TT Equation by Joby John

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Probing TT Equation to the Truth Table Method
A Study on Symbolic Logic
Academic Paper Submitted to Rev Dr Henry D
Almeida, SJ Faculty of Philosophy Head of the
Department Indian Studies Jnana- Deepa
Vidyapeeth Pune

• Submitted by
• Joby John
• NHMS
• II Year Philosophy (Exceptional)
• Jnana- Deepa Vidyapeeth
• Roll No. 17073
• jobykeelath_at_gmail.com
• Mobile[phone]

2
Truth Table
P q p q p?q p v q p?q pq
T T F F T T T T
T F F T F T F F
F T T F F T T F
F F T T F F T T
3
TT EquationTwo Methods
Method1
Method2

• TT Symbolic Description

TT Numerical Description
4
Method1TT Symbolic Description

• Step 1

Conversion of propositions to symbolic logical
form. Like p?(p?q)
5
Method1TT Symbolic Description

• Step 2

Find out the total number propositional variables
(eg. p,q,r,s etc.). p?(p?q) Here we have two
propositional variables
1
2
6
Method1TT Symbolic Description

• Step 3

Convert the variable to TT Equation
If variable negative E.g. p, q, r
If variable positive E.g. p, q,r
7
Method1TT Symbolic Description

• Step 3 Convert the variable to TT Equation

Total number of propositional variables(n)
8
Method1TT Symbolic Description

• Step 3 Convert the variable to TT Equation

Individual number of the variable(x)
x Individual number of the variable(x) Individual number of the variable(x) Individual number of the variable(x) Individual number of the variable(x) Individual number of the variable(x)
x p q r s t
x 1 2 3 4 5
9
Method1TT Symbolic Description

• Step 3 Convert the variable to TT Equation

Individual number of the variable(x)
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Method1TT Symbolic Description

• Step 3 Convert the variable to TT Equation

Multiply T and F according to this number
E.g. 21 (T,F) 2(T,F) 2T, 2F
11
Method1TT Symbolic Description

• Step 3 Convert the variable to TT Equation

Repeat the figures in the bracket according to
this number
Instead of writing this lengthy result just keep
the conversion like this 2(T,F)4
E.g. 21 (T,F)22 2(T,F)4 (2T, 2F) (2T, 2F)
(2T, 2F) (2T, 2F)
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Method1TT Symbolic Description

• Step 3Convert the variable to TT Equation

1
2
Here we have two positive variables So n2 X
(For P-1 and for Q-2)
p?(p?q)
p 22-1(T,F)21-1 2(T,F) (2T,2F)
q 22-2(T,F)22-1 (T,F)2 (T,F) (T,F)
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Method1TT Symbolic Description

• Step 3Convert the variable to TT Equation

p ? (p?q)
The final conversion symbol is put in the squire
brackets ?
(2T,2F) ? (2T,2F) ? (T,F)2
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Method1TT Symbolic Description

• Step4

Evaluate each combination by using TT Equation
Table
TT Equation Table TT Equation Table TT Equation Table
? T ? T T All other combinations F
T T T All other combinations F
F F T All other combinations F
v F v F F All other combinations T
? T ? F F All other combinations T
(2T,2F) ? (2T,2F) ? (T,F)2
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Method1TT Symbolic Description

• Step4Evaluate each combination by using TT
  Equation Table

TT Equation Table TT Equation Table TT Equation Table
? T ? T T All other combinations F
T T T All other combinations F
F F T All other combinations F
v F v F F All other combinations T
? T ? F F All other combinations T
T?T T
(2T,2F) ? (2T,2F) ? (T,F)2
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Method1TT Symbolic Description

• Step4Evaluate each combination by using TT
  Equation Table

TT Equation Table TT Equation Table TT Equation Table
? T ? T T All other combinations F
T T T All other combinations F
F F T All other combinations F
v F v F F All other combinations T
? T ? F F All other combinations T
T?F F
(2T,2F) ? (2T,2F) ? (T,F)2
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Method1TT Symbolic Description

