Truth tables¶

Show truth tables for connectives¶

Truth functions are defined as clauses in mathesis. You can show the truth tables for the clauses in HTML in JupyerLab/Jupyter Notebook:

In [1]:

from mathesis.system.classical.truth_table import ConditionalClause

conditional_clause = ConditionalClause()
conditional_clause

from mathesis.system.classical.truth_table import ConditionalClause conditional_clause = ConditionalClause() conditional_clause

Out[1]:

P	Q	Conditional(P, Q)
1	1	1
1	0	0
0	1	1
0	0	1

Outside Jupyer, you get a plain text table:

In [2]:

print(conditional_clause)

print(conditional_clause)

P        Q        Conditional(P, Q)        
1        1                1                
1        0                0                
0        1                1                
0        0                1                

Generate truth tables for classical logic¶

mathesis.semantics.truth_table.ClassicalTruthTable automatically generates the truth table for a given formula.

In [3]:

from mathesis.grammars import BasicGrammar
from mathesis.semantics.truth_table import ClassicalTruthTable

grammar = BasicGrammar()

fml = grammar.parse("(¬P∧(P∨Q))→Q")

table = ClassicalTruthTable(fml)
table

from mathesis.grammars import BasicGrammar from mathesis.semantics.truth_table import ClassicalTruthTable grammar = BasicGrammar() fml = grammar.parse("(¬P∧(P∨Q))→Q") table = ClassicalTruthTable(fml) table

Out[3]:

P	Q	¬P	P∨Q	¬P∧(P∨Q)	(¬P∧(P∨Q))→Q
0	0	1	0	0	1
0	1	1	1	1	1
1	0	0	1	0	1
1	1	0	1	0	1

table.is_valid() just returns whether the formula is valid.

In [4]:

f"Valid: {table.is_valid()}"

f"Valid: {table.is_valid()}"

Out[4]:

'Valid: True'

Generate truth tables for many-valued logics¶

Some many-valued logics are implemented out of the box. They are available from mathesis.semantics.truth_table.

Three-valued logic K₃ and Ł₃¶

Kleene's K₃¶

In [5]:

from mathesis.grammars import BasicGrammar
from mathesis.semantics.truth_table import K3TruthTable

grammar = BasicGrammar()

fml = grammar.parse("A∨¬A")

table = K3TruthTable(fml)
table

from mathesis.grammars import BasicGrammar from mathesis.semantics.truth_table import K3TruthTable grammar = BasicGrammar() fml = grammar.parse("A∨¬A") table = K3TruthTable(fml) table

Out[5]:

A	¬A	A∨¬A
i	i	i
0	1	1
1	0	1

In [6]:

f"Valid: {table.is_valid()}"

f"Valid: {table.is_valid()}"

Out[6]:

'Valid: False'

Łukasiewicz's Ł₃¶

WIP

Three-valued logic LP¶

In [7]:

from mathesis.grammars import BasicGrammar
from mathesis.semantics.truth_table import LPTruthTable

grammar = BasicGrammar()

fml = grammar.parse("(A∧¬A)→A")

table = LPTruthTable(fml)
table

from mathesis.grammars import BasicGrammar from mathesis.semantics.truth_table import LPTruthTable grammar = BasicGrammar() fml = grammar.parse("(A∧¬A)→A") table = LPTruthTable(fml) table

Out[7]:

A	¬A	A∧¬A	(A∧¬A)→A
0	1	0	1
1	0	0	1
i	i	i	i

In [8]:

f"Valid: {table.is_valid()}"

f"Valid: {table.is_valid()}"

Out[8]:

'Valid: True'

Four-valued logic FDE¶

WIP

Use custom symbols for truth values¶

Subclasses of ConnectiveClause() and TruthTable() can have truth_value_symbols attribute that is a dictionary mapping internal numeric truth values to arbitrary symbols like ⊤, ⊥, T, F, etc.

Define custom truth tables¶

WIP