propositions.deduction module¶
Useful proof manipulation maneuvers in Propositional Logic.

propositions.deduction.
prove_corollary
(antecedent_proof, consequent, conditional)¶ Converts the given proof of a formula
antecedent
to a proof of the given formulaconsequent
by using the given assumptionless inference rule of which'(
antecedent
>
consequent
)'
is a specialization. Parameters
antecedent_proof (
Proof
) – valid proof ofantecedent
.consequent (
Formula
) – formula to prove.conditional (
InferenceRule
) – assumptionless inference rule of which the assumptionless inference rule with conclusion'(
antecedent
>
consequent
)'
is a specialization.
 Return type
 Returns
A valid proof of
consequent
from the same assumptions as the given proof, via the same inference rules as the given proof and in additionMP
andconditional
.

propositions.deduction.
combine_proofs
(antecedent1_proof, antecedent2_proof, consequent, double_conditional)¶ Combines the given proofs of two formulas
antecedent1
andantecedent2
into a proof of the given formulaconsequent
by using the given assumptionless inference rule of which'(
antecedent1
>(
antecedent2
>
consequent
))'
is a specialization. Parameters
antecedent1_proof (
Proof
) – valid proof ofantecedent1
.antecedent2_proof (
Proof
) – valid proof ofantecedent2
from the same assumptions and inference rules asantecedent1_proof
.consequent (
Formula
) – formula to prove.double_conditional (
InferenceRule
) – assumptionless inference rule of which the assumptionless inference rule with conclusion'(
antecedent1
>(
antecedent2
>
consequent
))'
is a specialization.
 Return type
 Returns
A valid proof of
consequent
from the same assumptions as the given proofs, via the same inference rules as the given proofs and in additionMP
andconditional
.

propositions.deduction.
remove_assumption
(proof)¶ Converts the given proof of some
conclusion
formula, the last assumption of which is an assumptionassumption
, to a proof of'(
assumption
>
conclusion
)'
from the same assumptions exceptassumption
. Parameters
proof (
Proof
) – valid proof to convert, with at least one assumption, via some set of inference rules all of which have no assumptions except perhapsMP
. Return type
 Returns
A valid proof of
'(
assumption
>
conclusion
)'
from the same assumptions as the given proof except the last one, via the same inference rules as the given proof and in additionMP
,I0
,I1
, andD
.

propositions.deduction.
prove_from_opposites
(proof_of_affirmation, proof_of_negation, conclusion)¶ Combines the given proofs of a formula
affirmation
and its negation'~
affirmation
'
into a proof of the given formula.

propositions.deduction.
prove_by_way_of_contradiction
(proof)¶ Converts the given proof of
'~(p>p)'
, the last assumption of which is an assumption'~
formula
'
, to a proof offormula
from the same assumptions except'~
formula
'
. Parameters
proof (
Proof
) – valid proof of'~(p>p)'
to convert, the last assumption of which is of the form'~
formula
'
, via some set of inference rules all of which have no assumptions except perhapsMP
. Return type
 Returns
A valid proof of
formula
from the same assumptions as the given proof except the last one, via the same inference rules as the given proof and in additionMP
,I0
,I1
,D
, andN
.