- I've added a folder with the edited highlights from class.
- If NKU is closed Thursday, due to weather, then we will not have
class on Thursday (even though it seems that we should -- just
zoomed like we're doing today). That's the law, handed
down by our Department Chair!:)
- This session will be exclusively on Zoom,
because of a water issue with our building.
Here's a menu of our past zooms...
- I got the 1.1 assignment graded. For this assignment I graded the problems (randomly selected)
- 15cde
- 41ac
- 47
Take an implication, $A \rightarrow B$, which is clearly true, e.g. \[ (n>0) \rightarrow (n+1>0) \] Now write it using the "confusing" language, and see which is the correct one:
(n>0) only if (n+1>0) or(n+1>0) only if (n>0) Which is right?
The first: the second fails, for example, if $n=-0.5$, for example.
- 15cde
- For the assignment 1.2, which was due last night, I graded the
problems (randomly selected)
- 12
- 49 (the first trick is writing the theorem
correctly! If you don't get the theorem right, it's almost
impossible to prove it!:) It will look different, depending on
how you define the letters given: if you use
- R -- Russia was a superior power
- F -- France was strong
- N -- Napoleon made an error
- A -- Army failed
\[ R \land (F' \lor N) \land N' \land (A' \rightarrow F) \rightarrow (A \land R) \]
- 55 -- part c, they're hoping that you'll identify $P_1\land P_2 \land \ldots \land P_n$ with $A$, and away you go! Invoke parts a. or b.
- Saw this today, in a story from the Guardian about "tetrachromats"
-- people who have a fourth type of color receptor in their eyes (my
emphasis in the following):
According to Dr Kimberly Jameson, a University of California scientist who has studied Antico, just having the gene -- which around 15% of women have -- is not alone sufficient to be a tetrachromat, but it's a necessary condition. In Concetta's case ... one thing we believe is that because she's been painting sort of continuously since the age of seven years old, she has really enlisted this extra potential and used it. This is how genetics works: it gives you the potential to do things and if the environment demands that you do that thing, then the genes kick in.
$B$ is necessary for $A \iff A \rightarrow B$ $A$ is sufficient for $B \iff A \rightarrow B$ $B$ is necessary and sufficient for $A \iff A \leftrightarrow B$ Logic is all around you!
- We finished up Section 1.2, with a few remarks on highlight worksheet
1.2: Propositional Logic
We talked about the "Sheffer stroke", which shows that only one logical connective is necessary! We don't need all these "$\land$"s, and "$\lor$"s, and "$\rightarrow$"s, etc. We can get by with only the one logical connective:
So we showed that this can serve as negation: $A' \leftrightarrow A|A$:$A$ $B$ $A|B$ T T F T F T F T T F F T
And I meant to use a trick to finish off the demonstration, but we took the harder road last time of coming up with an explicit version of "$\land$". The easier path is to come up with a version of "$\land$" using negation, which we just showed we can generate with the Sheffer stroke: it's clear that $(A|B)' \leftrightarrow (A \land B)$:$A$ $A|A$ T F F T
And since we've already shown that we can write negation in terms of the Sheffer stroke, we're done (by our exercise #47)!$A$ $B$ $A|B$ $(A|B)'$ T T F T T F T F F T T F F F T F Big picture: once you've created a tool (e.g. we can write negation in terms of the Sheffer stroke), put it in your toolbox and bring it out as necessary.
- We'll wrap up Section 1.3
- Then it's on to Section 1.4 (Predicate Logic), using this highlight worksheet 1.4
- To prove you're smart, know some math!
- Maybe a little more than you want to know about Propositional Logic. "In 1917, French logician Jean Nicod discovered that it was possible to axiomatize propositional logic using the Sheffer stroke and only a single axiom schema and single inference rule."