1.2: Kinds of Inferences
- Page ID
- 223805
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Let’s look at two different inferences:
Inference i:
The sign says only 3 more miles to the coast, I suppose we’re getting close!
Inference d:
The definition of a scab is a union member who works during a strike, Manny is a union member who is working during a strike, so Manny is a scab.
Notice how even if we accept the premise of inference i, we need not accept the conclusion. Ten gazillion different things could make the sign inaccurate. Maybe the coast moved due to erosion or seismic activity, maybe the sign was stolen from its intended location and moved 10 miles inland, maybe the sign was a practical joke in the first place. Who knows?
For inference d, though, it’s not so open-ended. We have a definition and the claim that an individual meets that definition. If the definition of x is d and a is d, then a is an x. If we disagree with the conclusion, we either have to reject the definition, or the description of Manny. We can’t add new information to change the conclusion. Even if Manny is a good guy, or an alien in disguise, or a really pro-union guy, or supporting three kids and a wife with cancer, he’s still a scab if that is in fact the definition of a scab and if that is in fact a true description of Manny. Harsh, but certainly true if the premises are true.
The point of comparing inferences I and d is to see that there are two fundamentally different sorts of inference. We call an inference inductive if the support the premises provide for the conclusion is less than certain—if the premises don’t guarantee the conclusion. We call an inference deductive if the premises provide conclusive support for the conclusion—if they guarantee the conclusion or make the conclusion certain.
Deductive arguments are mathematical arguments like proofs and the like, logical arguments, arguments from definition, etc. If the premises are true and the argumentative structure is good, then the conclusion must be true.
Inductive arguments are arguments from analogy, arguments from qualified authority, causal inferences, scientific hypothetical reasoning, extrapolations from samples, and so on. Even if the argumentative structure is great, the truth of the premises only even makes the conclusion probably true at best.
There’s a third kind of argument where we select the best explanation from all of the available plausible explanations. We won’t spend time on it, but it’s worth noting its existence. It’s sometimes called “abduction.”
Deduction: arguments where the premises guarantee or necessitate the conclusion
- Mathematical Arguments, Logical Arguments, Arguments from Definition
Induction: arguments where the premises make the conclusion probable.
- Analogies, Authority, Causal Inferences, Scientific Reasoning, Extrapolations, etc.
Inference to the Best Explanation or Abduction: arguments where the best available explanation is chosen as the correct explanation.
Validity
Remember that truth is a property of propositions. That is, only propositions can be true or false. Arguments can never be true or false. It simply doesn’t make any sense to claim that an argument is true or false.
Okay, let’s talk about deductive arguments for a hot minute. Deductive argumentative structures are either valid or invalid. An invalid argument structure is one where the premises don’t guarantee the truth of the conclusion, but they should, given the type of argument involved. For instance, if it’s a mathematical argument, then it’s premises should guarantee its conclusion so it’s deductive. But if its premises don’t in fact guarantee its conclusion, then its an invalid deductive argument.
A valid argument structure is an argument structure where the premises guarantee the conclusion. That is, if the premises are true, the conclusion follows necessarily. It’s impossible for the premises to be true and the conclusion false. If 2+2=3, and 6-3=3, then necessarily, beyond any doubt, 2+2=6-3. No ifs, ands, or buts about it. It’s impossible for that conclusion to be false without at least one of those premises being false as well. All true premises and a valid argument means the conclusion must be true.
Keep in mind that validity is about structures. So the previous paragraph’s arithmetical argument has the structure a+b=c, d+e=c, therefore a+b=d+e. Anything we sub in for the letters, if we create two true premises, will necessitate a true conclusion.
What’s a valid argument that has true premises? That’s called a sound argument. Soundness is about both structure and truth: you have to have a good structure and true premises to be a sound argument. An unsound argument, conversely, is an argument that either is invalid or has at least one false premise.
