{"id":25245,"date":"2021-01-25T19:37:17","date_gmt":"2021-01-25T19:37:17","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=25245"},"modified":"2025-07-31T01:09:38","modified_gmt":"2025-07-31T01:09:38","slug":"formal-deductive-systems-in-propositional-logic","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/formal-deductive-systems-in-propositional-logic\/","title":{"rendered":"Formal Deductive Systems in Propositional Logic"},"content":{"rendered":"<p><center><\/p>\n<h1 style=\"text-align:center;\">Formal Deductive Systems in Propositional Logic<\/h1>\n<p style=\"text-align:center;\"><em><strong>Summary:<\/strong><\/br>In this class, we review formal deductive systems. We explain how these systems are used to decipher the relationships that may exist between different logical expressions, and the basic elements with which these proofs are built: the language, the axioms, and the inference rules. We mention \u0141ukasiewicz&#8217;s axioms and explain modus ponens as the deductive engine of propositional calculus. Additionally, we discuss reasoning, theorems, and premises, and explain how deductions are carried out in deductive systems.<\/em><\/p>\n<p style=\"text-align:center;\"><strong>Learning Objectives:<\/strong><\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Understand<\/strong> the concept of formal deductive systems in propositional logic.<\/li>\n<li><strong>Identify<\/strong> the basic components of formal deductive systems.<\/li>\n<li><strong>Learn<\/strong> \u0141ukasiewicz&#8217;s axioms in propositional calculus.<\/li>\n<li><strong>Understand<\/strong> modus ponens as the deductive engine of propositional calculus.<\/li>\n<li><strong>Comprehend<\/strong> how deductions are performed in deductive systems and the difference between premises, reasoning, and theorems.<\/li>\n<li><strong>Understand<\/strong> how deductions are generated through axiomatic schemes and inference rules.<\/li>\n<li><strong>Recognize<\/strong> the capacity of logic to connect expressions and replace them with usual language expressions.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>CONTENT INDEX<\/u>:<\/strong><br \/>\n<a href=\"#1\">WHAT IS A FORMAL DEDUCTIVE SYSTEM?<\/a><br \/>\n<a href=\"#2\">\u0141UKASIEWICZ&#8217;S AXIOMS FOR PROPOSITIONAL LOGIC<\/a><br \/>\n<a href=\"#3\">MODUS PONENS: THE DEDUCTIVE ENGINE OF PROPOSITIONAL CALCULUS<\/a><br \/>\n<a href=\"#4\">REASONING, THEOREMS, AND PREMISES<\/a><br \/>\n<a href=\"#5\">HOW TO PERFORM A PROOF IN PROPOSITIONAL LOGIC<\/a><br \/>\n<a href=\"#6\">THE CONCEPT OF PROVED EQUIVALENCE<\/a><br \/>\n<a href=\"#7\">THE DEDUCTION (META)THEOREM<\/a><br \/>\n<a href=\"#8\">THE RECIPROCAL OF THE DEDUCTION THEOREM<\/a><br \/>\n<a href=\"#9\">DEDUCTIONS ON EXPRESSIONS AND DEDUCTIONS ON DEDUCTIONS<\/a><br \/>\n<a href=\"#10\">MONOTONICITY RULE<\/a><br \/>\n<a href=\"#11\">SYNTHESIS AND REFLECTIONS ON DEDUCTIVE SYSTEMS AND PROPOSITIONAL LOGIC<\/a>\n<\/p>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/OvoEDefcSZg\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\n<\/center><\/p>\n<p style=\"text-align: justify;\">We have reached a turning point in our study of logic, as we now begin the review of Deductive Systems in Propositional Logic. This is where everything we&#8217;ve covered becomes operational and the true spirit of logic emerges, as we will study the essence of proofs. At this point, it is assumed that you have already learned how to write expressions and understand what propositional logic is about; if it&#8217;s not entirely clear, it&#8217;s recommended to review the previous classes.<\/p>\n<p style=\"text-align: justify;\">From here, we will review how propositional logic expressions relate to one another to form a deduction. The mechanism through which these relationships are constructed is the <strong>formal deductive system.<\/strong><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>What is a Formal Deductive System?<\/h2>\n<p style=\"text-align: justify;\">Formal deductive systems, or deductive calculus systems, have three basic components:<\/p>\n<ol style=\"color: #000000; text-align: justify;\">\n<li><strong>A Formal Language.<\/strong><\/li>\n<li><strong>An Axiomatic Scheme.<\/strong><\/li>\n<li><strong>Basic Inference Rules.<\/strong><\/li>\n<\/ol>\n<p style=\"text-align: justify;\">We have already reviewed formal languages. Now it&#8217;s time to introduce axiomatic schemes and basic inference rules.<\/p>\n<p style=\"text-align: justify;\">To build the deductive system for propositional calculus, we will start by constructing the deductive system from the <strong>Axioms of <a href=\"https:\/\/en.wikipedia.org\/wiki\/Jan_%C5%81ukasiewicz\" rel=\"noopener\" target=\"_blank\">\u0141ukasiewicz<\/a><\/strong>, and use <strong>Modus Ponens<\/strong> as the basic inference rule.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u0141ukasiewicz&#8217;s Axioms for Propositional Logic<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=206s\" target=\"_blank\" rel=\"noopener\"><strong>If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\beta<\/span><\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span> are propositional calculus expressions,<\/strong><\/a> then the following are axioms of propositional calculus:<\/p>\n<table>\n<tbody>\n<tr>\n<td>[A1]<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow (\\beta \\rightarrow \\alpha))<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>[A2]<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">((\\alpha \\rightarrow (\\beta \\rightarrow \\gamma))\\rightarrow ((\\alpha\\rightarrow \\beta)\\rightarrow(\\alpha \\rightarrow \\gamma)))<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>[A3]<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">((\\neg\\beta \\rightarrow \\neg\\alpha)\\rightarrow(\\alpha\\rightarrow \\beta))<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Modus Ponens: The Deductive Engine of Propositional Calculus<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=392s\" target=\"_blank\" rel=\"noopener\"><strong>If <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> are valid propositional calculus expressions, <\/strong><\/a>then modus ponens states that from <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow \\beta)<\/span><\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> is deduced. In reasoning form, this is written as follows:<\/p>\n<table style=\"text-align: justify;\">\n<caption>Modus Ponens Structure<\/caption>\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/td>\n<td>; Premise<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow \\beta)<\/span><\/span><\/td>\n<td>; Premise<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/td>\n<td>; MP(1,2)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Here, Modus Ponens is abbreviated between steps (1) and (2) using \u00abMP(1,2)\u00bb, and the synthesis of all this is represented by the notation:<\/p>\n<p style=\"text-align: center;\">Therefore, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{\\alpha, (\\alpha \\rightarrow \\beta)\\}\\vdash \\beta <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">We will soon see that all the deduction techniques of propositional calculus can be constructed from \u0141ukasiewicz&#8217;s axioms and Modus Ponens, which synthesize the basic rules of usual reasoning and form the foundational basis of <strong>classical logic.<\/strong><\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Reasoning, Theorems, and Premises<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=506s\" target=\"_blank\" rel=\"noopener\"><strong>In the deductive systems of propositional logic, reasoning<\/strong><\/a> (or deductions) are performed, and these are any sequence of expressions where each is either a premise or an expression obtained from the premises using only \u0141ukasiewicz&#8217;s axioms and modus ponens. A theorem is the result of a deduction without premises. A premise can be any expression that is neither an axiom nor derived from them. In general, when we have a set of premises <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span> and an expression <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> that is derived using some element of <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span>, the axioms, and modus ponens, we write \u00ab<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma \\vdash \\alpha<\/span><\/span>\u00bb and say<\/p>\n<p style=\"text-align: center;\"><em>from <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> is deduced<\/em><\/p>\n<p style=\"text-align: justify;\">If <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span> is an empty set, then instead of writing \u00ab<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\emptyset\\vdash \\alpha<\/span><\/span>\u00ab, we write \u00ab<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash \\alpha <\/span><\/span>\u00ab. This reads \u00ab<span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> is a theorem\u00bb. This form of representing theorems can be extended to represent axioms such that if <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span>, and <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span> are expressions, then \u0141ukasiewicz&#8217;s axioms can be written as follows:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>[A1]<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\alpha \\rightarrow (\\beta \\rightarrow \\alpha))<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>[A2]<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash((\\alpha \\rightarrow (\\beta \\rightarrow \\gamma))\\rightarrow ((\\alpha\\rightarrow \\beta)\\rightarrow(\\alpha \\rightarrow \\gamma)))<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>[A3]<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash((\\neg\\beta \\rightarrow \\neg\\alpha)\\rightarrow(\\alpha\\rightarrow \\alpha))<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Thus, axioms are said to be self-evident statements, or theorems are expressions inferred from nothing, meaning axioms and theorems are properties of propositional calculus.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>How to Perform a Proof in Propositional Logic<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=783s\" target=\"_blank\" rel=\"noopener\"><strong>At this point, we will move from theory to practice.<\/strong><\/a> And when it comes to performing a proof, many things can be said; however, no matter how brilliant the statements about deductive systems and propositional logic, understanding them does not necessarily imply the development of the necessary skills to perform a proof. For this reason, to teach how to make proofs, we will review the proof of a simple theorem.