PL-205: CH3&4 Truth Tables and Trees

conjunction **(P∧Q)** has the valuation of true when___

Both the antecedent and consequent are true

disjunction (**PvQ)** has the valuation of true when___

The antecedent or consequent, or both are true.

conditional **(P→Q)** has the valuation of true when____

When the consequent is not false! Given **P→Q),** it is true when P=T/Q=T; P=F/Q=T; P=F/Q=T

biconditional **(P↔Q)** has the valuation of true when_____

When both the antecedent and consequent are either true or false. They must have the same truth valuation for the biconditional to be true!

(Truth Table) When do propositions show a Tautology?

A proposition P is a tautology if and only if P is true under every valuation

(Truth Table) When do propositions show a Contradiction?

A proposition P is a contradiction if and only if P is false under every valuation

(Truth Table) When do propositions show a Contingency?

A proposition P is a contingency if and only if P is neither false nor true under every valuation

(Truth Table) When does a set of propositions show Equivalence?

A pair of propositions P, Q is ***logically equivalent*** if and only if P and Q have identical truth values under every valuation. No row where one of the pair has a different truth value than the other!

(Truth Table) When does a set of propositions show Consistency?

A set of propositions {P, Q, R, … Z} is ***logically consistent*** if and only if there is at least one valuation where P, Q, R, … Z are true. One row where P, Q, R, … Z are all true!

(Truth Table) When is argument valid?

Impossible for premises to be true and conclusion false. Valid if and only if there is NO row of the table where the premises are true and the conclusion is false.

(Truth Table) When is argument invalid?

An argument is invalid if and only if it is possible for the premises to be true and the conclusion false. Invalid if and only if there is a row of the table where premsises are ture and the conclusion is false!

(Truth Trees) What is a branch?

all the propositions obtained by starting from the bottom of the tree and reading upward through the tree

(Truth Trees) what is a fully decomposed branch?

all propositions in the branch that can be decomposed have been decomposed

(Truth Trees) what is a partially decomposed branch?

At least one proposition in the branch that has not been decomposed

(Truth Trees) what is a closed branch?

A branch containing a proposition **P** and its literal negation  **¬P. A closed branch is represented by an X.**

(Truth Trees) what is an open branch?

A branch that is not closed. The branch that does not contain a proposition P and its literal negation  **¬P!**

(Truth Trees) what is a Completed Open Branch?

A fully decomposed branch that is not closed. Does not contain a proposition and its literal negation. An open branch has **0** at the bottom of the tree!

(Truth Trees) what is a Completed Open Tree?

A tree that has at least one completed open branch. It must have at least one fully decomposed branch that is not closed.

(Truth Trees) what is a Closed Tree?

A tree with branches that are all closed. It will have an **X** under every branch

(Truth Trees) what is the **Descending Decomposition Rule**?

When decomposing a proposition **P,** decompose **P** under *every* open branch that descends from **P. (if it is decomposed above P, decompose P under all of the branches)**

What are the **Decomposable Proposition Types and their corresponding rule?**

* Conjunction - ∧**D**
* Disjunction - vD
* Conditional - **→D**
* Biconditional - **↔D**
* Negated conjunction - **¬**∧**D**
* Negated disjunction - **¬**vD
* Negated conditional - **¬→D**
* Negated biconditional - **¬↔D**
* Double-negation - **¬¬D**

(Truth Trees) What are the 4 strategic rules?

1. Use no more rules than needed
2. Use rules that *close* branches
3. Use *stacking rules* before *branching rules*
4. Decompose more complex propositions before simpler popositions

(Truth Trees: Analysis) When does a tree show it is consistent?

A set of propositions {P, Q, R, … Z} is ***consistent*** if and only if there is at least one valuation where P, Q, R, … Z are true.

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A tree shows consistency if it is a completed open tree (there is at least one completed open branch)

*Normal setup*

(Truth Trees: Analysis) When does a tree show it is inconsistent?

A set of propositions {P, Q, R, … Z} is ***inconsistent*** if and only if there is no valuation where P, Q, R, … Z are true.

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A tree shows consistency if it is a closed tree (all branches close)

***Normal setup***

(Truth Trees: Analysis) When does a tree show it is a Tautology ?

A proposition P, is true under every valuation. P is a tautology if the truth tree of the stack **¬P determines a closed tree**

***Abnormal setup: the negation of a proposition***

(Truth Trees: Analysis) When does a tree show it is a Contradiction?

A proposition **P** is false under every valuation. Contingency if and only if a tree of P determines closed tree.

***Normal setup***

(Truth Trees: Analysis) When does a tree show it is a Contingency?

A proposition **P** is neither always false nor always true under every valuation. Contingency if the tree of **¬P does not determine a closed tree and the** tree of **P does not determine a closed tree**

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(Truth Trees: Analysis) When does a tree show two propositions are equivalent?

A pair of propositions **P, Q** is equivalent if and only if P and Q have identical truth values under every valuation. A truth tree shows that P and Q are *equivalent* if and only if a tree of **¬(P↔Q) determines a closed tree!**

(Truth Trees: Analysis) When does a tree show an argument is valid?

An argument **P, Q, …R |- Z** is valid if and only if it is impossible for the premises to be true and the conclusion false. A truth tree shows that an argument **P, Q, …R |- Z** is valid if and only if **P, Q, …R, ¬Z determines a closed tree.**

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**NEGATE THE CONCLUSION OF THE ARGUMENT!**

Which of the 9 decomposition rules stack?

* Conjunction - ∧**D**
* Negated disjunction - **¬**vD
* Negated conditional - **¬→D**
* Double-negation - **¬¬D**

Which of the 9 decomposition rules branch?

* Negated conjunction - **¬**∧**D**
* Disjunction - vD
* Conditional - **→D**

Which of the 9 decomposition rules branch and stack?

* Biconditional - **↔D**
* Negated biconditional - **¬↔D**