393 lines
17 KiB
Markdown
393 lines
17 KiB
Markdown
---
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tags: []
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---
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## Rationale
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Like [truth-tables](Truth-tables.md), truth-trees are a means of graphically
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representing the logical relationships that may obtain between propositions.
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Truth-trees and truth-tables complement each other and which method you choose
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depends on which logical property you are seeking to derive.
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Whilst truth-tables have the benefit of being exhaustive - every possible truth
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assignment is factored into the representation - their complexity grows
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exponentially with each additional proposition they contain. This can make
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manually constructing truth tables long-winded and prone to mistakes.
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Truth-trees are less onerous but they lack the exhaustive scope of a
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truth-table. They are more targeted and are best used for demonstrating _that
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something is the case_ rather than _all the possible states that could be the
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case_. For example, a truth tree will tell us that a set _S is logically
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consistent_ whereas a truth-table will tell us that _S is consistent on the
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following three assignments._
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## Logical consistency
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Recall that a set of propositions is logically or truth-functionally
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[consistent](Consistency.md) just if there is at least one assignment of truth
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conditions which results in all members of the set being true. To identify
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consistency for a set of three propositions via the truth table approach we
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would need to construct a truth table with $2^3$ (8) rows. Assume that this set
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is consistent on one partial assignment only. This means that 87.5% of our rows
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are redundant, they are not required to prove the consistency of the set.
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However we can only know this and we can only be sure of consistency once we
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have gone through the process of generating an assignment for each row.
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Truth trees allow us to reduce the amount of work required and go straight to
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the assignment that proves consistency, disregarding the rest which are
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irrelevant.
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## Truth tree structure and key terms
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**When using a truth tree to derive logical consistency, the goal is to
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determine whether there is a truth-value assignment on which all of the
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sentences of a set are true. If the set is consistent we should be able to
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derive a partial assignment from the tree that demonstrates consistency.**
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Each truth tree begins with a series of sentences one on top of the other in a
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column. We call the sentences that comprise the initial column **set members**.
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In constructing the tree, we work downwards from the initial column decomposing
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set members into their atomic constituents. We a call an atomic sentence that
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has been decomposed a **literal.** A literal will either be an atomic sentence
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or the negation of an atomic sentence. If one of the set members is already a
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literal, there is no need to decompose it; it can remain as it is.
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Once every set member has been decomposed the truth tree is complete. It can
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then be interpreted in order to derive logical consistency or inconsistency. If
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the set is consistent, we are able to derive the partial assignment(s) that
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demonstrate consistency.
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The rules for decomposing compound sentences match the truth conditions of the
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logical connectives. There are rules for every possible connective and the
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negation of every possible connective however in terms of their tree shape they
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all correspond to either a conjunction or a disjunction. Disjunctive
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decomposition results in new branches being formed off the main column (or
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trunk). Conjunctive decomposition is non-branching which means the decomposed
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constituents are placed within the trunk of whichever tree or branch they are
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decomposed within.
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As we construct the tree we list each line in the left-hand margin and the
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decomposition rule in the right-hand margin. When we apply a decomposition rule
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we must cite the lines to which it applies.
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### Closed and open branches
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Any branch on which an atomic sentence ($P$) and the negation of that sentence
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($\sim P$) both occur is a **closed branch**. A branch that is not closed is an
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**open branch**. No partial assignment is recoverable from a closed branch. An
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open branch allows truth to ‘flow up’ to the original set members whereas a
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closed branch blocks this passage.
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### Completed open branch
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A completed open branch occurs when we have an open branch that has been fully
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decomposed: the branch is open and all molecular sentences have been ticked off
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such that it contains only literals.
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### Completed tree
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A tree where all its branches are either completed open branches or closed
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branches.
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### Closed tree
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A tree where all the branches are closed
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### Open tree
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A tree with at least one completed open branch
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## Deriving consistency
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Using the definitions above, we can now define truth-functional consistency and
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inconsistency in terms of truth trees:
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> A finite set ($\Gamma$ ) of sentences is truth-functionally inconsistent if
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> $\Gamma$ is a closed tree
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> A finite set ($\Gamma$ ) of sentences is truth-functionally consistent if
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> $\Gamma$ is an open tree
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## Examples
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### First example
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The following is a truth tree for the set ${P \lor Q, \sim P }$:
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### Interpretation
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- We decompose the disjunction at line 1 on line 3. We tick off the compound
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sentence to indicate that it is now decomposed and no longer under
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consideration.
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- Both P and its negation exist on a single branch (at line 2 and line 3). This
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makes it a closed branch. We indicate this by the X beneath the branch that is
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closed, citing the source of the closure by line number.
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- The rightward branch is a completed open branch given the decomposition at 3
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and the lack of negation of Q. Overall this makes the tree an open tree.
