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User:IssaRice/Computability and logic/Models symbol - Revision history
2026-07-14T19:56:01Z
Revision history for this page on the wiki
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https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Computability_and_logic/Models_symbol&diff=1437&oldid=prev
IssaRice: /* Notes */
2019-01-28T00:21:32Z
<p><span dir="auto"><span class="autocomment">Notes</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 00:21, 28 January 2019</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Some books only use the models symbol for one of the two use cases. E.g. Boolos/Burgess/Jeffrey only uses the symbol for truth in an interpretation. Therefore you might be really confused when you start reading other books and you start seeing stuff like <math>\Gamma \models \phi</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Some books only use the models symbol for one of the two use cases. E.g. Boolos/Burgess/Jeffrey only uses the symbol for truth in an interpretation. Therefore you might be really confused when you start reading other books and you start seeing stuff like <math>\Gamma \models \phi</math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* In the Boolos/Burgess/Jeffrey book, if a set of sentences comes after <math>\models</math>, then the sentences are taken disjunctively rather than conjunctively. (see p. 168)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* In the Boolos/Burgess/Jeffrey book, if a set of sentences comes after <math>\models</math>, then the sentences are taken disjunctively rather than conjunctively. (see p. 168)</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* https://math.stackexchange.com/a/2506938/35525</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==See also==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==See also==</div></td></tr>
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IssaRice
https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Computability_and_logic/Models_symbol&diff=1436&oldid=prev
IssaRice at 21:39, 27 January 2019
2019-01-27T21:39:50Z
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>! Part before "<math>\models</math>" !! <math>\models</math> !! Part after "<math>\models</math>" !! Possible pronunciations !! Meaning</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>! Part before "<math>\models</math>" !! <math>\models</math> !! Part after "<math>\models</math>" !! Possible pronunciations !! Meaning</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| A structure/interpretation <math>\mathfrak A</math> || <math>\models</math> || A sentence or formula <math>\phi</math> || The structure <math>\mathfrak A</math> satisfies the formula <math>\phi</math>.<ref name="goldrei">Derek Goldrei. ''Propositional and Predicate Calculus''. p. 134.</ref><br>The formula <math>\phi</math> is true in <math>\mathfrak A</math>.<ref name="goldrei" /> ||</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| A structure/interpretation <math>\mathfrak A</math> || <math>\models</math> || A sentence or formula <math>\phi</math> || The structure <math>\mathfrak A</math> satisfies the formula <math>\phi</math>.<ref name="goldrei">Derek Goldrei. ''Propositional and Predicate Calculus''. p. 134.</ref><br>The formula <math>\phi</math> is true in <math>\mathfrak A</math>.<ref name="goldrei" /><ins style="font-weight: bold; text-decoration: none;"><br>The structure <math>\mathfrak A</math> makes true the formula <math>\phi</math>. </ins>||</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| A set of sentences or formulas <math>\Gamma</math> || <math>\models</math> || A sentence or formula <math>\phi</math> || <math>\phi</math> is a logical consequence of <math>\Gamma</math>.<br><math>\Gamma</math> logically implies <math>\phi</math>.<br><math>\phi</math> is a semantic consequence of <math>\Gamma</math>.<br><math>\phi</math> is true in every model of <math>\Gamma</math>. ||</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| A set of sentences or formulas <math>\Gamma</math> || <math>\models</math> || A sentence or formula <math>\phi</math> || <math>\phi</math> is a logical consequence of <math>\Gamma</math>.<br><math>\Gamma</math> logically implies <math>\phi</math>.<br><math>\phi</math> is a semantic consequence of <math>\Gamma</math>.<br><math>\phi</math> is true in every model of <math>\Gamma</math>. ||</div></td></tr>
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IssaRice
https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Computability_and_logic/Models_symbol&diff=1435&oldid=prev
IssaRice at 21:38, 27 January 2019
2019-01-27T21:38:36Z
