Truth Tables and Modal Logic
Garry Goodwin 3/4/2015
[email protected]
Simple Account of Truth Tables
A basic account of truth tables is relatively simple. A proposition has just two truth
possibilities i.e. it may be True (T) or it may be False (F). We write these possibilities on a
table where p represents a proposition. There are four distinct truth functions for a single
proposition if we include contradiction and tautology.
Four Unary Truth Functions
p
¬p

Τ
T
F
F
Τ
F
T
F
Τ
Table 1
Truth tables and truth functions belong to that part of logic called semantics. Truth
functional logic is a Boolean logic. When the Boolean operations are interpreted truth
functionally the Boolean 1 is interpreted as True and the Boolean 0 as False. Thus the
Boolean operations may be presented as truth functional operations.
Boolean / Truth functional Operations
¬
v
F
T
&
F
T
→
F
T
↔
F
T
xor
F
T
T
F
T
T
T
T
T
T
T
F
T
T
F
T
T
T
F
F
T
F
F
T
F
F
T
F
T
T
F
T
F
F F T
Exclusive
Disjunction
Not
Or
And
Implication
Equivalence
Truth tables exhaust the combinations of truth possibilities and then compute the truth
functional relationship between one or more propositions using one or more of these
operations.
We now consider a less simple account of a truth table.
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Less Simple Account of Truth Tables
(i) Statements: {p, q, r, ….} are statements..
(ii) Truth Possibilities: {'p is the case', 'p is not the case', 'q is the case', 'q is not the
case',…} are truth possibilities.
(iii) Truth Values: {T, F} are truth evaluations.
(iv) Truth Tables Consider two truth possibilities i.e. when the statement is the case
and when the statement is not the case.
p
¬p

Τ
'p is the case' is…
T
F
F
Τ
'p is not the case' is…
F
T
F
Τ
Table 2
When truth values evaluate a proposition's truth possibilities they define its truth
conditions; there is redundancy but the definition is tautological. For example on Table 2 the
truth conditions for p state the possibility that 'p is the case' is True and the possibility that
'p is not the case' is False. If the truth possibilities are left implicit Table 2 looks exactly like
Table 1.
The less simple interpretation allows many valued logic to evaluate the two truth
possibilities. For example {F, C, NC, T} is a Boolean four valued logic. The C stands for
Contingent truth i.e. a state of affairs is true but it is accidental or extraneous, and NC is the
negation of C. The Boolean operations are expanded to account for four values.
Four Valued Boolean Operations
¬
T
F
NC C
C NC
F
T
Not
F
C NC
T
F
C NC
NC C C T
C NC T NC
F
T
T
T
Implication
→
T
T
T
T
T
V
F
T
T
NC NC
C
C
F
F
C NC
T
T
T NC
C
T
C NC
Or
T
T
T
T
T
F
C NC T
T
F
C NC T
NC C
F
T NC
C NC T
F
C
F
T NC C
F
Equivalence
↔
&
T
NC
C
F
F
F
F
F
F
C NC T
C NC T
F NC NC
C
F
C
F
F
F
And
F
C NC T
T
T NC C
F
NC NC T
F
C
C
C
F
T NC
F
F
C NC T
Exclusive Disjunction
xor
2
We are able to combine the four Boolean values and the less simple interpretation of a truth
table to define a Boolean modal logic.
p
¬p
◊p
¬◊p
□p
¬□p
'p is the case' is…
T
F
T
F
NC
C
'p is not the case' is…
F
T
C
NC
F
T
Table 3
As was the case with standard two valued propositions the definition of the modal truth
condition are also tautological but this time there is no redundancy. For example on Table 3
the truth conditions for □p evaluates the possibility that 'p is the case' as Not Contingent
and the possibility that 'p is not the case' as False. This is the definition for 'p is necessarily
the case' and both evaluations are needed.
Modal arguments are computed on truth tables. We compare a standard two valued table
with a modal table.
p
→
q
□p
→
q
T
T
T
NC
T
T
T
F
F
NC
C
F
F
T
T
F
T
T
F
T
F
F
T
F
Table 4
Table 5
Given the less simple interpretation tables 4 and 5 show a truth table is easily adapted for a
many valued logic. However it is important to note that a proposition remains bivalent. A
table only considers two truth possibilities and the four valued logic {F, C, NC, T} is not a
fuzzy logic. It has three values that unequivocally preserve truth and one value that
accounts for False. The values C and NC are not partial truth values. This system is a
conservative extension of classic two valued logic.
Historically Boolean modal logic has not been widely accepted as it proves arguments
generally found implausible. A lot more work is needed before this truth table method
produces a plausible modal logic. However this is possible and the system may be developed
into a thorough going semantic and decision procedure for modal logic that avoids the
problems associated with Boolean modal logic.
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