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As the set gives us an open tree, it must be truth-functionally consistent. If
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this is the case we should be able to determine the partial assignment in which
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each set member is true. Given that Q is not negated the assignment of
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consistency will contain Q but we have both P and ~P. This means there are two
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possible assignments where the set is consistent: $P, Q$ and $\sim P, Q$. This
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is confirmed by the truth-table:
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```
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P Q P ∨ ~ P Q
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T T T T *
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T F T F
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F T T T *
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F F T F
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```
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**Any time there is an open tree with a closed branch it will be the case that
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the negated sentences of the closed branch will appear both as** $S$ and
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$\sim S$ i**n the resultant assignment.**
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Invoking the truth-table highlights the differences between the two techniques.
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The values that are derived when we interpret a truth tree are not the
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truth-functions of the set members but the truth-values for when they are
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simultaneously true. With truth-tables in contrast, we are deriving the truth
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functions for every possible truth-value assignment. In other words the values
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derived from a truth tree correspond to the left hand side of the truth table
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not the right hand side.
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### Second example
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The following is a truth tree for the set
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${A & \sim B, C, \sim A \lor \sim B }$.
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### Interpretation
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- The two molecular set members are decomposed. The disjunction (line 3) results
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in a branching tree. The conjunction (line 1) results in the continuation of
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the trunk.
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- Both branches are completed making it a completed tree. As each branch is
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closed this is a closed tree.
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As this is a closed tree, the set is not truth-functionally consistent. This is
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confirmed by the truth table where there is no partial assignment where all set
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members are true.
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```
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A B C A & ~ B C ~ A ∨ ~ C
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T T T F T F
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T T F F F T
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T F T T T F
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T F F T F T
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F T T F T T
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F T F F F T
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F F T F T T
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F F F F F T
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```
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## Truth tree decomposition rules
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---
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So far we have encountered the decomposition rules for conjunction (`&D`) and
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disjunction (`vD`). We will now list all the rules. We will see that for each
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rule, the decomposition either branches or does not branch which is to say that
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each rule either has the shape of a conjunction or a disjunction (however the
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permitted values of the specific disjuncts/conjuncts obviously differ in each
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case). Moreover there is a parallel rule for the decomposition of the negation
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of each of the main connectives and these rules rely on logical equivalences
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### Negated negation decomposition: `~~D`
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Truth passes only if $P$ is true
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### Conjunction decomposition: `&D`
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Truth passes only $P$ and $Q$ are both true.
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### Negated Conjunction decomposition: `~&D`
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Truth passes if either $\sim P$ or $\sim Q$ is true. This rule is a consequence
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of the equivalence between $\sim (P & Q)$ and $\sim P \lor \sim Q$ , the first
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of DeMorgan’s Laws.
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### Disjunction decomposition: `vD`
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Truth passes if either $P$or $Q$ are true.
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### Negated Disjunction decomposition: `~vD`
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Truth passes if both $P$ and $Q$ are false. This rule is a consequence of the
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equivalence between $\sim (P \lor Q)$ and $\sim P & \sim Q$, the second of
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DeMorgan’s Laws.
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### Conditional decomposition: `⊃D`
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Truth passes if either $\sim P$ or $Q$ are true. This rule is a consequence of
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the equivalence between $P \supset Q$ and $\sim P \lor Q$ therefore this branch
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has the shape of a disjunction with $\sim P$ , $Q$ as its disjuncts.
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### Negated Conditional decomposition: `~⊃D`
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Truth passes if both $P$ and $\sim Q$ are true. This is a consequence of the
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equivalence between $\sim (P \supset Q)$ and $P & \sim Q$.
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### Biconditional decomposition: `≡D`
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Truth passes if either $P$ and $Q$ are true or $\sim P & \sim Q$ are true. This
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is an interesting rule because it combines the disjunction and conjunction tree
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shapes.
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### Negated biconditional decomposition: `~≡D`
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Truth passes if either $P$ and $\sim Q$ is true or if $\sim P$ and $Q$ is true.
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## Further examples and heuristics for complex truth trees
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With truth-trees regardless of which order you decompose the set members, the
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conclusion should always be the same. This said, there more are more efficient
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ways than others to construct the trees. You want to find the route that will
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demonstrate consistency or non-consistency with the shortest amount of work. The
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following heuristic techniques followed in order, facilitate this:
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1. Decompose those molecular sentences the decomposition of which does not
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produce new branches. In other words that are decompositions of double
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negations or pure conjunctions.
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1. Perform those decompositions that will rapidly generate closed branches.
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1. If neither (1) or (2) is applicable, decompose **the most complex** sentence
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first.
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Here are some examples of these rules applied:
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Observe that here we don’t bother to decompose the sentence on line 1. This is
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because, having decomposed the sentences on lines 2 and 3 we have arrived at a
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closed tree. It is therefore unnecessary to go any further for if two sentences
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in the set are inconsistent with each other, adding another sentence is not
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going to change the overall assignment of inconsistency.
