The Truth Tree Method for
QL is an extension of the Truth Tree Method for PL. We retain all of
our truth tree rules for the operators of PL. We add rules for
propositions with quantifiers. The rules for quantified propositions
are called instantiation rules. With the instantiation rules,
we convert a quantified proposition into truth-functional
propositions, i.e. a proposition whose main operator is one of our
propositional operators.
The truth tree instantation rule for the universal quantifier: If a wff is a universally quantified proposition, instantiate it with every individual constant that appears on tree. If there are no individual constants yet on the tree, introduce them as necessary. If new individual constants are introduced to the tree, go back and instantiate the universally quantified wff again with these newly introduced individual constants.
The truth tree instantation rule for the existential quantifier: If a wff is a existentially quantified proposition, instantiate it with a new individual constant, one that does not already appear on the path.
Finally, we need to deal with negations of quantified wff:
Quantifier Negation Rule: Negations of quantified wffs are rewritten as quantified negations using the following equivalences:
| ~(x)Fx | ≡ | (∃x)~Fx |
| ~(∃x)Fx | ≡ | (x)~Fx |
To see how QL Truth Trees works, let's test the following argument for validity:
| (x)(Ax ⊃ Bx) |
| (x)(Bx ⊃ Cx) |
| (x)(Ax ⊃ Cx) |
This is an AAA-1 syllogism, rewritten in QL. So you can test it independently for validity using Venn Diagrams or any of the other methods of testing syllogisms for validity. To test by the truth tree method, we set the tree up exactly as we did in PL. We write the premises and the denial of the conclusion at the top of the tree: (We use line numbers here just for convenient reference to the wffs. They are not required in truth-trees.)
| 1. | (x)(Ax ⊃ Bx) | premis | |||||
| 2. | (x)(Bx ⊃ Cx) | premise | |||||
| 3. | ~(x)(Ax ⊃ Cx) | ckecked | negation of conclusion | ||||
| 4. | (∃x)~(Ax ⊃ Cx) | checked | Used Quantifier Negation rule | ||||
| 5. | ~(Aa ⊃ Ca) | checked | instantiate esistential | ||||
| 6. | (Aa ⊃ Ba) | checked | instantiate the two universally quantified premises. | ||||
| 7. | (Ba ⊃ Ca) | checked | |||||
| Aa | apply rule for negation of conditional to wff on line 5. | ||||||
| ~Ca | |||||||
| / | \ | ||||||
| ~Aa | Ba | ||||||
| X | / | \ | |||||
| ~Ba | Ca | ||||||
| X | X | ||||||
All paths close, so the argument is valid. Notice that we did not use a check-mark for the universally quantified wffs. We never check universally quantified wffs, because we may have to go back and reinstantiate it. Typically, that's done after we've instantiated an existentially quantified wff.
Important tip: Instantiate the existentially quantified wffs before instantiating universally quantified wffs. That will reduce the number of constants on your tree, and the overall complexity of the tree.
The truth-tree examples in the next section will help you master this method.
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| table of contents | List of Exercises |
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