WFF and Truth Tables
PHIL102
October 4, 2016
Outline
Reminder of the Rules
Basic Truth Tables
Validity with Truth Tables
Syntax and Semantics
Last time we talked about the syntax and semantics rules
for P0.
Rules for Constructing WFFs
Here are the rules for constructing WFFs:
1. Any proposition letter by itself is well-formed.
2. If φ and ψ are well-formed, then the following are well
formed:
2.1
2.2
2.3
2.4
2.5
∼φ
(φ ∧ ψ)
(φ ∨ ψ)
(φ → ψ)
(φ ↔ ψ)
Note: sometimes authors omit the outermost pair of
parentheses for simplicity.
Rules for Constructing Truth Tables
Here are the rules for constructing truth tables:
1. First, we count the number of unique proposition
letters.
2. Second, we write out a table with all possible
combinations of truth values for those sentence
letters
3. Finally, we use the basic truth tables for our truth
functions to determine the truth value of the whole
sentence.
Basic Truth Tables: ∼
φ
∼φ
T
F
F
T
Basic Truth Tables: ∧
φ
ψ
(φ ∧ ψ)
T
T
T
T
F
F
F
T
F
F
F
F
Basic Truth Tables: ∨
φ
ψ
(φ ∨ ψ)
T
T
T
T
F
T
F
T
T
F
F
F
Basic Truth Tables: →
φ
ψ
(φ → ψ)
T
T
T
T
F
F
F
T
T
F
F
T
Basic Truth Tables: ↔
φ
ψ
(φ ↔ ψ)
T
T
T
T
F
F
F
T
F
F
F
T
Exercise
Validity
Recall our definition of validity: An argument is valid just
in case the conclusion follows necessarily from the
premises. That is, if the premises were true the
conclusion must be true as well.
Validity
We can use truth tables to check an argument’s validity!
Consider the following two arguments:
P
(P → Q)
Q
Q
(∼ P ∨ Q)
P
Validity
We can use truth tables to check an argument’s validity!
premise
P
T
T
F
F
conclusion
Q
T
F
T
F
premise
P→Q
T
F
T
T
Validity
We can use truth tables to check an argument’s validity!
conclusion
P
T
T
F
F
premise
Q
T
F
T
F
premise
(∼ P ∨ Q)
T
F
T
T
Next Time
Next time, we’ll begin looking at how to construct
deductive proofs in P0!
Reminders
You can get these slides and other
helpful materials at:
adamdedwards.com/phil102