Chapter 7.1! Symbolizing English Arguments! 5 Important Logical Operators!The Main Logical Operator Today s Lecture 1/28/10
Quiz State from memory (closed book and notes) the five famous valid forms and their names. --be sure your name is on your paper
Announcements Homework -- Ex 7.1 pgs. 298-299 Part A and B (All) Quiz on Tuesday (Feb 2nd) --state from memory each logical operator, its translation, and its corresponding type of compound statement. See the table on p. 279. (Example: ~ not negation) Book Issues Adding the Course
Validity v. Invalidity Again An argument is valid if and only if it s impossible for all of the premises to be true while the conclusion is false. An argument is invalid if and only if it is possible for all of the premises to be true while the conclusion is false.
Validity v. Invalidity Again --Our task shortly is to study a particular mechanical method for determining whether an argument is either valid or invalid. --It will be the method of Truth Tables.
--But before we can employ this mechanical method, it's essential that we be able to represent English arguments using a symbolic notation. --To do this we need to learn all that is involved with translating English statements into symbols.
Atomic v. Compound Statements An atomic statement is one that does not have any other statement as a component. Examples!Grass is green.!the library is adjacent to Sierra Tower.!Tennis is challenging.
Atomic v. Compound Statements A compound statement is one that has at least one atomic statement as a component. Examples! It is false that grass is red.! The library is adjacent to Sierra Tower and I am hungry.! If tennis is challenging, then tennis players have a reason to practice.
Atomic Statements: Symbolize Atomic Statements with a single upper case letter. B: Brass is a mixed metal. C: Cathy called in sick. N: Nadal wins trophies T: Thought is mysterious.
Compound Statements Symbolize Compound Statements by first symbolizing their Atomic Constituents, and then their logical words. (logical words: it s not the case that, and, or, if, then, if and only if, and their stylistic variants) (We will just symbolize the Atomic Constituents for now) Example: It is false that grass is red. It is false that G.
Compound Statements More examples of partially translated compound Statements: The library is adjacent to Sierra Tower and I am hungry -- L and I If tennis is challenging, then tennis players have a reason to practice -- If T, then P
Symbolizing the Logical Words Operator ~ Name tilde Translates not Compound Type negation dot and conjunction v vee or disjunction " arrow if, then conditional! double-arrow If and only if biconditional
~ NEGATIONS
Symbolizing Negations Grass is not red (R: Grass is red) is symbolized as this is our scheme of ~ R abbreviation It is not the case that grass is red ~ R It is false that grass is red ~ R
Negations of Compound Statements It is false that Dallas wins and Phoenix wins is symbolized as ~ (D P) (D: Dallas wins P: Phoenix wins) It s not true that if Dallas wins, then Phoenix wins ~ (D " P) The following is false: either Dallas wins or Phoenix wins. ~ (D # P)
Parentheses are Important It is false that Dallas wins and Phoenix wins ~ (D P) -- this a negation (this says that it s not the case that both of them win; it leaves us agnostic as to who actually wins; maybe both of them don t win) If we didn t use parentheses we would get ~ D P -- this is a conjunction, not a negation (this says that Dallas does not win and Phoenix wins)
Parentheses are Important It s not the case that if Dallas wins then Phoenix wins. ~(D " P) -- this is a negation (this says that Dallas winning does not entail Phoenix winning) If we didn t use parentheses we would get: ~D " P -- this is a conditional, not a negation (this says if Dallas doesn t win, then Phoenix wins)
Main Logical Operator The most important step in knowing where to place parentheses is finding the main logical operator (i.e. main connective) in the English statement. This lets you know what kind of compound statement it is, be it a negation or conditional, or disjunction, etc.
Main Logical Operator The main logical operator is, roughly, the operator that governs (or connects) the entire statement. Finding the main operator depends upon your ability to see what the statement is saying. This just takes practice. More will be said concerning main operators and parentheses later.
Back to Negations Again, all of these are examples of negations: ~ (D P) ~(D "P) ~(D # P) The ~ is the main logical operator
CONJUNCTIONS
Symbolizing Conjunctions Grass is purple and life is good. (G: grass is purple; L: life is good) G L
Stylistic Variants of and! Grass is purple but life is good.! Grass is purple; however life is good.! Grass is purple yet life is good.! Although grass is purple, life is good.! While grass is purple, life is good.! Grass is purple; nevertheless life is good. Still, G L
Not all uses of 'and' are conjunctions Not all uses of the English term 'and convey a conjunction. If they did, then we would be able to translate the statements in question into conjunctions and capture the essential meaning of the English statement. But we are unable to do this in all cases. For example
Not all uses of 'and' are conjunctions! Sometimes and indicates temporal order! Sometimes and indicates a relationship -Sarah cracked the safe and took the money. -You made a joke and I laughed. -Dave and Val are married. - Dave and Val moved the couch.
These are all Conjunctions P Q P ~(Q # R) (P " Q) (Q " P) The is the main operator ~P [Q # (R " S) ] [Q " (P # R)] S (P # Q) (R " S) The is the main operator
# DISJUNCTIONS
Symbolizing Disjunctions 1. Grass is green or pizza is edible. (G: Grass is green; P: pizza is edible) G # P 2. Alfred will not pass tomorrow or Alfred will study tonight (P: Alfred will pass tomorrow; S: Alfred will study tonight) ~P # S
Stylistic Variants of or Alfred will not pass tomorrow and/or Alfred will study tonight. Alfred will not pass tomorrow or Alfred will study tonight (or both). Alfred will not pass tomorrow unless Alfred will study tonight. Still ~P # S
Inclusive OR either P or Q (or both) Sometimes when people make a disjunctive claim, they intend the or to be read inclusively. e.g. If you want to live under my roof, either you get a job or you go to college. **The parent will not be bothered if you do both. Exclusive OR either P or Q (but not both) Sometimes when people make a disjunctive claim, they intend the or to be read exclusively. e.g. You may have the soup or you may have the salad. **The waitress will be bothered if you say both.