• Step4Evaluate each combination by using TT
  Equation Table

TT Equation Table TT Equation Table TT Equation Table
? T ? T T All other combinations F
T T T All other combinations F
F F T All other combinations F
v F v F F All other combinations T
? T ? F F All other combinations T
F?T T
(2T,2F) ? (2T,2F) ? (T,F)2
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Method1TT Symbolic Description

• Step4Evaluate each combination by using TT
  Equation Table

TT Equation Table TT Equation Table TT Equation Table
? T ? T T All other combinations F
T T T All other combinations F
F F T All other combinations F
v F v F F All other combinations T
? T ? F F All other combinations T
F?F F
(2T,2F) ? (2T,2F) ? (T,F)2
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Method1TT Symbolic Description

• Step4Evaluate each combination by using TT
  Equation Table

TT Equation Table TT Equation Table TT Equation Table
? T ? T T All other combinations F
T T T All other combinations F
F F T All other combinations F
v F v F F All other combinations T
? T ? F F All other combinations T
T,F
2T
(2T,2F) ? (2T,2F) ? (T,F)2
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Method1TT Symbolic Description

• Step4Evaluate each combination by using TT
  Equation Table

TT Equation Table TT Equation Table TT Equation Table
? T ? T T All other combinations F
T T T All other combinations F
F F T All other combinations F
v F v F F All other combinations T
? T ? F F All other combinations T
T,F
2T
(2T,2F) ? (2T,2F) ? (T,F)2
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Method1TT Symbolic Description

• Step4Evaluate each combination by using TT
  Equation Table

TT Equation Table TT Equation Table TT Equation Table
? T ? T T All other combinations F
T T T All other combinations F
F F T All other combinations F
v F v F F All other combinations T
? T ? F F All other combinations T
2T
T,F
(2T,2F) ? (2T,2F) ? (T,F)2
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Method1TT Symbolic Description

• Step4Evaluate each combination by using TT
  Equation Table

TT Equation Table TT Equation Table TT Equation Table
? T ? T T All other combinations F
T T T All other combinations F
F F T All other combinations F
v F v F F All other combinations T
? T ? F F All other combinations T
(2T,2F) ? (2T,2F) ? (T,F)2
2T
T,F
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Method1TT Symbolic Description

• Step4Evaluate each combination by using TT
  Equation Table

TT Equation Table TT Equation Table TT Equation Table
? T ? T T All other combinations F
T T T All other combinations F
F F T All other combinations F
v F v F F All other combinations T
? T ? F F All other combinations T
T,F
(2T,2F) ? (T,F ,2T)
T T
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Method1TT Symbolic Description

• Step4Evaluate each combination by using TT
  Equation Table

TT Equation Table TT Equation Table TT Equation Table
? T ? T T All other combinations F
T T T All other combinations F
F F T All other combinations F
v F v F F All other combinations T
? T ? F F All other combinations T
T,F
T T
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Method1TT Symbolic Description

• Step4Evaluate each combination by using TT
  Equation Table

Contingent INVALID
(T,F ,2T)
p ? (p ? q)
T T T T T
T F T F F
F T F T T
F T F T F
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Method1TT Symbolic Description
(T,F ,2T)

• Step5The final string (TT Equation Validity)of
  the evaluation will be same as the final result
  of Truth Table.
• Step6Evaluation result will be described as same
  as Truth Table (i.e. F and T combination will be
  Contingent, Total F will be Contradictory and
  Total T will be Tautology).
• StepPracticing TT Equation enables us to examine
  the validity of lengthy symbolic representation
  of the propositions.

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Method2TT Numerical Description

• Conversion of propositions to symbolic logical
  form.
• Find out the total number propositional variables
  (eg. p,q,r,s etc.).
• Convert the variable to TT Equation.
• Form the TT Symbolic Description.
• Convert the TT Symbolic Description to TT
  Numerical Description by using TTND Conversion
  Table1and2.
• The final string (TT Equation Validity) of the
  evaluation will be same as the final result of
  Truth Table. The TT Equation Validity will be
  based on TT Numerical Evaluation Table.
• Evaluation result will be described as same as
  Truth Table (i.e. -1 and 1 combination will be
  Contingent, Total -1 will be Contradictory and
  Total 1 will be Tautology).

