Philosophy 140 Chapter 7: Deductive Reasoning (Propositional Logic)

Compound Statements

a statement (proposition) that contains at least two simple statements as constituents, and is joined by logical connectives -Example: 'Jane is a girl and Jane is in grade 6' -Example: 'If you lied to me, then you are not being friendly'

Simple Statements

a statement (proposition) that does not contain any other statement as constituents. Also called 'atomic propositions' -Example: 'Jane is a girl', 'You lied to me'

Invalid reasoning Denying the consequent

if p then q. Not p. Therefore not q. -Premise 1 states 'if p then q' not 'without p there is no q'. So it is not necessarily the case that p entails not q. --"If today is Tuesday we have logic class. Today's not Tuesday (not p). Hence, we don't have logic today (not q)

Invalid reasoning Affirming the consequent

if p then q. q therefore p. -Premise 1 states 'if p then q', not 'if q then p;. So, it is not necessarily the case q entails p. --"If today is Tuesday we have logic class (p then q). We have logic today (q). Therefore it's Tuesday (therefore p)." --invalid, because it's possible that we have logic on other day so the week

Symbols: Variables

letters used to represent statements. -Example: "If it rains, I will bring an umbrella" can be turned into "if p, then q." -Example: "Alice rode her bide" can turn into "p"

Is this valid reasoning: "if today is Tuesday we have logic class. We have logic today. Therefore, it's Tuesday." A yes B no

nswer: no (we could have logic another day too)

Symbols: Logical connectives

symbols for the logical relations between statements -We will study 4 connectives: - & (and) - conjunction - V (or) - disjunction - ~ (not) - negation - -> (if...then...) - conditional

Logical Connectives: Disjunction

-A disjunction is an "or" statement -In a disjunction, we assert that either po or q is true, and that even if one of the disjuncts is false, the whole disjunction is still true. -This is the inclusive "or" -In propositional logic we assume the inclusive 'or'. The truth table we use in the inclusive or.

Modus Tollens

-A valid logical structure for conditionals where one denies the consequent -If p, then q, not q, therefore not p -The first premise says if p is true, then q must be true. The second premise says q is not true. The results: p cannot be true -If p were true, q would be true, but p is not true, so the only explanation is that p is not true either Examples: -If it rains, the sidewalk gets wet. But the sidewalk's not wet. So it must not have rained. -If you are 19 or more, you can drink alcohol. You cannot drink alcohol. So, you are not 19 or more

Valid Logical Structures: Hypothetical Syllogism

-A valid logical structure for conditionals where we string conditionals together -If p, the q, if q then r, therefore if p then r. -The first premise says if p is true, then q must be true, which means that r is true (by premise 2). The result: if p is true, r is true as well Example: If Guy steals the money, he will go to jail. If Guy goes to jail, his family will suffer. Therefore, if Guy steals the money, his family will suffer.

Valid Logical Structures: Disjunctive Syllogism

-A valid logical structure where we state that one of two options must be true, and the first option is false, so the second option must be true -Either p or q, not p, therefore q Example: -The number 3 is either even or odd. It is not even. Therefore, it is odd -Shawn will date either Sally or Mary. But Shawn would never date Sally. So, Shawn will date Mary. -Disjunctive Syllogism is valid reasoning, but it is sound only if there are only two alternatives. Recall: Fallacy of the False Dichotomy: asserting there are only two choices when there are more than two choices (So, Premise 1 is false) -Maybe Sally will reject Shawn, so it is not necessarily true that Shawn will date Sally or Mary, even if he wants to.

Disjunction examples

-Example: "Either 1+1=2 or 1=1" true -Examples: "Either 1+1=3 or Canada is in the southern hemisphere" False -Examples: "Either Montreal is in Ontario or Toronto is in Ontario" True

Truth Values and Conjunction

-For any given statement (simple or compound) there are several combinations of truth-values -Suppose I make the compound claim "a & b"; then "a" might be true or false and "b" might be true or false -The truth value of the whole statement depends on the truth values of the components -This is called truth-functionality - If there are two variables (a,b) with two possible values each (T or F), that means that there are four possibilities for the whole statement. (T/F), (T/T), (F/T), (F/F) Is this compound statement T/F? "Ontario is a province in Canada and Toronto is a city in Quebec" (one is right, one is false so the whole thing is false)

What are the parts of a conditional?

