5.1: Core Concepts
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Truth Preservation
An important concept in logic is Truth Preservation.
What does it mean? An inference is Truth Preservative if and only if its premises being true guarantees that its conclusion is true. Sound familiar? That’s because this is what Validity means. Truth Preservation means moving from premises to the conclusion (or from the antecedent to the consequent in an implication), we can’t “lose truth.” If the premises are true, then the conclusion is true and we haven’t “lost the truth” of the premises. The inference preserves truth.
This is important because a valid deductive inference will always be truth preservative.
For example:
All humans are mortal
Socrates was a human
Therefore Socrates was mortal
...is truth preservative, because if it’s true that All humans are mortal (true by definition, I’d think) and if it’s true that Socrates was human (a matter of historical fact), then “Socrates was Mortal” cannot possibly be false. The inference is truth preservative, so we can’t “lose” the truth in the premises in our inference to the conclusion (we can’t end up with a false conclusion).
It’s as if you begin a journey in premise base camp. You start loaded up with truth supplied by the premises. If it’s a good path from premises to conclusion, you still have truth when you get to the conclusion (the conclusion is true). More than that: you are guaranteed to still have the truth you start with. If it’s a bad path, then you may lose that truth you started with and end up with a false conclusion. If you happen to arrive at a true conclusion, it’s not guaranteed—it’s a mere accident. Truth preservative argument structures guarantee that you will have a true conclusion if you have true premises.
Deductive vs. Inductive
Remember also that we need to be able to distinguish between deductive and inductive arguments.
- Deductive arguments:
- Formally precise arguments, necessarily true
- Mathematical, logical, from definition,
- Inductive arguments:
- Informal support, probably true.
- Prediction, analogy, generalization, authority, causal inferences
Review section 1.2 in this textbook for a refresher on the difference between Deductive and Inductive arguments or inferences.
We will focus exclusively on deductive arguments for the next 2 chapters.
Form vs Content
Logic is often called Formal Logic because it is the study of how arguments work at the level of their form or structure. Many arguments can share one structure. Here’s an example:
No Pineapples are Trees
So, No Trees are Pineapples
Now let’s take the content away and just look at the structure:
No \(\square\) are \(\Delta\)
So, No \(\Delta\) are \(\square\)
We can see, since we’ve removed the terms (the content) and replaced them with symbols, what the structure of the argument is. Variables stand in for content in that they represent any possible term or proposition or the like that could be plugged into the space they occupy.
So, we can now produce a new argument with the same structure, by plugging in new terms.
No Friends are Enemies
So, No Enemies are Pineapples
Oooops! Missed a spot.
No Friends are Enemies
So, No Enemies are Friends
Can you see how this argument has new content while having the same structure as our inference about pineapples and trees from above? Logic is the study of argument structures and so when translating into a logical language, we’ll sometimes just replace the content with variables because we no longer care if the argument is about Pineapples or Friends or Quantum Fields!
Great, let’s move on to Categorical Logic: