Simple Truth Tables and some Harder Translations

(Originally written October 26, 2006 in Book 9)

Introduction to Logic
Harry J. Gensler

Simple Truth Tables (pg. 38)

Married in nine days! Stress, excitement, exhaustion. My studies have been suffering immensely; I hope to catch up a bit today though.

There are two possible truth values:
True
False

True is represented by '1'.
False is represented by '0'.

A truth table is a logical diagram for a Well-Formed Formula (WFF).

I went to Paris and Quebec = (P · Q)

P  Q  = (P · Q)
0   0      0
0   1      0
1   0      0
1   1      1

(P · Q) can only be true when both 'P' and 'Q' are true, as represented by the '1'.

A conjunction ("·") claims that both are true.

A disjunction claims that at least one is true. ("∨")

An inclusive disjunction = (P∨Q), meaning that at least one is true; but, both can be true.

An exclusive disjunction = (P∨Q)·~(P·Q), meaning that P or Q is true, but not both P and Q are true.

Inclusive Disjunction Truth Table
P  Q  = (P∨Q)
0  0      0
0  1      1
1  0      1
1  1      1

Exclusive Disjunction
P  Q  (P∨Q) = (P∨Q)·~(P·Q)
0  0      0                 0
0  1      1                 1
1  0      1                 1
1  1      1                 0

[There are many more truth tables confirming various propositional logic well formed formulas. I'll simply type the type of propositional well-formed formula and omit the truth table from here on out]

If-then statements are called a conditional.

If P then Q
P- the antecedent
Q- the consequent

The antecedent does not need to be true for the consequent to be true.

≡ is a biconditional.

[5 pages of truth values and truth tables omitted]

Contingent statements are true in some cases and false in others.

A tautology is true in all cases.

A self-contradiction ( P and not-P) is never true.

[1.5 pages of truth tables omitted]

The Truth Table Test (pg. 46)

The truth table test tests the validity of an argument. The argument is valid if and only if no line has all true premises and a false conclusion.

The Truth Assignment Test

PAY ATTENTION! This is why I failed the Logic Test!

"Take a propositional argument. Set each premise to 1 and the conclusion to 0. The argument is Valid if and only if no consistent way of assigning 1 and 0 to the letters will make this work - so we can't make the premises all true and conclusion false" (Gensler, 50).

[...more truth tests]

So the premises are true and the conclusion is false; thus, the argument is invalid.

[...even more truth tests]

The argument is invalid because of a true premise and a false conclusion.

Linehan - I think I get it! :)

[... yet more truth tests, 11.5 pages]

Harder Translations:

• Translate "yet", "however", "although" into "and" (·)
• Translate "unless" into "or" (∨)
• Translate "just if" into "iff" (if and only if) (≡)
• Translate "only if" into "if...then" (⊃)
• A is sufficient for B = If A then B
• A is necessary for B = If not-A then not-B
• A is necessary and sufficient for B = A if and only if B

[half a page of symbols and letters representing some answers to some questions from somewhere]