-In a conditional (p->) the first part (p) is called the antecedent, and the second part (q) is the consequent -A conditional says" If p is true, then q will also be true" -the only way we can call a conditional false is if p is true but q is false -in all other circumstances, we call the conditional true -If the antecedent is F, then regardless of how the consequent works out is can't disprove the conditional if p is T then q is T; hence we take the conditional as T when the antecedent is F! (We can't show its false so we can't say its false for sure)

The two meanings of "or"

-Most common meaning in logic: "one or the other or both" which is called the inclusive sense -"If he's sick or tired, he won't go jogging" -Both disjuncts could be true in this case -But "or" can also mean "either but not both" which is called the exclusive sense. Both disjuncts cannot be true. -"Either it's 12:00 noon or it's 12:00 midnight" -"Either the Rough Riders will win the game, or they will lose the game" -"I'll graduate in 2012 or in 2013" -In these cases each disjunct excludes the other

Propositions

-Some sounds that come out of our mouths have 'truth value', i.e., have content that can be evaluated as true/false. These are propositions. You are 'proposing' that something is or is not the case, or, is true/false -Propositions (approximately equal =) statements. We will use the term 'statements' -Example: The sky is blue. The grass is purple. Jane does not like John

Logical Connectives: Conditional

-The basic form of a conditional is "if...then..." -For example: "If the cat is on the mat, then the rat will stay home" -Symbolized a conditional looks like this: "p->q" where an arrow represents

Truth Tables: Conjunction

-The logical behaviour of the truth functional connectives is defined by their truth tables -These give the "net" value of a statement built up using the connective, for each possible combination of truth values of its component statements -Truth tables are a way of assessing the truth of the overall statement, given the component statements -Truth table for "and": if I claim "p and q" my claim is not true unless both conjuncts are in fact true

Logical Connectives: Conjunction

-Two simple statements joined by a connective to form a compound statement are known as a conjunction - Each of the component statements is called a conjunct - For example: "Julio is here, and Juan is here" -p & q -In propositional logic these are all logically equivalent ways to say 'and': and, but, yet, nevertheless, while, also, moreover -Make sure the connective really is conjoining two distinct statements and not a set of compound nouns as in "We went to Jack's Bar and Grill" or "Juanita and Maria were a team"

Modus Ponens

-a valid logical structure for conditional where one affirms the antecedent -if p, then q, p, therefore q. -The first premise says q is true if p is true. The second premise says p is true. The result: q is true -The is the most common and basic deductive form Examples: -If spot barks, a burglar is in the house. Spot is barking. Therefore, a burglar is in the house. -If you're a hockey fan and you live in Montreal, you cheer for the Habs. You're a hockey fan and you live in Montreal. Thus, you cheer for the Habs -Any such argument is valid

Non-Propositions

-some sounds that come out of our mouths have no 'truth value', i.e., are not true or false -Example: Utterance: 'goo goo ga ga' -Example: Question: 'Why are brussel sprouts so disgusting?' -Side Note: the proposal 'will you marry me?' is actually not a proposition! -Example: Exclamations: 'Horray! The Jays rock!'

Logical Connective: The "Wedge"

-the symbol for disjunction (v) is called a wedge and it is roughly equivalent to the word "or" "Unless" -the word "unless" is usually symbolized as inclusive or. -to translate "I will go to the movies unless I stay home" define symbols as follows: -Let p be "I will go to the movies" -Let q be "I stay home" -Translation: p v q

Logical Connectives: Negation

-the truth table for negation is very simple: it just reverses the truth value of any proposition to which it is applied -the negation of a statement p is symbolized ~p -the is read "not p" or "it is not the case that p: -The symbol ~ is called (missed word)

Truth Value

-the truth/falsity of a proposition. The truth value of a proposition is whether the proposition is true or false -Example: 'The sky is blue' has truth value = true (T) -Example: 'The grass is purple' has truth value = false (F)

Logical Connectives

A complex statement joins simple statements together by logical connectives such as: and, or, not, if...then -Example: 'Jane is a girl *and* Jane is in grade 6 -Example: "*if* you lied to me, *then* you are being friendly

Valid Logical Structures: Conditionals

Condition, also known as hypothetical, are 'if...them..." statements -Example: 'If I won the lottery, then I would pay my bills' -Generalized form: As with categorical logic, it is helpful to put the conditional in generalized form by replacing the content with letters, to focus on the logic -Generalized Form: 'If P, then Q' If p, then q. If p, : Antecedent: what must be the case in order for the consequent to be the case (ante in poker, before the deal) Then q.: Consequent: what is the case if the antecedent is the case (consequence, what happens after)

Truth Tables and Conjunctions Examples

Example: -"Earth is round and 1+1=2" True! -"1+2=2, and the earth is flat" False! -"Earth is flat, and sky is blue" False! -"Earth is flat and fire is cold" False!

Examples of Negation

Example: If "the sky is blue" is true, then the negation "the sky is not blue" is false -If "It is the case that the price of eggs in China is not steep" is true, then the negation "It is not the case that the price

Symbolic logic

Modern deductive logic that uses symbolic language to do its work.

Compare modus ponens to modus tollens

Modus Ponens: If a bear eats my head off, I will die. A bear ate my head off. I am dead. Modus Tollens: IF a bear eats my head off, I will die. I am not dead. So, obviously a bear didn't eat my head off.

Negation Truth Table

P Q T F F T

Truth Table for Conjunction

P Q P & Q T T T T F F F T F F F F

Truth Table: Disjunction

P Q P v Q T T T T F T F T T F F F

Truth table for conditional

P Q p->q T T T T F F F T T F F t

Propositional logic studies

Studies the logical relation between propositions -A proposition is not asking someone on a date, to hook up or to get married

What are the two types of symbols in truth tables?

Variables and logical connectives