5: What is Logic?
- Page ID
- 223754
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Deductive logic can be used to solve logical puzzles, to understand how inferences and arguments hang together, and—most practically—to build computers. Without deductive logic, your phone would just be an expensive brick.
What is deductive logic, though?
Well, it’s a method of expression and analysis of the logical form of statements, groups of statements, and inferences.
Expression: Logic is a language in which we can express logical relations between basic elements—we can express or represent logical structures. We do this by abstracting away from the English or ordinary language sentence to a statement that only includes the logical relations. We, in other words, take stuff away until we’ve only got a few basic elements and then we’re ready to manipulate those basic elements in order to analyze the statements and arguments.
Analysis: Once we’ve got our statement, group of statements, or inference translated into a logical language, we can then use the math-like methods of analysis we’ll learn to figure out:
a. What does this statement entail or imply? Is this statement necessarily true? Is it self-contradictory?
b. Is this set of beliefs or statements consistent? Could you believe them all at once?
c. Is this argument valid? Do the premises entail the conclusion?
Here’s a example:
If you want a ride to school, you’ll need to be ready by 8.
Translates to:
(Ride to School) \(\rightarrow\) (Ready by 8)
And finally into:
S \(\rightarrow\) R
When we get to propositional logic, you’ll learn how the arrow operator works. With that knowledge and the methods of logical analysis, we can prove that this statement implies the following:
If you aren’t ready by 8, then you aren’t getting a ride to school
Either you aren’t getting a ride to school, or you’ll be ready by 8.
And doesn’t imply the following:
If you don’t get a ride to school, that means you weren’t ready by 8.
Why study logic? Well, there are many answers to that question, but here are a few I like the best:
1. Let’s be honest, you’ll never build a truth table again after leaving this class, so why waste our time doing it here? Good Question!
Because once we “look under the hood” of arguments by exploring their structure through logical analysis, we gain a deeper appreciation for what makes arguments tick and how arguments demonstrate their conclusions. This is just like understanding how a car goes. It’s fine to go without much knowledge of what happens under the hood of a car until something goes wrong and you need to fix the car yourself. You don’t want to end up stranded!
2. We also begin to internalize the distinctions between premises and conclusions, truth and validity, validity and soundness, consistency and logical equivalence, and others. Over time working with these concepts, thinking in these terms becomes natural. Furthermore, we begin to internalize certain simple and common logical relations and logical forms. These become second nature and we naturally and habitually think in more precise ways.
3. Finally, we get a window into one of the key tools that philosophers use to understand and critique arguments and positions. We also open the door to more complex logical analysis, which is important to mathematicians, linguists, computer scientists and programmers, artificial intelligence researchers, and philosophers alike.