Anki Deck Changes

Note 1: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: Lg8Zv7Tp4J

modified

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Front

An interpretation or structure in predicate logic is a tuple \(\mathcal{A} = (U, \phi, \varphi, \xi)\) where:
- \(U\) is a non-empty universe
- \(\phi\) assigns function symbols to functions \(U^k \rightarrow U\)
- {{c3::\(\varphi\) assigns predicate symbols to functions \(U^k \rightarrow \{0,1\}\)}}
- \(\xi\) assigns variable symbols to values in \(U\)

Back

After

Front

An interpretation or structure in predicate logic is a tuple \(\mathcal{A} = (U, \phi, \varphi, \xi)\) where:
- \(U\) is a non-empty universe
- \(\phi\) (phi) assigns function symbols to functions \(U^k \rightarrow U\)
- {{c3::\(\psi\) (psi) assigns predicate symbols to functions \(U^k \rightarrow \{0,1\}\)}}
- \(\xi\) (xi) assigns variable symbols to values in \(U\)

Back

Field-by-field Comparison

Field	Before	After
Text	An <i>interpretation</i> or <i>structure</i> in predicate logic is a tuple \(\mathcal{A} = (U, \phi, \varphi, \xi)\) where:<br>- {{c1::\(U\) is a <b>non-empty</b> universe}}<br>- {{c2::\(\phi\) assigns function symbols to functions \(U^k \rightarrow U\)}}<br>- {{c3::\(\~~varph~~i\) assigns predicate symbols to functions \(U^k \rightarrow \{0,1\}\)}}<br>- {{c4::\(\xi\) assigns variable symbols to values in \(U\)}}	An <i>interpretation</i> or <i>structure</i> in predicate logic is a tuple \(\mathcal{A} = (U, \phi, \varphi, \xi)\) where:<br>- {{c1::\(U\) is a <b>non-empty</b> universe}}<br>- {{c2::\(\phi\) (phi) assigns function symbols to functions \(U^k \rightarrow U\)}}<br>- {{c3::\(\psi\) (psi) assigns predicate symbols to functions \(U^k \rightarrow \{0,1\}\)}}<br>- {{c4::\(\xi\) (xi) assigns variable symbols to values in \(U\)}}

Tags: ETH::1._Semester::DiskMat::6._Logic::6._Predicate_Logic_(First-order_Logic)::3._Semantics

Note 2: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: Qn4Vs7Ck2H

modified

Before

Front

An interpretation consists of {{c1::a set \(\mathcal{Z} \subseteq \Lambda\) of \(\Lambda\)}}, {{c2::a domain (a set of possible values) for each symbol in \(\mathcal{Z}\)}}, and {{c3::a function that assigns to each symbol in \(\mathcal{Z}\) a value in the associated domain}}.

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Often the domain is defined in terms of the universe \(U\) where a symbol can be a function, predicate or element of \(U\).

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Often the domain is defined in terms of the universe \(U\) where a symbol can be a function, predicate or element of \(U\).

A set of symbols \(\mathcal{Z} \subseteq \Lambda\)
- \(\Lambda\) is the "alphabet" or collection of all available symbols
- \(\mathcal{Z}\) is the subset of symbols we're actually interpreting
A domain for each symbol
- For each symbol in \(\mathcal{Z}\), there's a set of possible values it could take
- This is the "universe of discourse" for that symbol
An assignment function
- For each symbol in \(\mathcal{Z}\), the function picks one specific value from its domain
- This gives meaning to each symbol

Field-by-field Comparison

Field	Before	After
Extra	Often the domain is defined in terms of the <i>universe</i> \(U\) where a symbol can be a function, predicate or element of \(U\).	Often the domain is defined in terms of the <i>universe</i> \(U\) where a symbol can be a function, predicate or element of \(U\).<br><ol><li><strong>A set of symbols</strong> \(\mathcal{Z} \subseteq \Lambda\)<ul> <li>\(\Lambda\) is the "alphabet" or collection of all available symbols </li> <li>\(\mathcal{Z}\) is the subset of symbols we're actually interpreting </li> </ul> </li> <li><strong>A domain for each symbol</strong> <ul> <li>For each symbol in \(\mathcal{Z}\), there's a set of possible values it could take </li> <li>This is the "universe of discourse" for that symbol</li> </ul> </li> <li><strong>An assignment function</strong> <ul> <li>For each symbol in \(\mathcal{Z}\), the function picks one specific value from its domain </li> <li>This gives meaning to each symbol</li> </ul> </li> <h2></h2></ol>

Tags: ETH::1._Semester::DiskMat::6._Logic::3._Elementary_General_Concepts_in_Logic::3._Semantics::Interpretation