<\/p>\n<p style=\"text-align: justify; color: #880000;\"><strong>Theorem<\/strong><\/p>\n<p style=\"text-align: justify;\">If <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> is a propositional logic expression, then it follows that<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\alpha\\rightarrow \\alpha)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000088;\"><strong>Proof<\/strong><\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> (\\alpha\\rightarrow ( \\alpha \\rightarrow \\alpha)) <\/span><\/span><\/td>\n<td>; A1<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> (\\alpha\\rightarrow ((\\alpha\\rightarrow \\alpha)\\rightarrow\\alpha)) <\/span><\/span><\/td>\n<td>; A1<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ( (\\alpha\\rightarrow((\\alpha\\rightarrow\\alpha)\\rightarrow\\alpha)) \\rightarrow ((\\alpha\\rightarrow (\\alpha\\rightarrow\\alpha))\\rightarrow( \\alpha\\rightarrow \\alpha))) <\/span><\/span><\/td>\n<td>; A2<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ((\\alpha\\rightarrow (\\alpha\\rightarrow\\alpha))\\rightarrow( \\alpha\\rightarrow \\alpha)) <\/span><\/span><\/td>\n<td>; MP(2,3)<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ( \\alpha\\rightarrow \\alpha) <\/span><\/span><\/td>\n<td>; MP(1,5)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\">Therefore, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vdash (\\alpha\\rightarrow\\alpha)<\/span><\/span><\/p>\n<p>End of the proof.<\/p>\n<p style=\"text-align: justify;\">As you can see, in deductive systems and propositional logic, proofs are far from trivial, but once built, they are easy to replicate.<\/p>\n<p style=\"text-align: justify;\">Now, before diving into making deductions with these techniques, we will first develop some properties and definitions that will be extremely useful for this task, as reasoning only with this will lead to terrible problems.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h2>The Concept of Proved Equivalence<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=1191s\" target=\"_blank\" rel=\"noopener\"><strong>If <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> are any expressions, and it holds<\/strong><\/a> that both <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\}\\vdash \\beta<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\beta\\} \\vdash \\alpha<\/span><\/span>, then <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> are said to be proved equivalent, and this is written as <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha \\dashv \\vdash \\beta<\/span><\/span>. This is symbolically summarized as:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\{\\alpha\\}\\vdash\\beta \\wedge \\{\\beta\\}\\vdash\\alpha \\right) \\Leftrightarrow \\left(\\alpha\\dashv\\vdash\\beta\\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">This is a meta-property of propositional logic.<\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h2>The Deduction (Meta)Theorem<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=1355s\" target=\"_blank\" rel=\"noopener\"><strong>If <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> are propositional calculus expressions<\/strong><\/a> and <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span> is a set of premises, then it follows that if from <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma \\cup \\{\\alpha\\}<\/span><\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> is deduced, then from <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span> it is deduced <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow \\beta)<\/span><\/span>. Symbolically, this is expressed as:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\Gamma \\cup \\{\\alpha\\}\\vdash \\beta \\right) \\Rightarrow \\left( \\Gamma\\vdash(\\alpha\\rightarrow\\beta)\\right)<\/span>\n<p style=\"text-align: justify; color: #880000;\"><strong>Proof:<\/strong><\/p>\n<p style=\"text-align: justify;\">To satisfy <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma \\cup \\{\\alpha\\}\\vdash \\beta<\/span><\/span>, it is necessary to have a deduction of the form:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_1<\/span><\/span><\/td>\n<td>; Premise 1 of <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>(n)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_n<\/span><\/span><\/td>\n<td>; Premise n of <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span><\/td>\n<\/tr>\n<tr>\n<td>(n+1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\overline{\\gamma}_1<\/span><\/span><\/td>\n<td>; Modus Ponens between some previous steps<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>(n+m)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\overline{\\gamma}_m<\/span><\/span><\/td>\n<td>; Modus Ponens between some previous steps<\/td>\n<\/tr>\n<tr>\n<td>(n+m+1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/td>\n<td>; Premise<\/td>\n<\/tr>\n<tr>\n<td>(n+m+2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/td>\n<td>; Modus Ponens (n+m+1, previous steps, except n+m+1)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\">Therefore, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\Gamma\\cup\\{\\alpha\\} \\vdash \\beta <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">To achieve this, at least one of the expressions <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_1, \\cdots \\gamma_n,\\overline{\\gamma_1},\\cdots,\\overline{\\gamma_m}<\/span><\/span> must be of the form <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha\\rightarrow \\beta)<\/span><\/span>, but all these steps only involve elements of <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span> and \u0141ukasiewicz&#8217;s axioms in their deduction. Therefore, it must be that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma\\vdash (\\alpha \\rightarrow \\beta)<\/span><\/span>, thus proving the theorem.