Substitutability in First-Order Logic











up vote
0
down vote

favorite












Here is a definition for substitutability found in a PDF of logic notes by Eric Pacuit:



enter image description here



I am more concerned with the part of the definition squared in red. My question is: Given $(forall y) psi$, does this mean that, since $y$ does not occur free in $(forall y) psi$, is $tau$ substitutable for $y$ in $(forall y) psi$?






logic first-order-logic substitution







share|cite|improve this question


















  • 1




    Yes, because the term substitutable means that performing the operation we "cause no troubles". In that case, the operation is allowed simply because it does not change the original formula.
    – Mauro ALLEGRANZA
    Nov 18 at 12:49










  • @MauroALLEGRANZA I found another definition for substitutability from the textbook by Arindama Singh: $y$ is free for $x$ in $phi$ iff $x$ does not occur free within the scope of any $forall y$ or $exists y$. Is the negation: $y$ is not free for $x$ in $phi$ iff $x$ occurs free within the scope of any $forall y$ or $exists y$?
    – numericalorange
    Nov 19 at 4:53















up vote
0
down vote

favorite












Here is a definition for substitutability found in a PDF of logic notes by Eric Pacuit:



enter image description here



I am more concerned with the part of the definition squared in red. My question is: Given $(forall y) psi$, does this mean that, since $y$ does not occur free in $(forall y) psi$, is $tau$ substitutable for $y$ in $(forall y) psi$?






logic first-order-logic substitution







share|cite|improve this question


















  • 1




    Yes, because the term substitutable means that performing the operation we "cause no troubles". In that case, the operation is allowed simply because it does not change the original formula.
    – Mauro ALLEGRANZA
    Nov 18 at 12:49










  • @MauroALLEGRANZA I found another definition for substitutability from the textbook by Arindama Singh: $y$ is free for $x$ in $phi$ iff $x$ does not occur free within the scope of any $forall y$ or $exists y$. Is the negation: $y$ is not free for $x$ in $phi$ iff $x$ occurs free within the scope of any $forall y$ or $exists y$?
    – numericalorange
    Nov 19 at 4:53













up vote
0
down vote

favorite









up vote
0
down vote

favorite











Here is a definition for substitutability found in a PDF of logic notes by Eric Pacuit:



enter image description here



I am more concerned with the part of the definition squared in red. My question is: Given $(forall y) psi$, does this mean that, since $y$ does not occur free in $(forall y) psi$, is $tau$ substitutable for $y$ in $(forall y) psi$?






logic first-order-logic substitution







share|cite|improve this question













Here is a definition for substitutability found in a PDF of logic notes by Eric Pacuit:



enter image description here



I am more concerned with the part of the definition squared in red. My question is: Given $(forall y) psi$, does this mean that, since $y$ does not occur free in $(forall y) psi$, is $tau$ substitutable for $y$ in $(forall y) psi$?






logic first-order-logic substitution




logic first-order-logic substitution






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 18 at 4:42









numericalorange

1,619311




1,619311








  • 1




    Yes, because the term substitutable means that performing the operation we "cause no troubles". In that case, the operation is allowed simply because it does not change the original formula.
    – Mauro ALLEGRANZA
    Nov 18 at 12:49










  • @MauroALLEGRANZA I found another definition for substitutability from the textbook by Arindama Singh: $y$ is free for $x$ in $phi$ iff $x$ does not occur free within the scope of any $forall y$ or $exists y$. Is the negation: $y$ is not free for $x$ in $phi$ iff $x$ occurs free within the scope of any $forall y$ or $exists y$?
    – numericalorange
    Nov 19 at 4:53














  • 1




    Yes, because the term substitutable means that performing the operation we "cause no troubles". In that case, the operation is allowed simply because it does not change the original formula.
    – Mauro ALLEGRANZA
    Nov 18 at 12:49










  • @MauroALLEGRANZA I found another definition for substitutability from the textbook by Arindama Singh: $y$ is free for $x$ in $phi$ iff $x$ does not occur free within the scope of any $forall y$ or $exists y$. Is the negation: $y$ is not free for $x$ in $phi$ iff $x$ occurs free within the scope of any $forall y$ or $exists y$?
    – numericalorange
    Nov 19 at 4:53








1




1




Yes, because the term substitutable means that performing the operation we "cause no troubles". In that case, the operation is allowed simply because it does not change the original formula.
– Mauro ALLEGRANZA
Nov 18 at 12:49




Yes, because the term substitutable means that performing the operation we "cause no troubles". In that case, the operation is allowed simply because it does not change the original formula.
– Mauro ALLEGRANZA
Nov 18 at 12:49












