinate claim.
Think in terms of betting: Which would you bet on: that either Russia or China would win the most Gold medals at the next Olympics, or that Russia would win the most? If you’re wise to the logic of the situation, you’d know to bet on the disjunction or “or” statement since you’d have a higher change of winning. This is what we mean by “stronger claim” and “weaker claim”: a stronger claim is a claim that is “harder to make true” in the sense that the world has to be one particular way to conform to that claim. A weaker claim, conversely, is relatively easier to make true in that the world could be a couple of different ways for the claim to be true.
Logic is the science of logical consequences: these relationships between statements such that if one is true, then the other must be true.
But logic consequence isn’t like just any consequence: think about the following example:
If the Russians invaded the United States,
then the US would have to defend itself.
The second statement: that the US would have to defend itself, seems to follow in some sense from the first statement. Nevertheless, this is not a logical consequence: it’s not an entailment that holds (or is valid) in virtue of the structure or form of the two statements involved. It’s an inferential relationship that these propositions have in virtue of the way the world works or the way war and international relations works. That, also, isn’t strict entailment. Logical entailment is deductive entailment in that if the premise(s) is true, the conclusion must be true. It couldn’t possibly be false given the truth of the premise(s). For instance, if it is in fact true that:
If one gets properly poisoned, then one will get sick.
That is, if one can’t possibly be poisoned without getting sick, then it follows by logical entailment that:
If one doesn’t get sick, then one wasn’t properly poisoned.
And it also follows that:
Either one didn’t get properly poisoned, or one will get sick.
These are logical entailments in that these inferences hold in virtue of form. Any statement with the logical form:
If A, then B
Will entail corresponding statements with the corresponding logical forms of:
If not B, then not A
And
Either not A or B.
These are relationships that hold between any statements at all that have these logical structures. If the first one is true, then the other ones are necessarily true. Logical entailment is entailment that holds because of the logical form of the statements involved.