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Method2TT Numerical Description
TTND Conversion Table-1 TTND Conversion Table-1 TTND Conversion Table-1 TTND Conversion Table-1
TTSD Value Symbol TTND Conversion Equations ( x1and y -1) Conversion Value of equation
TT ? v ? T
FF ? v F
FF ? T
TF ? ? F
TF   v   T
FT ? F
FT v ? T
TTND Conversion Table-2 TTND Conversion Table-2 TTND Conversion Table-2
? 1 ? 1 1 All other combinations -1
1 1 1 All other combinations -1
-1 -1 1 All other combinations -1
v -1 v -1 -1 All other combinations 1
? 1 ? -1 -1 All other combinations 1
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Method2TT Numerical Description
TT Numerical Evaluation Table TT Numerical Evaluation Table TT Numerical Evaluation Table TT Numerical Evaluation Table
Value Result T,F representation Value of the result
Tautology 1,1 T,T T
Contradictory -1, -1 F,F F
Contingent -1, 1 / 1, -1 TF/FT F
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Method2TT Numerical Description
 
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Method2TT Numerical Description
Rewriting 2T2F ?(2T2F?(TF, TF)
 
(1, -1) ?(1, -1?-1,-1 )
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Method2TT Numerical Description
TTND Conversion Table-2 TTND Conversion Table-2 TTND Conversion Table-2
? 1 ? 1 1 All other combinations -1
1 1 1 All other combinations -1
-1 -1 1 All other combinations -1
v -1 v -1 -1 All other combinations 1
? 1 ? -1 -1 All other combinations 1
Rewriting 2T2F ?(2T2F?(TF, TF)
-1
1
(1, -1) ?(1, -1?-1,-1 )
(1, -1) ?(-1, 1)
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Method2TT Numerical Description
TTND Conversion Table-2 TTND Conversion Table-2 TTND Conversion Table-2
? 1 ? 1 1 All other combinations -1
1 1 1 All other combinations -1
-1 -1 1 All other combinations -1
v -1 v -1 -1 All other combinations 1
? 1 ? -1 -1 All other combinations 1
Rewriting 2T2F ?(2T2F?(TF, TF)
-1
1
(1, -1) ?(-1, 1)
-1 1
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Method2TT Numerical Description
TT Numerical Evaluation Table TT Numerical Evaluation Table TT Numerical Evaluation Table TT Numerical Evaluation Table
Value Result T,F representation Value of the result
Tautology 1,1 T,T T
Contradictory -1, -1 F,F F
Contingent -1, 1 / 1, -1 TF/FT F
Rewriting 2T2F ?(2T2F?(TF, TF)
-1, 1
Contingent INVALID
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Example Method1-TTSD

• (p ?q) ?(q ?r) ?(r ?s) ? (p ?s)

