Substitution is a fundamental operation performed on terms and formulas. It occurs in all inference systems in first-order logic. The substitution is complex in the presence of quantifiers in FOL. If we write F[a\/x]<\/strong>, so it refers to substitute a constant “a<\/strong>” in place of variable “x<\/strong>“.<\/p>\n\n\n\n Equality:<\/strong><\/p>\n\n\n\n First-Order logic does not only use predicate and terms for making atomic sentences but also uses another way, which is equality in FOL. For this, we can use equality symbols<\/strong> which specify that the two terms refer to the same object.<\/p>\n\n\n\n Example: Brother (John) = Smith.<\/strong><\/p>\n\n\n\n As in the above example, the object referred by the Brother (John)<\/strong> is similar to the object referred by Smith<\/strong>. The equality symbol can also be used with negation to represent that two terms are not the same objects.<\/p>\n\n\n\n Example: \uffe2(x=y) which is equivalent to x \u2260y.<\/strong><\/p>\n\n\n\n As propositional logic we also have inference rules in first-order logic, so following are some basic inference rules in FOL:<\/p>\n\n\n\n 1. Universal Generalization:<\/strong><\/p>\n\n\n\n Example:<\/strong> Let’s represent, P(c): “A byte contains 8 bits<\/strong>“, so for \u2200 x P(x)<\/strong> “All bytes contain 8 bits<\/strong>.”, it will also be true.<\/p>\n\n\n\n 2. Universal Instantiation:<\/strong><\/p>\n\n\n\n Example:1.<\/strong><\/p>\n\n\n\n IF “Every person like ice-cream”=> \u2200x P(x) so we can infer that Example: 2.<\/strong><\/p>\n\n\n\n Let’s take a famous example,<\/p>\n\n\n\n “All kings who are greedy are Evil.” So let our knowledge base contains this detail as in the form of FOL:<\/p>\n\n\n\n \u2200x king(x) \u2227 greedy (x) \u2192 Evil (x),<\/strong><\/p>\n\n\n\n So from this information, we can infer any of the following statements using Universal Instantiation:<\/p>\n\n\n\n 3. Existential Instantiation:<\/strong><\/p>\n\n\n\n Example:<\/strong><\/p>\n\n\n\n From the given sentence: \u2203x Crown(x) \u2227 OnHead(x, John),<\/strong><\/p>\n\n\n\n So we can infer: Crown(K) \u2227 OnHead( K, John),<\/strong> as long as K does not appear in the knowledge base.<\/p>\n\n\n\n 4. Existential introduction<\/strong><\/p>\n\n\n\n For the inference process in FOL, we have a single inference rule which is called Generalized Modus Ponens. It is lifted version of Modus ponens.<\/p>\n\n\n\n Generalized Modus Ponens can be summarized as, ” P implies Q and P is asserted to be true, therefore Q must be True.”<\/p>\n\n\n\n According to Modus Ponens, for atomic sentences pi, pi’, q<\/strong>. Where there is a substitution \u03b8 such that SUBST (\u03b8, pi’,) = SUBST(\u03b8, pi)<\/strong>, it can be represented as:<\/p>\n\n\n\n Example:<\/strong><\/p>\n\n\n\n We will use this rule for Kings are evil, so we will find some x such that x is king, and x is greedy so we can infer that x is evil.<\/strong><\/a><\/a><\/a><\/p>\n\n\n\n Inference in First-Order Logic is used to deduce new facts or sentences from existing sentences. Before understanding the FOL inference rule, let’s understand some basic terminologies used in FOL. Substitution: Substitution is a fundamental operation performed on terms and formulas. It occurs in all inference systems in first-order logic. The substitution is complex in the […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[885],"tags":[],"_links":{"self":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/2290"}],"collection":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/comments?post=2290"}],"version-history":[{"count":0,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/2290\/revisions"}],"wp:attachment":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/media?parent=2290"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/categories?post=2290"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/tags?post=2290"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Note: First-order logic is capable of expressing facts about some or all objects in the universe.<\/h4>\n\n\n\n
FOL inference rules for quantifier:<\/h2>\n\n\n\n
![\"Inference](\"https:\/\/static.javatpoint.com\/tutorial\/ai\/images\/ai-inference-in-first-order-logic.png\")
.<\/li>![\"Inference](\"https:\/\/static.javatpoint.com\/tutorial\/ai\/images\/ai-inference-in-first-order-logi2.png\")
.<\/li><\/ul>\n\n\n\n
“John likes ice-cream” => P(c)<\/p>\n\n\n\n![\"Inference](\"https:\/\/static.javatpoint.com\/tutorial\/ai\/images\/ai-inference-in-first-order-logic3.png\")
<\/li><\/ul>\n\n\n\n![\"Inference](\"https:\/\/static.javatpoint.com\/tutorial\/ai\/images\/ai-inference-in-first-order-logic4.png\")
<\/li>
“Priyanka got good marks in English.”
“Therefore, someone got good marks in English.”<\/li><\/ul>\n\n\n\nGeneralized Modus Ponens Rule:<\/h2>\n\n\n\n
![\"Inference](\"https:\/\/static.javatpoint.com\/tutorial\/ai\/images\/ai-inference-in-first-order-logic5.png\")
<\/figure>\n\n\n\n