hausman

Skills to be acquired in this tutorial:

To become familiar with the notions of closed and complete trees. To be able to use trees to test for satisfiability and invalidity.

Reading

A. Hausman, H.Kahane, P.Tidman, [2007] Logic and Philosophy Chapter 6

Tutorial:

Closure and completeness

In Tutorial 1, we met the notions of closed and open branches (a closed branch was one containing a formula and also the negation of that formula, an open branch was a branch that was not closed).

10/20/26

Skills to be acquired in this tutorial:

To become familiar with the rules for sentential truth trees.

Reading

A. Hausman, H.Kahane, P.Tidman, [2007] Logic and Philosophy Chapter 6

Tutorial:

Truth Tree Rules

Trees are going to be used to 'picture' the truth conditions or requirements for a formula (then, as the technique is developed, for several formulas at once).

Any easy way to remember the rules is to regard them as having been built from the truth tables.

The truth table for 'and' is

Prerequisities

You need to know some sentential logic to be able to understand this. In particular, you need to know about the symbols used in sentential logic, truth tables, satisfiability, consistency, and semantic invalidity (by counter example). You do not need to know sentential rules of inference and derivations. [The tutorials on sentential logic give the required background.]

Skills to be acquired in this tutorial:

To become familiar with the notions of tree, branch, open branch, closed branch.

10/19/26

Tutorial 2 Example: How Experts Symbolize

10/12/26

You may have derivations of your own that you wish to try. Just type, paste, or drag and drop, them into the panel, select your derivation, and click 'Start from selection'.

You will need to use the correct logical symbols. Here they are

F ∴ F . G ∼ ∨ ⊃ ≡ ∃ ∴

And the right syntax (the premises separated by commas and then a 'therefore' followed by the conclusion).

So, a selection needs to have a form similar to this

F⊃G,G⊃H∴F⊃H

10/13/26

You now have to tools to appraise arguments to the level of detail offered by predicate logic.

Let us run through how these might be used with a long and difficult example.

Consider the argument

10/13/26

The Rule of Universal Instantiation UI

If a derivation contains a line of the form

n (<variable>)<scope> <any justification>

then a line of the form

<<scope>[<term>/<variable>]> 'n UI'

may be added to the derivation, provided that <variable> is free for <term> in the <scope> ie that the substitution [<term>/<variable>] is legitimate .

10/13/26

The material explained here is conceptually at a higher level than the earlier tutorials; it also covers relatively rare cases.

Reading

Hausman[2007] Logic and Philosophy Chapter 11

Skills to be acquired:

To understand the concepts of scope, free, bound, and free for. To learn of the restrictions on the quantificational rules

The Tutorial

The semantics of relations proceeds in much the way one would expect-- the new item that has to be taken account of is the order of the terms (because, for example, Tab is not at all the same thing as Tba -- Arthur being taller than Beryl is not the same as Beryl being taller than Arthur).

Let us start with an Interpretation

Interpretation 1

Universe= {a,b}
F={a}