STATEMENT LOGIC: TRUTH TABLES (Charles Sanders Peirce, 1839-1914)
Statement logic is the logic of statements. In statement logic, statements—as opposed to predicates, subject terms, and quantifiers—are the basic unit on which to do logic. SYMBOLIZING ENGLISH ARGUMENTS
Symbolizing an argument helps us to see its logic and to determine its validity. To symbolize an argument we need to symbolize the statements that compose it. Toward that end, let’s distinguish two kinds of statements: atomic and compound.
Atomic statement: a statement that does not have any other statement as a component For example…
• Peter likes to play soccer. • William adores his mother. • Frances’s favorite city is Siena.
We symbolize an atomic statement with a capital letter; use a different letter for each different atomic statement.
• A • B • C
Compound statement: a statement that has at least one atomic statement as a component For example…
• Peter likes to play soccer and William adores his mother. • Either Peter likes to play soccer or William adores his mother.
• If Frances’s favorite city is Siena, then Frances’s favorite city is in Italy. • Frances’s favorite city is Siena if and only Frances’s favorite city is in Italy. • It’s false that William is a lousy chess player.
We symbolize a compound statement by assigning a different capital letter to each different atomic statement that is a component.
• A and B. • Either A or B. • If C, then D.
Logical operators (key logical words):
OPERATOR NAME TRANSLATES TYPE OF COMPOUND
~ tilde not negation
● dot and conjunction
v vee or disjunction
→ arrow if…, then… conditional
↔ double arrow if and only if biconditional
A and B. A ● B
Either A or C. A v C
If A, then B. A → B
A if and only if B. A ↔ B
It’s false that D. ~D
Some Comments on Types of Compounds Negations
• Stylistic variations:
It is not the case that Peter has a broken thumbnail. It is false that Peter has a broken thumbnail.
It is not true that Peter has a broken thumbnail. Peter fails to have a broken thumbnail.
Peter does not have a broken thumbnail. All of these are symbolized the same way: ~P • Negations can involve other logical operators:
~P ~(P ● B) ~(P v B) ~(P → B) ~(P ↔ B)
Conjunctions
• Stylistic variations:
Peter has a broken thumbnail, but he’s brave about it. Peter has a broken thumbnail; however, he’s brave about it. While Peter has a broken thumbnail, he’s brave about it. Although Peter has a broken thumbnail, he’s brave about it. Peter has a broken thumbnail, yet he’s brave about it.
Peter has a broken thumbnail; nevertheless, he’s brave about it. Peter has a broken thumbnail though he’s brave about it. All of these are symbolized the same way: P ● B
• Not every use of “and” is properly translated with the dot. For example…
I climbed Mt. Baker and looked inside the sulfur cone. I got in my truck and turned the key.
The point: The word “and” sometimes means “and then”, indicating temporal order; the dot does not indicate temporal order.
Another example…
Peter and William are brothers. Ryan and Christine are married.
The point: The word “and” sometimes indicates a relationship; the dot does not indicate any relationship.
• Parentheses
Oftentimes, you need to use parentheses in order to make it clear what is being said. For example…
It’s false that both Obama and McCain are president. A = Obama is president
B = McCain is president Incorrect translation: ~A ● B
• Conjunctions can involve other logical operators A ● (B → C)
A ● (B v C) (~A ↔ B) ● C
In each of these cases, the main operator is the dot. Disjunctions
• Stylistic variations
Either Peter broke the window or William did. Peter broke the window or William did. Peter broke the window and/or William did. Peter broke the window or William did (or both). All of these are symbolized in the same way: A v B • Inclusive vs. exclusive disjunction
Two rules to remember:
Rule #1. Assume inclusive disjunction unless explicitly stated otherwise For example:
Either he loves Mary or he loves Shelly, but not both.
Either he scored 21 points or he scored 22 points, but not both. Rule #2. To express exclusive disjunction, use this symbolization:
(A v B) ● ~(A ● B) • “neither…nor….”
Neither you nor I know for sure exactly when we are going to die. (Y = You know for sure exactly when you are going to die; I = I know for sure exactly when I am going to die.) Permissible symbolizations:
~Y ● ~I ~(Y v I)
• Disjunctions can involve other logical operators A v (B → C)
A v (B ● C) ~B v (C ↔ D)
In each of these cases, the main operator is the vee. Conditionals
• Stylistic variations (page 286) • Sufficient condition
For example
(1) If Pat is a bachelor, then Pat is unmarried. If A, then B.
(2) Pat’s being a bachelor is a sufficient condition for Pat’s being unmarried. A is a sufficient condition for B.
Symbolized the same way: A Æ B • Necessary condition
For example…
(1) If there’s a fire in the building, then there is air in the building. If A, then B.
(2) There being air in the building is a necessary condition for there being a fire in the building.
B is a necessary condition for A. Symbolized the same way: A Æ B
• …unless…
There are two ways to translate statements with “unless”: Use the conditional
Illustration
(1) We will lose on Saturday unless Dykstra has a super game. (L: We will lose on Saturday; D: Dykstra has a super game)
L unless D
Note: (1) means the same thing as
(2) Dykstra’s having a super game is a necessary condition for our not losing on Saturday D is a necessary condition for ~L
(3) Alternative: if Dykstra does not have a super game, then we will lose If ~D, then L
Symbolize: ~L Æ D
So, whenever you see something of the form L unless D
A natural way to symbolize it as: ~L Æ D
However, it might be easier to simply remember to replace “unless” with a vee: L v D
• Conditionals can involve other logical operators (A v B) Æ C
D Æ (E ● ~F) G Æ (H ↔ L)
In each case, the main operator is the arrow. Biconditionals
• Stylistic variants A if and only if B A just in case B
• Conjunction of A → B and B → A. For example
It will rain if and only if the atmospheric conditions are just right for rain. R ↔ A
(R → A) ● (A → R) Another example
There’s fire if and only if there’s air, combustion, and fuel. F ↔ (A ● (C ● U))
(F → (A ● (C ● U))) ● (((A ● C) ● U) → F)
STATEMENT LOGIC Our Symbolic Language
Vocabulary: (1) parentheses, (2) logical operators (3) statement capital letters An expression in statement logic is any string using this vocabulary.
A well-formed formula (WFF) is a grammatically correct expression.
What counts as a WFF? Let lowercase letters, e.g. p, q, r, etc., stand for statement variables, which can stand for any statement.
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 v q). 5. If p and q are WFFs, then so is (p Æ q). 6. If p and q are WFFs, then so is (p ↔ q).
Nothing is a WFF unless it can be demonstrated to be one by 1-6.
Is it a WFF, strictly speaking? (~B) n (M Æ ~~N) y ((Q Æ S) Æ T) y ~(~W v ~Z) y ~(m ● h) n (~E ● ~F ● ~~G) n (~U Æ (W)) n ((~H ● ~~F) Æ ~(~K ↔ ~N)) y (A Æ B Æ C) n (E Æ (~F Æ G) n ((L v M) Æ ~S) y (~P v Q v ~R) n
Permissible departures from strict grammar in our symbolic language • Dropping parentheses (without creating ambiguity)
• Using brackets Is it a permissible departure? ~G ● ~H y (A v B) ↔ (C v D) y [~Z ● ~W v ~~Y] n ~J ↔ ~K y (~Q v ~R v ~~S) n ~A Æ (~C Æ F) y
Indicating an argument in our symbolic language • Comma ( , ) used to separate premises • Triple-dot ( :. ) used to indicate conclusion So, for example,…