<\/p>\n<p>End of the proof.<\/p>\n<p><a name=\"8\"><\/a><\/p>\n<h2>The Reciprocal of the Deduction Theorem<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=1668s\" target=\"_blank\" rel=\"noopener\"><strong>Under the same conditions as the deduction theorem<\/strong><\/a>, it follows that:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\left(\\Gamma\\vdash(\\alpha \\rightarrow \\beta)\\right) \\Rightarrow \\left( \\Gamma \\cup \\{\\alpha\\}\\vdash \\beta \\right)\n\n<\/span>\n<p style=\"text-align: justify; color: #880000;\"><strong>Proof:<\/strong><\/p>\n<p style=\"text-align: justify;\">If it is satisfied that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma\\vdash (\\alpha\\rightarrow \\beta)<\/span><\/span>, then a deduction of the form is obtained:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_1<\/span><\/span><\/td>\n<td>; Premise 1 of <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>(n)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma_n<\/span><\/span><\/td>\n<td>; Premise n of <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span><\/td>\n<\/tr>\n<tr>\n<td>(n+1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow \\beta)<\/span><\/span><\/td>\n<td>; Modus Ponens between some previous steps<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Now, if we add <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> as a premise to this reasoning, we will have the following steps:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(n+2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/td>\n<td>; Additional premise<\/td>\n<\/tr>\n<tr>\n<td>(n+3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/td>\n<td>; MP(n+1,n+2)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\">Therefore, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\Gamma \\cup \\{\\alpha\\} \\vdash \\beta<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Which is what we wanted to prove.<\/p>\n<p>End of the proof.<\/p>\n<p><a name=\"9\"><\/a><\/p>\n<h2>Deductions on Expressions and Deductions on Deductions<\/h2>\n<p style=\"text-align: justify;\">Proofs like the one we previously performed to obtain the result <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\alpha\\rightarrow \\alpha)<\/span><\/span> are cases of deductions based on expressions, where each step contains a specific expression. Similarly, deductions can be made based on other deductions, where each step is itself a deduction. In practice, both are done analogously, but the latter allows us to use the deduction theorem and its reciprocal, providing great flexibility to reasoning techniques. To see this, let&#8217;s prove again that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\alpha \\rightarrow \\alpha)<\/span><\/span>, but this time using deductions instead of expressions. An alternative approach is as follows:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\alpha \\rightarrow (\\alpha \\rightarrow \\alpha))<\/span><\/span><\/td>\n<td>; A1<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\}\\vdash ( \\alpha \\rightarrow \\alpha)<\/span><\/span><\/td>\n<td>; RTD(1)<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\}\\cup \\{\\alpha\\}\\vdash \\alpha<\/span><\/span><\/td>\n<td>; RTD(2)<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\}\\vdash \\alpha<\/span><\/span><\/td>\n<td>; Note that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\alpha\\}\\cup\\{\\alpha\\}=\\{\\alpha\\}<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash (\\alpha\\rightarrow \\alpha)<\/span><\/span><\/td>\n<td>; TD(4)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Note that this reasoning is not shorter than the one we had done earlier, but it is much easier to perform; we only rely on the deduction theorem, its reciprocal, and the A1 axiom scheme to build the proof.<\/p>\n<p style=\"text-align: justify;\">It may seem that in the reasoning we just performed, we used only one of \u0141ukasiewicz&#8217;s axioms and ignored both the other axioms and modus ponens. Does this imply that by reasoning in this way, we forget the other axioms and modus ponens? The answer is both yes and no. On one hand, we can act as if we&#8217;ve forgotten some axioms and modus ponens simply because we are not explicitly using them; however, it must be remembered that both the deduction theorem and its reciprocal are direct consequences of \u0141ukasiewicz&#8217;s axioms and modus ponens, which implies that by using them, as we did in the reasoning above, we are implicitly using those axioms and modus ponens.<\/p>\n<p><a name=\"10\"><\/a><\/p>\n<h2>Monotonicity Rule<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=1972s\" target=\"_blank\" rel=\"noopener\"><strong>If <span class=\"katex-eq\" data-katex-display=\"false\">\\tau<\/span> is a theorem,<\/strong><\/a> then for any expression <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span>, it holds that<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\beta\\}\\vdash\\tau<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">This is a very easy rule to prove since <span class=\"katex-eq\" data-katex-display=\"false\">\\tau<\/span> being a theorem means that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash \\tau<\/span><\/span>. That is, there exists a reasoning that leads to the expression <span class=\"katex-eq\" data-katex-display=\"false\">\\tau<\/span> without needing any additional premises, so adding another expression to the premises (empty) makes no difference.