(8F,8T)?(4T,4F)2) ?((4T,4F)2?(2T,2F)4)
?((2F,2T)4? (T,F)8) ? ((8F,8T)?(T,F)8)
Conversion P (8T,8F) P (8F,8T) Q(4T,4F)2 R
(2T,2F)4 R (2F,2T)4 S (T,F)8
(8T,4T,4F)?((2T,2F,4T)2) ?((3T,F)4 ?
(8T),(T,F)4
(2T,2F,4T,2T,2F,4F)?(3T,F)4 ? (8T),(T,F)4
(2T,2F,3T,F,2T,6F) ? (8F,(F,T)4
(2F,2T,3F,T,F,T,6T) 2F,2T,3F,T,F,7T Contingent
INVALID!
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No (p ? q) ? (q   ? r) ? (r ? s) ? ? (p ? s)
1 F T T T T T T T F T T F 2F F F T T
2 F T T T T T T T F T F F 2F F F T F
3 F T T F T F F F T T T T 2T F F T T
4 F T T F T F F F T F F T 2T F F T F
5 F T F T F T T T F T T F 3F F F T T
6 F T F T F T T T F T F F 3F F F T F
7 F T F T F T F T T T T F 3F F F T T
8 F T F T F T F F T F F T 1T F F T F
9 T T T T T T T T F T T F 1F F T T T
10 T T T T T T T T F T F T 7T T T F F
11 T T T F T F F F T T T T 7T F T T T
12 T T T F T F F F T F F T 7T T T F F
13 T F F F F T T F F T T T 7T F T T T
14 T F F F F T T F F T F T 7T T T F F
15 T F F F F T F F T T T T 7T F T T T
16 T F F F F T F F T F F T 7T T T F F
(2F,2T,3F,T,F,T,6T) 2F,2T,3F,T,F,7T Contingent INVALID!   (2F,2T,3F,T,F,T,6T) 2F,2T,3F,T,F,7T Contingent INVALID!   (2F,2T,3F,T,F,T,6T) 2F,2T,3F,T,F,7T Contingent INVALID!   (2F,2T,3F,T,F,T,6T) 2F,2T,3F,T,F,7T Contingent INVALID!   (2F,2T,3F,T,F,T,6T) 2F,2T,3F,T,F,7T Contingent INVALID!   (2F,2T,3F,T,F,T,6T) 2F,2T,3F,T,F,7T Contingent INVALID!   (2F,2T,3F,T,F,T,6T) 2F,2T,3F,T,F,7T Contingent INVALID!   (2F,2T,3F,T,F,T,6T) 2F,2T,3F,T,F,7T Contingent INVALID!   (2F,2T,3F,T,F,T,6T) 2F,2T,3F,T,F,7T Contingent INVALID!   (2F,2T,3F,T,F,T,6T) 2F,2T,3F,T,F,7T Contingent INVALID!   (2F,2T,3F,T,F,T,6T) 2F,2T,3F,T,F,7T Contingent INVALID!   (2F,2T,3F,T,F,T,6T) 2F,2T,3F,T,F,7T Contingent INVALID!   (2F,2T,3F,T,F,T,6T) 2F,2T,3F,T,F,7T Contingent INVALID!   (2F,2T,3F,T,F,T,6T) 2F,2T,3F,T,F,7T Contingent INVALID!   (2F,2T,3F,T,F,T,6T) 2F,2T,3F,T,F,7T Contingent INVALID!   (2F,2T,3F,T,F,T,6T) 2F,2T,3F,T,F,7T Contingent INVALID!   (2F,2T,3F,T,F,T,6T) 2F,2T,3F,T,F,7T Contingent INVALID!   (2F,2T,3F,T,F,T,6T) 2F,2T,3F,T,F,7T Contingent INVALID!  
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Example Method2-TTND

• (p ?q) ?(q ?r) ?(r ?s) ? (p ?s)

(8F,8T)?(4T,4F)2) ?((4T,4F)2?(2T,2F)4)
?((2F,2T)4? (T,F)8) ? ((8F,8T)?(T,F)8)
(-8,8)?(4,-4)2) ?((4,-4)2?(2,-2)4) ?((-2,2)4?
(1,-1)8) ? ((-8,8)?(1,-1)8)
(8,4,-4)?((2,-2,4)2) ?((3,-1)4 ? (8),(1,-1)4
Conversion P (8T,8F) P (8F,8T) Q(4T,4F)2 R
(2T,2F)4 R (2F,2T)4 S (T,F)8
(2,-2,6,-6)?(3,-1)4 ? (8),(1,-1)4
(2,-2,3,-1,2,-6) ? (-8,(-1,1)4
(-2,2,-3,1,-1,7)
Contingent INVALID! (Mixture of positive and
negative)
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Conclusion The main use of the TT Equation is
that symbolized propositions can be examined.
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Reference

• Alexander, Peter, An Introduction to Logic,
  London George Allen and Unwin Ltd, 1971. Print
• Copi, Irving. Introduction to Logic, New York
  Macmillan Publishing Co., Sixth Edition, 1982.
  Print.
• Crystal, David, Cambridge Encyclopedia of
  Language, UK Cambrige University Press, Second
  Edition,1997. Print.
• Latta, Robert and Alexander Macbeath, The
  Elements of Logic, London Macmillan, 1956. Print
  

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Thank you