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## Deriving properties other than logical consistency from truth trees
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So far truth trees have been discussed purely in terms of logical consistency
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however they can be used to derive all the other key truth-functional properties
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of propositional logic. Given the foundational role of consistency to logic,
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these properties are expressible in terms of consistency which is what makes
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them amenable to formulation in terms of truth trees.
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### Logical falsity
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For a given finite set $\Gamma$, $\Gamma$ is logically consistent just if all of
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its members can be true at once. Expressed in terms of truth trees, this is
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equivalent to an open tree. Contrariwise, $\Gamma$ is inconsistent if it is not
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possible for every member of the set to be true at once. This is the same as a
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tree where all of the branches are closed (i.e. a closed tree).
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When we wish to assess
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[logical falsity](Logical%20truth%20and%20falsity.md#logical-falsity) we are not
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focused on sets however, we are interested in a property of a sentence. However
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we can easily construe single sentences as unit sets: sets with a single member.
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With this in mind and the above accounts of consistency and logical falsity we
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are equipped to express logical falsity in terms of truth-trees with the
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following rule:
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> A sentence $P$ is logically false if and only if the unit set ${ P }$ has a
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> closed tree
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A logically false sentence cannot be true on any assignment. This is the same
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thing as an inconsistent set. Thus it will be represented in a truth tree as
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inconsistency which is disclosed via a closed tree.
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### Logical truth
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For a sentence $P$ to be
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[logically true](Logical%20truth%20and%20falsity.md#logical-truth), there must
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be no possible assignment in which $P$ is false. We express this informally by
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saying _it is not possible to consistently deny $P$._ We know that in terms of
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truth trees an inconsistent set is a closed tree therefore a unit set of ${ P }$
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is logically true if ${ \sim P }$ is a closed tree. This is to say: if the
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negation of $P$ is inconsistent.
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> A sentence $P$ is logically true if and only if the set ${ \sim P }$ has a
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> closed tree
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### Logical indeterminacy
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[Indeterminacy](Indeterminacy.md) follows from the two definitions above; we do
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not require any additional apparatus. We recall that a sentence $P$ is logically
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indeterminate just if it is neither logically true or logically false. Thus the
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truth tree for an indeterminate sentence is straightforward:
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> A sentence $P$ is logically indeterminate if and only if neither the set
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> ${ P }$ nor the set ${ \sim P }$ has a closed tree
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This follows because a closed tree for ${ P }$ means it is not logically false
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and an open tree for ${ \sim P }$ means it is not logically true. So if it is
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neither of these things, $P$ must be indeterminate.
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### Logical equivalence
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Recall that $P$ and $Q$ are [logically equivalent](Logical%20equivalence.md)
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just if there is no truth assignment on which one is true and the other is
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false. We know from the
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[material biconditional shorthand](Corresponding%20material%20and%20biconditional.md#corresponding-material-biconditional)
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that this state of affairs can be expressed as $P \equiv Q$ and that if this
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compound sentence is true on every assignment then both simple sentences are
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equivalent. But ‘true on every assignment’ is another way of saying _logically
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true_ since there is no possibility of a false assignment. We already know what
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logical truth looks like as a truth tree: it is a closed tree for the negation
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of the sentence being tested. Therefore, to test the logical equivalence of two
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sentences it is necessary to construct a truth tree for the negation of the
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sentences conjoined by the biconditional (i.e. $\sim (P \equiv Q)$ )and see if
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this results in a closed tree. If it does, the two sentences are logically
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equivalent.
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> Sentences $P$ and $Q$ are truth-functionally equivalent if and only if the set
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> $\sim (P \equiv Q)$ has a closed tree
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### Logical entailment and validity
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Let’s remind ourselves of the meaning of truth-functional
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[entailment](Validity%20and%20entailment.md#entailment) and
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[validity](Validity%20and%20entailment.md#validity) and the relation between the
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two. $\Gamma$ $\vdash$ $P$ is true if and only if there is no truth-assignment
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in which every member of $\Gamma$ is true and $P$ is false. Entailment is
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closely related to validity; it is really just a matter of emphasis: we say that
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$\Gamma$ are the premises and $P$ is the conclusion and that this is a valid
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argument if there is no assignment in which every member of $\Gamma$ is true and
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$P$ is false.
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As with the previous properties, to express validity and entailment in terms of
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truth trees we need to express these concepts in the language of logical
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consistency. $\Gamma$ entails $P$ just if one cannot consistently assert
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$\Gamma$ whilst denying $P$. This is to say that the set $\Gamma \cup {\sim P}$
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is inconsistent. So we just need a closed truth tree for $\Gamma \cup {\sim P}$
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to demonstrate the validity of this set.
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> A finite set of sentences $\Gamma$ truth-functionally entails a sentence $P$
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> if and only if the set $\Gamma \cup {\sim P}$ has a closed truth tree.
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> An argument is truth functionally valid if and only if the set consisting of
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> the premises and the negation of the conclusion has a closed truth tree.
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