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>! Part before "<math>\models</math>" !! <math>\models</math> !! Part after "<math>\models</math>" !! Possible pronunciations !! Meaning</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>! Part before "<math>\models</math>" !! <math>\models</math> !! Part after "<math>\models</math>" !! Possible pronunciations !! Meaning</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| A structure/interpretation <math>\mathfrak A</math> || <math>\models</math> || A sentence or formula <math>\phi</math> || The structure <math>\mathfrak A</math> satisfies the formula <math>\phi</math>.<ref name="goldrei">Derek Goldrei. ''Propositional and Predicate Calculus''. p. 134.</ref<del style="font-weight: bold; text-decoration: none;">><br</del>><br>The formula <math>\phi</math> is true in <math>\mathfrak A</math>.<ref name="goldrei" /> ||</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| A structure/interpretation <math>\mathfrak A</math> || <math>\models</math> || A sentence or formula <math>\phi</math> || The structure <math>\mathfrak A</math> satisfies the formula <math>\phi</math>.<ref name="goldrei">Derek Goldrei. ''Propositional and Predicate Calculus''. p. 134.</ref><br>The formula <math>\phi</math> is true in <math>\mathfrak A</math>.<ref name="goldrei" /> ||</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| A set of sentences or formulas <math>\Gamma</math> || <math>\models</math> || A sentence or formula <math>\phi</math> || <math>\phi</math> is a logical consequence of <math>\Gamma</math>.<del style="font-weight: bold; text-decoration: none;"><br></del><br><math>\Gamma</math> logically implies <math>\phi</math>.<del style="font-weight: bold; text-decoration: none;"><br></del><br><math>\phi</math> is a semantic consequence of <math>\Gamma</math>.<del style="font-weight: bold; text-decoration: none;"><br></del><br><math>\phi</math> is true in every model of <math>\Gamma</math>. ||</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| A set of sentences or formulas <math>\Gamma</math> || <math>\models</math> || A sentence or formula <math>\phi</math> || <math>\phi</math> is a logical consequence of <math>\Gamma</math>.<br><math>\Gamma</math> logically implies <math>\phi</math>.<br><math>\phi</math> is a semantic consequence of <math>\Gamma</math>.<br><math>\phi</math> is true in every model of <math>\Gamma</math>. ||</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| Nothing || <math>\models</math> || A sentence or formula <math>\phi</math> || <math>\phi</math> is valid.<ref>Boolos; Burgess; Jeffrey. ''Computability and Logic''. p. 168.</ref<del style="font-weight: bold; text-decoration: none;">><br</del>><br><math>\phi</math> is a tautology (especially in the case of propositional logic). ||</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| Nothing || <math>\models</math> || A sentence or formula <math>\phi</math> || <math>\phi</math> is valid.<ref>Boolos; Burgess; Jeffrey. ''Computability and Logic''. p. 168.</ref><br><math>\phi</math> is a tautology (especially in the case of propositional logic). ||</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td></tr>
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IssaRice
https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Computability_and_logic/Models_symbol&diff=1434&oldid=prev
IssaRice: /* Notes */
2019-01-27T21:34:27Z
<p><span dir="auto"><span class="autocomment">Notes</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:34, 27 January 2019</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Some books only use the models symbol for one of the two use cases. E.g. Boolos/Burgess/Jeffrey only uses the symbol for truth in an interpretation. Therefore you might be really confused when you start reading other books and you start seeing stuff like <math>\Gamma \models \phi</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Some books only use the models symbol for one of the two use cases. E.g. Boolos/Burgess/Jeffrey only uses the symbol for truth in an interpretation. Therefore you might be really confused when you start reading other books and you start seeing stuff like <math>\Gamma \models \phi</math>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* In the Boolos/Burgess/Jeffrey book, if a set of sentences comes after <math>\models</math>, then the sentences are taken disjunctively rather than conjunctively. (see p. 168)</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==See also==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==See also==</div></td></tr>
</table>
IssaRice
https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Computability_and_logic/Models_symbol&diff=1433&oldid=prev
IssaRice at 21:29, 27 January 2019
2019-01-27T21:29:41Z
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| A set of sentences or formulas <math>\Gamma</math> || <math>\models</math> || A sentence or formula <math>\phi</math> || <math>\phi</math> is a logical consequence of <math>\Gamma</math>.<br><br><math>\Gamma</math> logically implies <math>\phi</math>.<br><br><math>\phi</math> is a semantic consequence of <math>\Gamma</math>.<br><br><math>\phi</math> is true in every model of <math>\Gamma</math>. ||</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| A set of sentences or formulas <math>\Gamma</math> || <math>\models</math> || A sentence or formula <math>\phi</math> || <math>\phi</math> is a logical consequence of <math>\Gamma</math>.<br><br><math>\Gamma</math> logically implies <math>\phi</math>.<br><br><math>\phi</math> is a semantic consequence of <math>\Gamma</math>.<br><br><math>\phi</math> is true in every model of <math>\Gamma</math>. ||</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">|-</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| Nothing || <math>\models</math> || A sentence or formula <math>\phi</math> || <math>\phi</math> is valid.<ref>Boolos; Burgess; Jeffrey. ''Computability and Logic''. p. 168.</ref><br><br><math>\phi</math> is a tautology (especially in the case of propositional logic). ||</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td></tr>
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IssaRice
