Module 4

A symbolic expression is a WFF under the following conditions:

1. capital letters (which stand for atomic statements) are WFFs 2. if p is a WFF, then so is ∼ p 3. if p and q are WFFs, then so is ( p • q ) 4. if p and q are WFFs, then so is ( p ∨ q ) 5. if p and q are WFFs, then so is ( p → q ) 6. if p and q are WFFs, then so is ( p ↔ q ) nothing counts as a WFF unless it can be demonstrated to be one by applications of the previous conditions

Well-Formed Formula

WFF a grammatically correct symbolic expression

Truth-Functional

a compound statement is truth-functional if its truth value is completely determined by the truth value of the atomic statement that compose it

Sufficient Condition

a condition that guarantees that a statement is true or that a phenomenon will occur the antecedent of a true conditional statement provides sufficient condition for the truth of the consequent

Necessary Condition

a condition that, if lacking, guarantees that a statement is false or that a phenomenon will not occur the consequent of a true conditional statement provides a necessary condition for the truth of the antecedent

Material Conditional

a conditional that is false only when its antecedent is true and its consequent is false; otherwise, it is true

Truth-Functional

a connective is truth-functional when the truth-value of a complex statement formed using it is determined solely by the truth-value of its parts english has many non-truth-functional connectives (because, etc.), but all of the connectives of statement logic are truth-functional

Statement Variables

a lowercase letter that serves as a placeholder for any statement

Validity

a valid argument is an argument which is such that it is impossible for the premises to be all true and the conclusion false

Connective

allows us to form complex statements using simple statements - statement logic has the five connectives

Invalidity

an invalid argument is an argument which is such that it is possible for all the premises to be true and the conclusion false

Biconditionals

conjunctions of conditionals treated as conjunctions of material conditionals

What does a truth table tell us about an argument?

each row in the table describes a possible situation in very abstract term what we are looking for is a row, and hence a possible situation, in which the premises are all true but the conclusion is false if we can find such a row (or situation), then the argument form is invalid

Symbolizing an Argument

enables us to apply certain powerful techniques to determine its validity to do so, we must distinguish atomic from compound statements

First Step in Using Truth Tables to Establish (In)Validity

generate all the possible truth-value assignments for L and S there are two truth values (truth and falsehood), so our truth table must have 2ⁿ rows, where n is the number of statement letters in the symbolic argument it is important to generate them mechanically both to avoid error and to facilitate communication

Negations

has the opposite truth value of the statement negated

Parentheses and Brackets

important when translating English into symbols used the same way as in math we are allowed to drop the outermost pair of parentheses to avoid clutter provided that we do not create ambiguity or change the meaning we are allowed to alternate parentheses with brackets in long expressions, as this sometimes makes statements a bit easier to read and clarifies the logical form

Commas and Triple Dot ∴

in symbolizing arguments, commas are used to separate premises triple dot marks the conclusion

Second Step in Using Truth Tables to Establish (In)Validity

in the column nearest to the vertical line simply alternate Ts and Fs in the next column to the left, alternate couples (two Ts, followed by two Fs)

Notes about Truth Tables

indicate which rows show invalidity with a star: * list the statement letters in the order in which they appear in our symbolization you must at least provide a column of truth values under every logical operator and under every atomic statement that stands alone as a premise or the conclusion once the table is complete, we focus on the columns under the main operator of each premise and the conclusion - if we are doing the truth table on paper, it might help to circle these columns it's helpful to write "Ps" above your premises and "C" above the conclusion

Disjunctions

is false if both its disjuncts are false; otherwise, it is true

Conjunctions

is true if both its conjuncts are true; otherwise, it is false

Limitation of the Truth Table Method

it becomes unwieldy as arguments become longer

Reasons to Learn Language of Statement/Propositional Logic

logical form of statements, and thus arguments, will be made explicit - formal fallacies will be avoided potential ambiguities will be avoided - avoid the fallacy of equivocation ease of use

→

name arrow translates if-then type of compound conditional

•

name dot translates and type of compound conjunction

↔

name double-arrow translates if and only if type of compound biconditional

∼

name tilde translates not type of compound negation

∨

name vee/wedge translates or type of compound disjunction

Atomic Statement

one that does not have any other statement as a component

Compound Statement

one that has at least one atomic statement as a component we can symbolize the atomic statements in compounds with capital letters, providing a scheme of abbreviation

Minor Logical Operator

operator that governs smaller components of a compound statement

Double-Arrow

symbolizes biconditionals; "if and only if" statements and their stylistic variants

Arrow

symbolizes conditionals; if-then and its stylistic variants including "unless"

Vee/Wedge

symbolizes disjunctions; "or" and its stylistic variants meaning the inclusive sense of "or," unless it makes the argument invalid when we want to communicate something of the form "Either A or B (but not both)", we can represent it in our symbol system as the conjunction of two statements: "Either A or B, and not both A and B"

Logical Operators

symbols that stand for key logical words

Three Aspects of Language

syntax the symbols of language along with the rules for proper formulation semantics the meanings of the symbols - what allows us to determine truth/falsity pragmatics the use of language - implicature

Truth Tables

the main idea behind truth tables is that the truth value of certain compound statements is a function of the truth value of the atomic statements that make them up a representation of all the possible ways in which the truth-value of a complex statement is determined by the truth-value of its component statements

Main Logical Operator

the operator that governs the largest component or components of a compound statement

Conjuncts

the statements composing a conjunction

Dot

used to translate "and" as well as its stylistic variants - remember that this doesn't cover every use of the word "and" conjunctions

Tilde

used to translate "not" and its stylistic variants negations

What good does it do to have a truth table for conditionals if it gives a questionable picture of the relationship between the truth value of English conditionals (in general) and the truth value of their constituent parts?

when the truth table method is applied to arguments, it nicely corroborates our belief in the validity of such intuitive inference rules it confirms our belief in the invalidity of such common, formal fallacies as denying the antecedent and affirming the consequent

Third Step in Using Truth Tables to Establish (In)Validity

write the steps of the argument out on the line at the top of the table and fill in the columns under each step of the argument, row by row

Steps to Making a Truth Tables

1. determine the number of rows you will need in your truth table - the number of simple statements in your complex statement determines the number of rows - number of rows needed = 2ⁿ, with n = number of simple statements 2. determine the number of columns you will need - column needed for each simple statement - column needed for statement itself - column needed for each complex statement that is a component of the final statement 3. fill in the truth-table - start with the first simple statement, divide number of rows by 2 and list that number of Ts, followed by that number of Fs - go to the next simple statement and divide by 2 again; list the number of Ts followed by this number of Fs - continue until correct number of rows is reached - if there is another simple statement, divide by 2 again and continue - use the truth-tables for the connectives to complete the table

Steps in Developing Truth Tables for Arguments

1. identify the number of simple statements, and use this information to determine the number of rows 2. identify any complex components of premises and conclusion 3. create a full truth-table 4. read the table - if there are no rows (possibilities) in which all the premises are true and the conclusion false, then the argument is valid - if there is at least one row where all the premises are true and the conclusion false, then the argument is invalid (you only need one)

Material Biconditional

A conjunction of two material conditionals; it is true when its constituent statements have the same truth value and false when they differ in truth value