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# Sphinx build info version 1 | ||
# This file hashes the configuration used when building these files. When it is not found, a full rebuild will be done. | ||
config: 7f2341602339f5787e05519e6e1924b2 | ||
config: 32dd555d207fe9d43466c601a4446270 | ||
tags: 645f666f9bcd5a90fca523b33c5a78b7 |
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************ | ||
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.. bibliography:: bibliography_bibtex.bib | ||
:style: plain | ||
:cited: |
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disjunction-elimination | ||
========================= |
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formula-statement | ||
=================== | ||
|
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hypothesis | ||
============ | ||
|
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inconsistency-introduction-1 | ||
======================================== | ||
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Definition | ||
---------- | ||
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*inconsistency-introduction-1* is the :doc:`inference_rule`: | ||
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.. math:: | ||
\left( P, \neg \left(P\right) \right) \vdash Inc\left(\mathcal{T}\right) | ||
In straightforward language, if we prove a proposition and its negation, it follows that the theory is inconsistent. | ||
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Quotes | ||
------ | ||
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Python sample usage | ||
---------------------- | ||
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.. admonition:: Source code | ||
:class: tip, dropdown | ||
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.. literalinclude :: ../../sample/code/inconsistency_introduction_1.py | ||
:language: python | ||
:linenos: | ||
.. admonition:: Unicode output | ||
:class: note, dropdown | ||
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.. literalinclude :: ../../sample/output/inconsistency_introduction_1_unicode.txt | ||
:language: text | ||
.. admonition:: Plaintext output | ||
:class: note, dropdown | ||
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.. literalinclude :: ../../sample/output/inconsistency_introduction_1_plaintext.txt | ||
:language: text |
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inconsistency-introduction-2 | ||
======================================== | ||
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Definition | ||
---------- | ||
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*inconsistency-introduction-2* is the :doc:`inference_rule`: | ||
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.. math:: | ||
\left(\left(x = y\right), \left(x \neq y\right)\right) \vdash Inc\left(\mathcal{T}\right) | ||
In straightforward language, if we prove both the equality and inequality of two terms, it follows that the theory is inconsistent. | ||
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Quotes | ||
---------- | ||
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A proof of consistency will have to show, by appealing to contentual considerations which are completely unproblematic, that in the formalism in question it is never possible to derive the formula 𝑎 ≠ 𝑎, or alternatively it is not possible to prove both 𝑎 = 𝑏 and 𝑎 ≠ 𝑏. :cite:p:`mancosu_2021_introductionprooftheorynormalization{p. 5}` | ||
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Python sample usage | ||
---------------------- | ||
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.. admonition:: Source code | ||
:class: tip, dropdown | ||
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.. literalinclude :: ../../sample/code/inconsistency_introduction_2.py | ||
:language: python | ||
:linenos: | ||
.. admonition:: Unicode output | ||
:class: note, dropdown | ||
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.. literalinclude :: ../../sample/output/inconsistency_introduction_2_unicode.txt | ||
:language: text | ||
.. admonition:: Plaintext output | ||
:class: note, dropdown | ||
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.. literalinclude :: ../../sample/output/inconsistency_introduction_2_plaintext.txt | ||
:language: text |
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inconsistency-introduction-3 | ||
======================================== | ||
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Definition | ||
---------- | ||
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*inconsistency-introduction-3* is the :doc:`inference_rule`: | ||
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.. math:: | ||
\left( P \neq P \right) \vdash Inc\left(\mathcal{T}\right) | ||
In straightforward language, if we prove that an object is not equal to itself, it follows that the theory is inconsistent. | ||
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Quotes | ||
---------- | ||
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A proof of consistency will have to show, by appealing to contentual considerations which are completely unproblematic, that in the formalism in question it is never possible to derive the formula 𝑎 ≠ 𝑎, or alternatively it is not possible to prove both 𝑎 = 𝑏 and 𝑎 ≠ 𝑏. :cite:p:`mancosu_2021_introductionprooftheorynormalization{p. 5}` | ||
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Python sample usage | ||
---------------------- | ||
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.. admonition:: Source code | ||
:class: tip, dropdown | ||
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.. literalinclude :: ../../sample/code/inconsistency_introduction_3.py | ||
:language: python | ||
:linenos: | ||
.. admonition:: Unicode output | ||
:class: note, dropdown | ||
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.. literalinclude :: ../../sample/output/inconsistency_introduction_3_unicode.txt | ||
:language: text | ||
.. admonition:: Plaintext output | ||
:class: note, dropdown | ||
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.. literalinclude :: ../../sample/output/inconsistency_introduction_3_plaintext.txt | ||
:language: text |
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proof-by-contradiction-1 | ||
========================= | ||
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Definition | ||
----------- | ||
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*proof-by-contradiction-1* is the :doc:`inference_rule`: | ||