<\/p>\n<p style=\"text-align: justify;\">Similarly, we can propose the following result: if from a set of premises <span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span> is inferred, then it holds that:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma\\cup\\{\\alpha\\}\\vdash\\gamma<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Where <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> is any expression.<\/p>\n<p><a name=\"11\"><\/a><\/p>\n<h2>Synthesis and Reflections on Deductive Systems and Propositional Logic<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=OvoEDefcSZg&amp;t=1933s\" target=\"_blank\" rel=\"noopener\"><strong>When we provide the language of propositional logic with an inference rule and basic expressions:<\/strong><\/a> Modus Ponens and \u0141ukasiewicz&#8217;s Axioms, we are doing something analogous to building a \u00abdeductive machine\u00bb and a \u00abmotor that powers it.\u00bb From here, all the basic deduction rules naturally emerge, and we will begin to review them in the lessons immediately following this one.<\/p>\n<p style=\"text-align: justify;\">One more detail. The expressions of propositional logic are, in fact, meta-expressions of the two-symbol language we saw earlier. Remember, the beauty of these meta-expressions lies in their ability to substitute their meta-variables with any expression of the language to obtain a new one that satisfies that structure. When we endow propositional logic language with axiomatic schemes and inference rules, we construct the Deductive Systems of propositional logic, allowing for deductions that connect expressions. The result is a deductive framework capable of encompassing infinite deductions: all those that can be obtained by substituting meta-variables with any expressions we choose. The power of logic truly unleashes when we realize that beyond the two-symbol language expressions we initially used, we see what happens when we substitute them with expressions from our usual language.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Formal Deductive Systems in Propositional Logic Summary:In this class, we review formal deductive systems. We explain how these systems are used to decipher the relationships that may exist between different logical expressions, and the basic elements with which these proofs are built: the language, the axioms, and the inference rules. We mention \u0141ukasiewicz&#8217;s axioms and [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":33793,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":10,"footnotes":""},"categories":[567,619],"tags":[],"class_list":["post-25245","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics","category-propositional-logic"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Formal Deductive Systems in Propositional Logic - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Explore the foundations of formal deductive systems in propositional logic, including axioms, inferences, and theorems\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/en\/formal-deductive-systems-in-propositional-logic\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Formal Deductive Systems in Propositional Logic\" \/>\n<meta property=\"og:description\" content=\"Explore the foundations of formal deductive systems in propositional logic, including axioms, inferences, and theorems\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/en\/formal-deductive-systems-in-propositional-logic\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-01-25T19:37:17+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-07-31T01:09:38+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/01\/sitemasdeductivos-2.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1536\" \/>\n\t<meta property=\"og:image:height\" content=\"698\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Formal Deductive Systems in Propositional Logic\" \/>\n<meta name=\"twitter:description\" content=\"Explore the foundations of formal deductive systems in propositional logic, including axioms, inferences, and theorems\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/01\/sitemasdeductivos-2.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"9 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/formal-deductive-systems-in-propositional-logic\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/formal-deductive-systems-in-propositional-logic\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Formal Deductive Systems in Propositional Logic\",\"datePublished\":\"2021-01-25T19:37:17+00:00\",\"dateModified\":\"2025-07-31T01:09:38+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/formal-deductive-systems-in-propositional-logic\\\/\"},\"wordCount\":2196,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/formal-deductive-systems-in-propositional-logic\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2021\\\/01\\\/sitemasdeductivos-2.jpg\",\"articleSection\":[\"Mathematics\",\"Propositional Logic\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/formal-deductive-systems-in-propositional-logic\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/formal-deductive-systems-in-propositional-logic\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/formal-deductive-systems-in-propositional-logic\\\/\",\"name\":\"Formal Deductive Systems in Propositional Logic - 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