https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Computability_and_logic/Models_symbol&diff=1432&oldid=prev
IssaRice at 21:26, 27 January 2019
2019-01-27T21:26:07Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:26, 27 January 2019</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| A set of sentences or formulas <math>\Gamma</math> || <math>\models</math> || A sentence or formula <math>\phi</math> || <math>\phi</math> is a logical consequence of <math>\Gamma</math>.<br><br><math>\Gamma</math> logically implies <math>\phi</math>.<br><br><math>\phi</math> is a semantic consequence of <math>\Gamma</math>.<br><br><math>\phi</math> is true in every model of <math>\Gamma</math>. ||</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| A set of sentences or formulas <math>\Gamma</math> || <math>\models</math> || A sentence or formula <math>\phi</math> || <math>\phi</math> is a logical consequence of <math>\Gamma</math>.<br><br><math>\Gamma</math> logically implies <math>\phi</math>.<br><br><math>\phi</math> is a semantic consequence of <math>\Gamma</math>.<br><br><math>\phi</math> is true in every model of <math>\Gamma</math>. ||</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Notes==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Other tricky things:</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* Some books only use the models symbol for one of the two use cases. E.g. Boolos/Burgess/Jeffrey only uses the symbol for truth in an interpretation. Therefore you might be really confused when you start reading other books and you start seeing stuff like <math>\Gamma \models \phi</math>.</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==See also==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==See also==</div></td></tr>
</table>
IssaRice
https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Computability_and_logic/Models_symbol&diff=1431&oldid=prev
IssaRice at 21:20, 27 January 2019
2019-01-27T21:20:55Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:20, 27 January 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l6">Line 6:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In either of the above two purposes, we are talking about the semantics (rather than syntax) of a logical system.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In either of the above two purposes, we are talking about the semantics (rather than syntax) of a logical system.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* If <math>\mathfrak A</math> is a structure/interpretation and <math>\phi</math> is a sentence, then <math>\mathfrak A \models \phi</math> means ...</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* If <math>\mathfrak A</math> is a structure/interpretation and <math>\Gamma</math> is a set of sentences, then <math>\mathfrak A \models \Gamma</math> means ...</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* If <math>\mathfrak A</math> is a structure/interpretation and <math>\Gamma</math> is a set of sentences, then <math>\mathfrak A \models \Gamma</math> means ...</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* If <math>T</math> is a theory and <math>\phi</math> is a sentence, then <math>T \models \phi</math> means ...</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* If <math>T</math> is a theory and <math>\phi</math> is a sentence, then <math>T \models \phi</math> means ...</div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{| class="sortable wikitable"</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{| class="sortable wikitable"</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>! Part before "<math>\models</math>" !! <math>\models</math> !! Part after <math>\models</math> !! Possible pronunciations !! Meaning</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>! Part before "<math>\models</math>" !! <math>\models</math> !! Part after <ins style="font-weight: bold; text-decoration: none;">"</ins><math>\models</math><ins style="font-weight: bold; text-decoration: none;">" </ins>!! Possible pronunciations !! Meaning</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| A structure/interpretation <math>\mathfrak A</math> || <math>\models</math> || A sentence or formula <math>\phi</math> || The structure <math>\mathfrak A</math> satisfies the formula <math>\phi</math>.<ref name="goldrei">Derek Goldrei. ''Propositional and Predicate Calculus''. p. 134.</ref><br><br>The formula <math>\phi</math> is true in <math>\mathfrak A</math>.<ref name="goldrei" /> ||</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| A structure/interpretation <math>\mathfrak A</math> || <math>\models</math> || A sentence or formula <math>\phi</math> || The structure <math>\mathfrak A</math> satisfies the formula <math>\phi</math>.<ref name="goldrei">Derek Goldrei. ''Propositional and Predicate Calculus''. p. 134.</ref><br><br>The formula <math>\phi</math> is true in <math>\mathfrak A</math>.<ref name="goldrei" /> ||</div></td></tr>
</table>
IssaRice
https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Computability_and_logic/Models_symbol&diff=1430&oldid=prev
IssaRice at 21:19, 27 January 2019