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.. math:: | ||
\left( \mathcal{H} assume \not \mathbf{P}, Inc\left(\mathcal{H}\right) \right) \vdash \mathbf{P} | ||
Python implementation | ||
---------------------- | ||
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Sample usage | ||
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ | ||
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.. literalinclude :: ../../sample/code/proof_by_contradiction_1.py | ||
:language: python | ||
.. literalinclude :: ../../sample/output/proof_by_contradiction_1_unicode.txt | ||
:language: text | ||
Documentation | ||
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ | ||
.. module:: core | ||
:noindex: | ||
.. autoclass:: InferenceRuleInclusionDict | ||
:members: proof_by_contradiction_1 | ||
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Bibliography | ||
------------ | ||
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.. footbibliography:: |
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proof-by-contradiction-2 | ||
========================= | ||
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Definition | ||
---------- | ||
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*proof-by-contradiction-2* is the :doc:`inference_rule`: | ||
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.. math:: | ||
\left( \mathcal{H} assume \not \mathbf{P}, Inc\left(\mathcal{H}\right) \right) \vdash \mathbf{P} | ||
Python implementation | ||
--------------------- | ||
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ProofByContradiction2Declaration | ||
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ | ||
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.. module:: core | ||
:noindex: | ||
.. autoclass:: ProofByContradiction2Declaration | ||
:members: | ||
:special-members: __init__ | ||
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ProofByContradiction2Inclusion | ||
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ | ||
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.. autoclass:: ProofByContradiction2Inclusion | ||
:members: | ||
:special-members: __init__ | ||
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Bibliography | ||
------------ | ||
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.. footbibliography:: |
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proof-by-refutation-1 | ||
========================= | ||
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Definition | ||
---------- | ||
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*proof-by-refutation-1* is the :doc:`inference_rule`: | ||
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.. math:: | ||
\left( \boldsymbol{\mathcal{H}} \: \textit{assume} \: \neg \boldsymbol{P}, \boldsymbol{P}, Inc\left( \boldsymbol{\mathcal{H}} \right) \right) \vdash \boldsymbol{P} | ||
Where: | ||
* :math:`\boldsymbol{\mathcal{H}}` is an :doc:`hypothesis` | ||
* :math:`\boldsymbol{P}` is a :doc:`formula_statement` | ||
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In plain language, it consists in posing the hypothesis that a proposition :math:`\boldsymbol{P}` is not true, refuting that hypothesis by proving :math:`\boldsymbol{P}`, inferring the inconsistency of that hypothesis from this contradiction, and finally inferring :math:`\boldsymbol{P}`. | ||
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Python implementation | ||
--------------------- | ||
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ProofByRefutation1Declaration | ||
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ | ||
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.. module:: core | ||
:noindex: | ||
.. autoclass:: ProofByRefutation1Declaration | ||
:members: | ||
:special-members: __init__ | ||
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ProofByRefutation1Inclusion | ||
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ | ||
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.. autoclass:: ProofByRefutation1Inclusion | ||
:members: | ||
:special-members: __init__ | ||
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Bibliography | ||
------------ | ||
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.. footbibliography:: |
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proof-by-refutation-2 | ||
========================= | ||
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Definition | ||
---------- | ||
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*proof-by-refutation-2* is the :doc:`inference_rule`: | ||
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.. math:: | ||
\left( \mathcal{H} assume \not \mathbf{P}, Inc\left(\mathcal{H}\right) \right) \vdash \mathbf{P} | ||
Python implementation | ||
--------------------- | ||
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ProofByRefutation2Declaration | ||
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ | ||
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.. module:: core | ||
:noindex: | ||
.. autoclass:: ProofByRefutation2Declaration | ||
:members: | ||
:special-members: __init__ | ||
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ProofByRefutation2Inclusion | ||
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ | ||
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.. autoclass:: ProofByRefutation2Inclusion | ||
:members: | ||
:special-members: __init__ | ||
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Bibliography | ||
------------ | ||
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.. footbibliography:: |
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minimalist example | ||
===================== | ||
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.. literalinclude :: ../../sample/output/proof_by_contradiction_1_unicode.txt | ||
:language: text |
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/* | ||
Fix to assure text wrapping in code and code output inclusions implemented with the literalinclude sphinx directive. | ||
References: | ||
- https://docs.readthedocs.io/en/stable/guides/adding-custom-css.html | ||
- https://stackoverflow.com/questions/66971299/sphinx-documentation-with-readthedocs-theme-text-block-wrapping | ||
- https://stackoverflow.com/questions/46800045/how-to-wrap-a-long-literal-block-in-restructuredtext | ||
*/ | ||
pre { | ||
white-space: pre-wrap !important; | ||
word-break: break-all; | ||
} |
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