Logicians Treat or as Inclusive We will treat or as inclusive in the absence of a context that suggests an exclusive reading. There is, however, a way of translating an exclusive or which is, again, P or Q (but not both) Consider: You may have the soup or you may have the salad, but not both. Q: How would you translate this?
Exclusive OR You may have soup or you may have salad, but not both. (S: you may have soup; L: you may have salad) (S # L) ~(S L)
'Neither-Nor' is Not a Disjunction! Neither Simon nor Garfunkel is sad. S: Simon is sad. G: Garfunkel is sad. Two Equivalent Readings 1. ~ (S # G) 2. ~S ~G
These are all Disjunctions P # Q P # ~(Q R) (P " Q) # (Q P) The # is the main operator ~P #[Q (R " S) ] [Q " (P # R)] # S (P # Q) #(R " S) The # is the main operator
" CONDITIONALS
Symbolizing Conditionals If Lisa is identical to an immaterial soul, then Lisa is essentially invisible. L: Lisa is identical to an immaterial soul I: Lisa is essentially invisible L " I If Lisa is identical to a material body, then Lisa is not essentially invisible. M: Lisa is identical to a material body I: Lisa is essentially invisible M " ~I
Some Stylistic variants of 'if-then' If Gizmo is a cat, then Gizmo is a mammal! Gizmo is a cat only if Gizmo is a mammal.! Assuming that Gizmo is a cat, Gizmo is a mammal.! Gizmo is a mammal if Gizmo is a cat G: Gizmo is a cat; M: Gizmo is a mammal G " M (Note: there are other stylistic variants. See p. 286)
A note on only if The term Only if (unlike if ) introduces a consequent; the antecedent precedes the only if Remember ANTECEDENT only if CONSEQUENT The term only if intuitively (naturally) introduces a necessary condition (or a requirement). Since the consequent of a conditional is a necessary condition for the antecedent, it s a bit easier to see how only if introduces a consequent.
Sufficient and Necessary Conditions Sufficient Conditions " P " Q is claiming that the occurrence of P is sufficient condition for Q. " A sufficient condition is a condition that guarantees that a statement is true (or that a phenomenon will occur). Necessary Conditions " P " Q is also claiming that the occurrence of Q is a necessary condition for P. " A necessary condition is a condition that, if lacking, guarantees that a statement is false (or that a phenomenon will not occur).
Some Examples --If Alex knows he has hands, then Alex believes he has hands. Knowing something is sufficient for believing it. --Alex knows he has hands only if Alex has good reason to believe he has hands. Having good reason to believe something is a necessary condition on having knowledge of it. --Given that one has a conscious pain, one is aware of the pain. Being conscious of pain is sufficient for being aware of pain. --You can legally drink only if you are at least 21. Being at least 21 is a necessary condition on being able to legally drink.
These are all Conditionals P " Q P " ~(Q # R) (P " Q) " (Q " P) ~P "[Q # (R " S) ] [Q (P # R)] " S (P Q) " (R " S) The " is the main operator The " is the main operator
Unless can be translated by the " as well as the # Depending on the intent of the speaker, 'Alfred will not pass tomorrow unless Alfred will study tonight' could read: If Alfred will study tonight, then it s false that he won t pass tomorrow (i.e. he will pass) S" ~~P (or S " P). This corresponds to the exclusive reading of 'or' in that the intent is not to say that Alfred could very well study and not pass. For if he studies tonight, he will pass tomorrow.
Unless can be translated by the " as well as the # 'Alfred will not pass tomorrow unless Alfred will study tonight' could read: If Alfred will not study tonight, then Alfred will not pass tomorrow ~S " ~P This corresponds to the inclusive reading of 'or' in that it leaves open the possibility that he could study and not pass. The conditional just says that if he doesn't study tonight, he won't pass. It doesn't automatically follow from this that if he does study, he will pass.
Unless can be translated by the " as well as the # Since we are sticking with an inclusive reading of or (and unless), p unless q (where p and q stand for any statement, compound or atomic) should be symbolized as: ~q " p
! Biconditionals
Symbolizing Biconditionals Leslie is in her 30 s if and only if Leslie is between the ages of 30-39. L: Leslie is in her 30 s A: Leslie is between the ages of 30-39 L! A
Symbolizing Biconditionals Jon is a bachelor if and only if Jon is not married and Jon is a male. B: Jon is a bachelor M: Jon is a male R: Jon is married Q: How would you symbolize this?
Symbolizing Biconditionals Jon is a bachelor if and only if Jon is not married and Jon is a male. B: Jon is a bachelor M: Jon is a male R: Jon is married B! (~R M)
Stylistic variant of if and only if Leslie is in her 30 s just in case Leslie is between the ages of 30-39. L: Leslie is in her 30 s A: Leslie is between the ages of 30-39 M! A
These are all Biconditionals P! Q P! ~(Q R) (P " Q)! (Q P) ~P! [Q (R " S) ] [Q " (P # R)]! S (P # Q)! (R " S)