2019-01-27T21:19:33Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:19, 27 January 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l3">Line 3:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># When the symbol that comes before "<math>\models</math>" is a structure/interpretation, then it says something about truth in that structure/interpretation.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># When the symbol that comes before "<math>\models</math>" is a structure/interpretation, then it says something about truth in that structure/interpretation.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># When the symbol that comes before "<math>\models</math>" is a sentence or set of sentences, then it says something about semantic consequence (also called logical consequence, logical implication, semantic implication). In this case, we are talking about all possible structures/interpretations.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># When the symbol that comes before "<math>\models</math>" is a sentence or set of sentences, then it says something about semantic consequence (also called logical consequence, logical implication, semantic implication). In this case, we are talking about all possible structures/interpretations.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In either of the above two purposes, we are talking about the semantics (rather than syntax) of a logical system.</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* If <math>\mathfrak A</math> is a structure/interpretation and <math>\phi</math> is a sentence, then <math>\mathfrak A \models \phi</math> means ...</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* If <math>\mathfrak A</math> is a structure/interpretation and <math>\phi</math> is a sentence, then <math>\mathfrak A \models \phi</math> means ...</div></td></tr>
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IssaRice
https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Computability_and_logic/Models_symbol&diff=1429&oldid=prev
IssaRice at 21:15, 27 January 2019
2019-01-27T21:15:54Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:15, 27 January 2019</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| A structure/interpretation <math>\mathfrak A</math> || <math>\models</math> || A sentence or formula <math>\phi</math> || The structure <math>\mathfrak A</math> satisfies the formula <math>\phi</math>.<ref name="goldrei">Derek Goldrei. ''Propositional and Predicate Calculus''. p. 134.</ref><br><br>The formula <math>\phi</math> is true in <math>\mathfrak A</math>.<ref name="goldrei" /> ||</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| A structure/interpretation <math>\mathfrak A</math> || <math>\models</math> || A sentence or formula <math>\phi</math> || The structure <math>\mathfrak A</math> satisfies the formula <math>\phi</math>.<ref name="goldrei">Derek Goldrei. ''Propositional and Predicate Calculus''. p. 134.</ref><br><br>The formula <math>\phi</math> is true in <math>\mathfrak A</math>.<ref name="goldrei" /> ||</div></td></tr>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| A set of sentences or formulas <math>\Gamma</math> || <math>\models</math> || A sentence or formula <math>\phi</math> || <math>\phi</math> is a logical consequence of <math>\Gamma</math>.<br><br><math>\Gamma</math> logically implies <math>\phi</math>.<br><br><math>\phi</math> is a semantic consequence of <math>\Gamma</math>. ||</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| A set of sentences or formulas <math>\Gamma</math> || <math>\models</math> || A sentence or formula <math>\phi</math> || <math>\phi</math> is a logical consequence of <math>\Gamma</math>.<br><br><math>\Gamma</math> logically implies <math>\phi</math>.<br><br><math>\phi</math> is a semantic consequence <ins style="font-weight: bold; text-decoration: none;">of <math>\Gamma</math>.<br><br><math>\phi</math> is true in every model </ins>of <math>\Gamma</math>. ||</div></td></tr>
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IssaRice
https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Computability_and_logic/Models_symbol&diff=1428&oldid=prev
IssaRice at 21:15, 27 January 2019
2019-01-27T21:15:13Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:15, 27 January 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The "models" symbol, <math>\models</math>, is used for several purposes in mathematical logic.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The "models" symbol, <math>\models</math>, is used for several purposes in mathematical logic<ins style="font-weight: bold; text-decoration: none;">. Roughly, there are two basic purposes:</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"># When the symbol that comes before "<math>\models</math>" is a structure/interpretation, then it says something about truth in that structure/interpretation.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"># When the symbol that comes before "<math>\models</math>" is a sentence or set of sentences, then it says something about semantic consequence (also called logical consequence, logical implication, semantic implication). In this case, we are talking about all possible structures/interpretations</ins>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* If <math>\mathfrak A</math> is a structure/interpretation and <math>\phi</math> is a sentence, then <math>\mathfrak A \models \phi</math> means ...</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* If <math>\mathfrak A</math> is a structure/interpretation and <math>\phi</math> is a sentence, then <math>\mathfrak A \models \phi</math> means ...</div></td></tr>
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IssaRice