Module 5

Contingent Statements' Important Logical Relations to Tautologies and Contradictions

any argument that has a tautology as its premise but a contingent statement as its conclusion is invalid - the premise will be true in every row of the truth table, while the conclusion will be false in at least one row suppose that the premises of an argument, when made into a conjunction, form a contingent statement; then, if the conclusion of the argument is a contradiction, the argument is invalid - the conclusion will be false in every row, while the premise will be true in at least one row

Logical Equivalence's Important Relationship to Tautologies

if a biconditional statement is a tautology, then its two constituent statements (joined by the double-arrow) are logically equivalent

Law of Non-Contradiction

no statement of the form P • ~P is true

How to Write Invalidating Assignments

premise - T/F

Truth tables can be used to sort statements into three logically significant categories:

tautologies contradictions contingent statements each statement of our language meets one and only one of these conditions

How will you know if the premises of an argument are inconsistent?

there will be no row in the truth table in which all of the premises are true

Logical Consistency

two (or more) statements are logically consistent if and only if they are both (all) true on some assignment of truth values to their atomic components same truth value on at least one row

Logical Inconsistency

two (or more) statements are logically inconsistent if and only if they are never both (all, at least one?) true on any assignment of truth values to their atomic components any argument with logically inconsistent premises will be valid yet unsound

Logical Contradiction

two statements are logically contradictory if and only if they disagree in truth value on every assignment of truth values to their atomic components different truth value on every row if two statements are logically contradictory, then the two statements must also be inconsistent

Logical Equivalence

two statements are logically equivalent if and only if they agree in truth value on every assignment of truth values to their atomic components same truth value on each row

Contradiction

a statement is a contradiction if and only if it is false on every assignment of truth values to its atomic components false on every row any argument that has a contradiction among its premises is a valid argument

Tautology

a statement is a tautology if and only if it is true on every assignment of truth values to its atomic components true on every row every argument whose conclusion is a tautology is valid—regardless of the content of the premises

Contingent Statements

a statement is contingent if and only if it is true on some assignments of truth values to its atomic components and false on others true on some rows and false on others

Full Truth Table Review

a truth table is a representation of all the possible ways in which the truth-value of a complex statement is determined by the truth-values of its component statements a truth table with a row where all the premises are true and the conclusion false is required to prove an argument invalid thus, rows where either one or more of the premises are false, or rows where the conclusion is true cannot help us prove an argument invalid

Steps to the Abbreviated Truth Table Method

1. after placing the argument in a truth table, determine whether there are multiple ways in which the conclusion can be false 2. if there is just one way, place an F under the (main operator of the) conclusion and a T under (the main operator of) each premise --- to show invalidity, uniformly assign Ts and Fs to all of the components of the conclusion and the premises; write Ts and Fs under the atomic statements on the left of the table --- to show validity, uniformly assign Ts and Fs to all of the components of the conclusion and the premises; write a backslash under (the main operator of) the premise you were led to say was both true and false --- do not writeTs and Fs under the atomic statements on the left 3. if there is more than one way for the conclusion to be false, place one F under the (main operator of the) conclusion for each way it can be false, thereby creating as many rows as there are ways for the conclusion to be false --- on each row, place a T under (the main operator of) each premise --- to show invalidity, follow instruction 1b for at least one row --- to show validity, follow instruction 2b for every row

Steps to the Abbreviated/Indirect Truth Table Method (Class)

1. assume the premises are true and the conclusion false 2. continue with any forced assignments - if an invalidating assignment is found, stop - if a contradiction is found, stop 3. if at some time an assignment is not forced, each possible assignment must be checked - if none reveal an invalidating assignment, the argument is valid - if an invalidating assignment is found, stop - if a contradiction is found, check next assignment

Abbreviated/Indirect Truth Table Method

if there is an assignment of truth values to one row of a truth table, making all the premises true while the conclusion is false , then the argument form in question is invalid the central strategy of the abbreviated truth table method is to hypothesize that there is such a row, and then to confirm the hypothesis, thereby showing that the argument is invalid, or disconfirm it, thereby showing that the argument is valid we create a truth table with all true premises and a false conclusion, and then work backward to see if we can make it invalid we indicate that we are forced to assign both a T and an F to A ∨ B by writing the symbol "/" taking away true conclusion and false premise rows assuming true premises and false conclusions will result in forced truth value assignments for simple statements if there's multiple ways you can potentially have true premises and false conclusion, you have to try them all this method gives us the means to deal with arguments with more than three simple statements

When the conclusion is a conjunction or a biconditional, remember two principles:

if there is any assignment of values in which the premises are all true and the conclusion is false, then the argument is invalid if more than one assignment of truth values will make the conclusion false, then consider each such assignment; if each assignment that makes the conclusion false makes at least one premise false, then the argument is valid

Tautologies and Contradictions

if you add a tilde to a tautology, it becomes a contradiction if you add a tilde to a contradiction, it becomes a tautology

Four Logically Significant Relationships

logical equivalence logical contradiction logical consistency logical inconsistency