12. Truth Functions

Question 1

Truth Functionality (3)

Answer 1

• Meaning is exhaustively described by a truth table
• Connectives ∧, ∨ and ¬ are truth functional
• Natural Words And, Or, Not are truth functional (If we ignore all their non-truth-conditional meanings)

Question 2

Truth Functional Connectives (2)

Answer 2

• Connectives are truth functional only when they take the truth values of their inputs and nothing else.
• Connective standardize truth values (see truth tables)
  EX: A and B are true, if we use ¬ then ¬A and ¬B are false.

Question 3

Non-Truth-Functional Connectives (2)

Answer 3

• Natural Language Connectives such as because are non-truth-functional:
  (1) they do not have a truth table (not standardized)
  (2) they consider things beyond the truth values

EX: All squares are square BC they have four right angles
Vs All squares are square BC all circles are round

Question 4

Truth Function (3)

Answer 4

A function that takes truth values as inputs and outputs a truth value.
Truth functions grow doubly exponentially, with each input added inputs increase 2^2^n.
Truth function outputs Unary: 4, Binary: 16, Ternary: 256

Question 5

Monotone FUnctions

Answer 5

(1) Go up in value or (2) stay the same, (3) never go down.
(4) 0 is Higher than 1.
EX:
✔A) 1 0 0 X E) 0 1 0
✔B) 1 1 0 X F) 0 0 1
✔C) 1 1 1 X G) 0 1 1
✔D) 0 0 0 X H) 1 0 1

Question 6

Truth Tables for Wff

Answer 6

A) WFF α is built using n atomic propositions, there are 2^n valuations so the truth table for α requires 2^n rows: the number of rows should always be a multiple of 2.
B) Write all rows at the same time
C) The first set should be divided in half, the second set should be in four, and the third set should be intercalated.

EX:
A) 1 1 1 1 0 0 0 0
B) 1 1 0 0 1 1 0 0
C) 1 0 1 0 1 0 1 0

Question 7

Inention

Answer 7

The truth conditions of x, its entire column in a truth table.

Question 8

Extension

Answer 8

The actual value of x in the world, its truth value in a particular row.