@MauroALLEGRANZA I found another definition for substitutability from the textbook by Arindama Singh: $y$ is free for $x$ in $phi$ iff $x$ does not occur free within the scope of any $forall y$ or $exists y$. Is the negation: $y$ is not free for $x$ in $phi$ iff $x$ occurs free within the scope of any $forall y$ or $exists y$?
– numericalorange
Nov 19 at 4:53




@MauroALLEGRANZA I found another definition for substitutability from the textbook by Arindama Singh: $y$ is free for $x$ in $phi$ iff $x$ does not occur free within the scope of any $forall y$ or $exists y$. Is the negation: $y$ is not free for $x$ in $phi$ iff $x$ occurs free within the scope of any $forall y$ or $exists y$?
– numericalorange
Nov 19 at 4:53










1 Answer
1






active

oldest

votes

















up vote
1
down vote



accepted










By definition, the answer is yes, $tau$ is substitutable for $y$ in $(forall y)phi$. This happens because there is nothing that to substitute and, therefore, you not change the original formula.






share|cite|improve this answer





















    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "69"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














     

    draft saved


    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3003141%2fsubstitutability-in-first-order-logic%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    1
    down vote



    accepted










    By definition, the answer is yes, $tau$ is substitutable for $y$ in $(forall y)phi$. This happens because there is nothing that to substitute and, therefore, you not change the original formula.






    share|cite|improve this answer

























      up vote
      1
      down vote



      accepted










      By definition, the answer is yes, $tau$ is substitutable for $y$ in $(forall y)phi$. This happens because there is nothing that to substitute and, therefore, you not change the original formula.






      share|cite|improve this answer























        up vote
        1
        down vote



        accepted







        up vote
        1
        down vote



        accepted






        By definition, the answer is yes, $tau$ is substitutable for $y$ in $(forall y)phi$. This happens because there is nothing that to substitute and, therefore, you not change the original formula.






        share|cite|improve this answer












        By definition, the answer is yes, $tau$ is substitutable for $y$ in $(forall y)phi$. This happens because there is nothing that to substitute and, therefore, you not change the original formula.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 18 at 5:38









        Gödel

        1,351319




        1,351319






























             

            draft saved


            draft discarded



















































             


            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3003141%2fsubstitutability-in-first-order-logic%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            Ellipse (mathématiques)

            Image
            Pour les articles homonymes, voir Ellipse. L'ellipse est le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante. Section du cône ou projection du cercle. En géométrie, une ellipse est une courbe plane fermée obtenue par l’intersection d’un cône de révolution avec un plan, à condition que celui-ci coupe l'axe de rotation du cône ou du cylindre : c'est une conique d'excentricité strictement comprise entre 0 et 1. On peut également la définir comme le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante (sa construction par la méthode du jardinier est très simple). Dans la vie courante, l’ellipse est la forme qu'on perçoit en regardant un cercle en perspective, ou la figure formée par l’ombre d'un disque sur une surface plane. On retrouve aussi, en première approximation [ 1 ] , des ellipses dans les trajectoires des corps célestes (planètes, comètes ou satellites arti...
            Read more

            Quarter-circle Tiles

            Image
            up vote 0 down vote favorite I have $99$ identical square tiles, each with a quarter-circle drawn on it like this: [asy] size(1.5cm); draw(Arc((2,0),1,90,180),red+1); draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)); [/asy] When I arrange the tiles in a $9times 11$ rectangular grid, each with a random orientation, what is the expected value of the number of full circles I form? I think this problem has to do with finding the chance any given 2x2 square has a circle, but I can't find it. expected-value share | cite | improve this question asked Nov 20 at 15:03 6minecraftninja 1 2 ...
            Read more

            Mont Emei

            Image
            Mont Emei.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Vue du sommet Wanfo du mont Emei Géographie Altitude 3 099  m , Wanfo Coordonnées 29° 31′ 11″ nord, 103° 19′ 57″ est Administration Pays   Chine Province Sichuan Ville-préfecture Leshan Géolocalisation sur la carte : Sichuan Mont Emei Géolocalisation sur la carte : Chine Mont Emei modifier   Paysage panoramique du mont Emei, incluant le paysage panoramique du grand Bouddha de Leshan *   Patrimoine mondial de l'UNESCO Passerelle en bois sur le versant ouest Pays   Chine Subdivision Sichuan Type Mixte Critères (iv) (vi) (x) Superficie 15 400 ha Numéro d’identification 779 Zone géographique Asie et Pacifique  ** Année d’inscription 1996 (20 e session) * Descriptif officiel UNESCO ** Classification géog...
            Read more