Truth, Validity, Soundness
What’s Truth? A proposition makes a statement about the world and the world either is or isn’t the way the proposition describes it to be. One proposition claims that the Gross Domestic Product of the United States of America is approximately $14 Trillion. To find out whether this is true or false, go figure out what the GDP of the US is. Is it approximately $14 Trillion? Another proposition claims that there’s a brown cat on the front porch of your house. Is this true? To find out, just go look at the world: is there in fact a cat on your front porch? Is it a brown cat?
The propositions that make up an argument (the premises and the conclusion) are all either true or false. As with all things in philosophy, there is a lot more to say about the complexities here. Some of the earliest philosophy in the Western philosophical tradition is philosophy of logic or language. Aristotle, for instance, asked whether it’s true or false that there will be a sea battle tomorrow. Isn’t it contingent? Aren’t there lots of indeterminant factors involved in determining whether or not a sea battle will in fact take place? If so, it seems like that proposition is neither true nor false yet. So not every proposition is either true or false. We need not, though, deal with such issues. We can proceed as if every proposition is determinately either true or false.
Remember: Validity is a property of argument structure: it means “this structure is such that if the premises of any argument with this structure are true, then the conclusion of that argument must be true.”
It means: arguments of this structure will never have all true premises and a false conclusion. The structure guarantees the truth of the conclusion given the truth of the premises. Almost like the structure carries the truth of the premises directly to the conclusion without fail. A reliable one-way transporter of truth.
A sound argument is an argument that has a valid structure but then also has true premises. If an argument is sound, and if validity means the conclusion must be true if the premises are true, then the conclusion must be true, then what do we know about the truth of the conclusion of any sound argument? Yes! You’re so smart: the conclusion of any sound argument is guaranteed to be true.
Truth: propositions are either true or false
Validity: good deductive argument structure: True premises make the conclusion necessarily true. (if not, it’s an Invalid structure)
Soundness: Valid deductive argument, all True premises. (If not, it’s an Unsound argument)
\[\fbox{All True Premises} + \fbox{Valid Structure} = \fbox{Sound Argument}\nonumber\]
Truth, Strength, Cogency
Switching over to inductive arguments, we find an analogous set of properties. Again, inductive arguments are made up of propositions, which can be true or false.
The biggest difference is that even good inductive arguments only offer probabilistic support for their conclusions. Meaning accepting all of the premises doesn’t necessitate that one accept the conclusion, it merely gives one more or less strong reason for accepting the conclusion. So the argumentative structure of an inductive argument isn’t either good or bad, it’s a matter of degree (and often a matter of what the actual content is). An inductive argument can therefore offer stronger or weaker inductive support for its conclusion.
“Cogent” and “uncogent” are the words we use in place of “sound” and “unsound” for inductive arguments since inductive arguments cannot be sound or unsound. Cogent, therefore, means all true premises and the premises give strong inductive support for the conclusion.
Consider these two arguments:
I saw a black cat Therefore all cats are black |
I saw the Sun rise in the East every day of my life and everyone I know reports the same and history books and ancient astronomers report the same, so the Sun will rise in the East tomorrow. |
Notice how the argument on the left provides pretty weak support for the conclusion. I can believe that the speaker in fact saw a black cat and still think that’s a bad reason for concluding that all cats are black. The right argument, though, is much stronger. There’s much more evidence and the nature of the evidence makes the conclusion much more probable given the truth of all of the premises.
Truth: propositions are either true or false
Strength: Inductive argument: true premises make conclusion probably true.
Cogency: Strong inductive argument, all True premises.
\[\fbox{All True Premises} + \fbox{Strong Inductive Support} = \fbox{Cogent Argument}\nonumber\]
Again, inductive arguments are collections of propositions (the premises and the conclusion(s) are all propositions. And each of these propositions might be either true or false depending on whether it accurately describes reality.
An inductive argument cannot be valid. Why? Because a valid argument guarantees the truth of the conclusion. But an inductive argument only justifies its conclusion to some level of probability.