hausman

Tree Tutorial 2 More Rules

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.

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.

For example, the truth table for 'and' is

Conjunction

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}

10/13/26

Reading

Hausman[2007] Logic and Philosophy Chapter 10

[The core of what follows is from Leblanc and Wisdom [1972] p.117 and f.]

Symbolization using the Universal Quantifier (of non relational English)

‘Everything is lovely’ we understand as meaning ‘Whatever thing you may choose, it is lovely’ and this becomes

(x)Lx