Twootie Conversions
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.
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*** |
A^{1}, B^{1}, C^{1}, … A^{2}, … A^{3}, … |
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 ¬∀∃ |
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, &
X
Twootie requires a step 4 before the tree can close:
4) Q 2, &
X
- 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.