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Topic: Material Implication: Deriving the Truth Table
Replies: 13   Last Post: Jan 4, 2018 7:41 AM

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 Dan Christensen Posts: 8,219 Registered: 7/9/08
Material Implication: Deriving the Truth Table
Posted: Jan 3, 2018 12:20 AM

Truth Table for IMPLIES

A B A => B

1 T T T
2 T F F
3 F T T
4 F F T

where A and B are logical, true-or-false propositions.

In mathematics, implication is usually defined as:

A =>B <=> ~[A & ~B]

This "definition" can be actually be derived using the deduction (conclusion) and detachment of properties of the implication operator:

http://www.dcproof.com/DeriveImplies.html (formal proof in DC Proof format)

Using these same properties, we can also use this definition to derive the entries of the above truth table.

Line 1: A & B => [A => B] http://www.dcproof.com/ImpliesLine1.html

Line 2: A & ~B => ~[A => B] http://www.dcproof.com/ImpliesLine2.html

Lines 3-4: ~A => [A => B] http://www.dcproof.com/ImpliesLines3-4.html

Dan

Visit my Math Blog at http://www.dcproof.wordpress.com