Tutorial Six
SD: Deductions in Oth Order Logic

Real-life logical thinking doesn't involve truth tables! We just draw conclusions...one at a time. Here's a very simple example.

Say you know that Chris will get an 'A' or 'B' this term. And that if she gets a 'B' she'll be unhappy. Later we discover that she's not getting an 'A'. So, we conclude first that she'll get only a 'B' and so, second, that she'll be unhappy getting a B.

We'll formalize this reasoning as follows. First we'll use this symbolization key,

c: Chris, Ax: x gets an 'A', Bx: x gets a 'B', Hx: x is happy.

Next, we'll write down the three premises.

    1. Ac v Bc
    2. Bc > ~Hc
    3. ~Ac

Then, we'll draw the obvious conclusions one step at a time:

Statement 1. (Chris gets either an 'A' or 'B') and 3. (Chris doesn't get an 'A') lead to the statement that Chris got a B:

  1. Bc

So, 2. (Getting a 'B' will make Chris unhappy) together with 4. (Chris did get a 'B') lead to

  1. ~Hc.

Finally, 4. (Chris get a 'B') and 5. (Chris is unhappy) allow us to draw our final conclusion:

    1. Bc&~Hc