OLI - 3. Semantics

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30 Terms

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Truth-Value Assignment
AĀ ***_____***Ā specifies a unique truth-value (eitherĀ TĀ orĀ F) for each atomic formula.
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Characteristic Truth-table for Conjunction
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Characteristic Truth-table for Disjunction
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Characteristic Truth-table for the Conditional
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Characteristic Truth-table for Negation
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Truth and Falsity Relative to a Truth-Value Assignment
* \n IfĀ Ļ†Ā is an atomic formula (sentential letter) of sentential logic, thenĀ Ļ†Ā is true onĀ Ļƒāˆ—Ā just in caseĀ ĻƒĀ assigns the valueĀ TĀ toĀ Ļ†, and false otherwise.
* IfĀ Ļ†Ā is a formula of the formĀ Ā¬Ļˆ, thenĀ Ļ†Ā is true onĀ Ļƒāˆ—Ā just in caseĀ ĻˆĀ is false onĀ Ļƒāˆ—, and false otherwise.
* IfĀ Ļ†Ā is a formula of the formĀ (Ļˆ&Ļ), thenĀ Ļ†Ā is true onĀ Ļƒāˆ—Ā just in case bothĀ ĻˆĀ andĀ ĻĀ are true onĀ Ļƒāˆ—, and false otherwise.
* IfĀ Ļ†Ā is a formula of the formĀ (ĻˆāˆØĻ), thenĀ Ļ†Ā is true onĀ Ļƒāˆ—Ā just in case eitherĀ ĻˆĀ is true onĀ Ļƒāˆ—Ā orĀ ĻĀ is true onĀ Ļƒāˆ—, and false otherwise.
* IfĀ Ļ†Ā is a formula of the formĀ (Ļˆā†’Ļ), thenĀ Ļ†Ā is true onĀ Ļƒāˆ—Ā just in case eitherĀ ĻˆĀ is false onĀ Ļƒāˆ—Ā orĀ ĻĀ is true onĀ Ļƒāˆ—, and false otherwise.
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Tautology
A formula is calledĀ ***logically true***Ā or aĀ ***_____***Ā just in case it is true on every truth-value assignment.
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Contradictory Formula
A formula is called aĀ ***_____***Ā just in case it is false on every truth-value assignment.
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Contingent Formula
A formula is called aĀ ***_____***Ā just in case it is true on some truth-value assignments, and false on others.
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Logical Consequence
The conclusion of an argument is aĀ ***_____***Ā of its premises if and only if any truth-value assignment that makes all the premises true also makes the conclusion true.
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Validity
An argument isĀ ***_____***Ā if and only if its conclusion is a logical consequence of its premises.
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Invalidity
An argument isĀ ***_____***Ā in case it is not valid, that is, if there is some truth-value assignment that makes the premises true, but the conclusion false.
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Counterexample
A truth-value assignment that makes the premises of an argument true and its conclusion false is called aĀ ***_____***Ā to the argument.
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Conditional Analogue
TheĀ ***_____***Ā of an argument with premisesĀ Ļ†1,...,Ļ†nĀ and conclusionĀ Ļ‡Ā is the formulaĀ ((Ļ†1&(...&Ļ†n))ā†’Ļ‡).
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Truth-Tree Construction Rules
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Truth-Tree: Closed Branch
A branch of truth-tree isĀ ***_____***Ā if and only if it contains both any formulaĀ Ļ†Ā and the negationĀ Ā¬Ļ†Ā of that formula.
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Truth-Tree: Open Branch
A branch of truth-tree isĀ ***_____***Ā if and only if it is not closed.
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Truth-Tree: Completed Branch
A branch of truth-tree is _____Ā if no further decomposition rules can be applied on that branch. All closed branches are completed. An open branch is completed if every formula on the branch is either a literal or already decomposed.
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Completed Truth-Tree
A truth-tree isĀ ***_____***Ā if every branch of the tree is completed.
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Procedure for Generating Truth-Trees

1. Start by writing down the formula for which you want to generate a truth-tree.
2. Based on the syntactic form of the expression, apply the appropriate truth-tree rule, putting a check mark next to the formula to indicate that it has been analyzed.
3. For each open branch, determine whether the branch contains both any formulaĀ Ļ†Ā and its negationĀ Ā¬Ļ†. If any branch does contain a formula and its negation, mark the branch closed. If all branches in the tree are closed, you are done. Otherwise, continue to the next step.
4. If the only formulae on open branches that do not have check marks next to them are atomic formulae or negations of atomic formulae, you are done. Otherwise, continue to the next step.


1. Choose an unchecked formula on an open branch in the truth-tree that is not atomic and not the negation of an atomic formula, and apply this procedure to that formula, starting with step 2.
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Truth-values
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Truth-functional
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Function
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Truth-tables
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Truth-conditions
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Characteristic truth-table
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Logically true
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Logical consequence
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Contradictory
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Truth-trees