Meaning and Argument

The truth tree system in Twootie matches closely with Meaning and Argument, but not exactly. The notes below should make switching between the two easier. If you find any other differences, please email Sarah-Jane.

Corresponding Logics

Meaning and Argument	Twootie
Propositional Logic (PL)	Sentential Problems
Property Predicate Logic (PPL)	No real equivalent*
Relational Predicate Logic (RPL)	Predicate Logic Problems
Relational Predicate Logic with Identity (RPL=)	Identity/Function Problems

*You could try the first few Predicate Logic problems, if you don't mind having variables in there!

Symbol Changes

	Meaning and Argument	Twootie
Negation	~	¬
Conditional		-->
Biconditional*	( ) & ( )
Negation of Identity		¬ =
Variables**	x, y, z, x1,...x2, …	w, x, y ,z
Predicates***	A¹, B¹, C¹, … A², … A³, …	A, B, C, …

*The biconditional (If and Only If) can be written in terms of two conjoined conditionals, thus Meaning and Argument does not assign it a new symbol (see section 7.10). Thus, "P Q " is equivalent to "(P Q) & (Q P )".

**Variables in Twootie cannot take subscripts.

***Twootie predicates do not take place numbers.

Terminology

Meaning and Argument	Twootie
Singular terms	Constants
Unbound variables	Free variables

Truth Tree Rules

Meaning and Argument	Twootie
N/A	All rules end in "d" or "D"
Branches done by hand	Automatic Branching: Just enter the first branch's contents and the rule.
Quantifier Exchange (QE)	Either ¬ or ¬
Universal Quantifier (UQ)
Existential Quantifier (EQ)
Identity Out (IO)	=
Identity In (II)	No rule -- branch can be closed if it contains, e.g., "Ø a=a"
Biconditionals: see below.

Biconditionals

As M&A treats biconditionals as conjoined conditionals, there are no special rules for dealing with them (& and rules are enough). Twootie's bicondional () is a new symbol requiring its own rules:

P  Q     Ø(P  Q)
  /\          / \
 P ¬P        P  ¬P
 Q ¬Q       ¬Q   Q

Habits

Twootie doesn't let you take the short cuts that you probably do on paper. Meaning and Argument supports these short cuts, so Twootie will not let you replicate some of the book's solutions exactly. The differences aren't substantial though.

You must decompose the entire sentence once you've started it. Thus this, e.g., is unacceptable:

1) ~P Premise

2) P & Q Negation of conclusion

3) P 2, &

Twootie requires a step 4 before the tree can close:

4) Q 2, &

Double negations must be removed explicitly with the double negation rule (~~ in M&A, ¬ ¬ in Twootie). Meaning and Argument introduces this rule, but it's pretty easy to skip in practice. It's required by Twootie, though.

Rules can't be applied multiple times in one step. This shows up most frequently with Quantifier Exchanges. Meaning and Argument treats the following inference as acceptable:

~(x)(y)(Q2xy)
(x)(y)~(Q2xy) 1, QE twice

Twootie won't let you do this, though. The rule must be applied to the first quantifier, then it must be instantiated before the rule can apply to the second quantifier. For example:

¬ (x)(y)(Qxy)
(x)¬ (y)(Qxy) 1, ¬ d (i.e. QE)
¬ (y)(Qay) 2, d (i.e. EQ)
(y)¬ (Qay) 3, ¬ d (i.e. QE)

M&A lets you perform all the quantifier exchanges before instantiating; with Twootie, you have to start instantiating before completing the exchanges.

The Identity Out rule in Twootie (=) works a little differently than in M&A. You can only put the first singular term in place of the second, but not vice versa. So a=b let's us substitute a for b, but not b for a.

Completing the trees

If a branch closes, put an "X" under it. If it is open but complete, hit F10 then use the branch menu to declare it open.