# Weak-strong four valued logic question

I've got a simple problem with logic. I haven't found any solution, I hope someone here is familiar with this system.

Consider the 4-valued logic, where every value consists of two bits:

• first one denotes the truth value (0=false, 1=true)
• second one denotes the strength (0=weak, 1=strong)

We can define negation as follows:

+---+----+----+----+----+
|   | 00 | 01 | 10 | 11 |
+---+----+----+----+----+
| ¬ | 10 | 11 | 00 | 01 |
+---+----+----+----+----+


And conjunction:

+----+----+----+----+----+
| ∧  | 00 | 01 | 10 | 11 |
+----+----+----+----+----+
| 00 | 00 | 01 | 00 | 01 |
| 01 | 01 | 01 | 01 | 01 |
| 10 | 00 | 01 | 10 | 10 |
| 11 | 01 | 01 | 10 | 11 |
+----+----+----+----+----+


This is how my teacher showed it. Everything is clear, except I cann't know how can a strong true and a weak false make a strong false.

(00 ∧ 11) <=> 01

why?

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Before deleting the question could you just say a bit more about the (lattice-theoretic) nature and the history of this system? Because, as far as I'm concerned, I've never come across it. At first sight it reminded me of the product system resulting from Dunn / Belnap's 4-valued logic. But then one would expect that the value of $(\phi \wedge \psi)$ is $\langle min\lbrace v(\phi), v'(\phi) \rbrace, min\lbrace v(\psi), v'(\psi)\rbrace \rangle$, where $v, v'$ are any classical propositional valuations. But this does not work for the arguments $\langle 0, 0 \rangle$, $\langle 0, 1 \rangle$. –  Jon Feb 2 at 16:32