Discussion:
Forming a Bridge between Proof Theory and Model Theory
(too old to reply)
Pete Olcott
2017-12-20 18:24:19 UTC
Permalink
Raw Message
<html>
<head>

<meta http-equiv="content-type" content="text/html; charset=utf-8">
</head>
<body text="#000000" bgcolor="#FFFFFF">
<a
href="https://en.wikipedia.org/wiki/Logical_consequence#Syntactic_consequence"
target="_blank" rel="noopener"
data-mce-href="https://en.wikipedia.org/wiki/Logical_consequence#Syntactic_consequence"
style="box-shadow: currentcolor 0px 1px 0px 0px; color: rgb(0,
122, 204); text-decoration: none;">Syntactic versus Semantic
Logical Consequence</a><br>
Formalizing the internal semantic meaning of propositional
(sentential) variables using <u>Meaning Postulates</u> Rudolf
Carnap (1952) bridges the gap between syntactic consequence (formal
proof) and semantic consequence (logical entailment). <br>
<br>
<a
href="http://www.thatmarcusfamily.org/philosophy/Course_Websites/Readings/Carnap%20-%20Meaning%20Postulates.pdf"
target="_blank" rel="noopener"
data-mce-href="http://www.thatmarcusfamily.org/philosophy/Course_Websites/Readings/Carnap%20-%20Meaning%20Postulates.pdf"
style="box-shadow: currentcolor 0px 1px 0px 0px; color: rgb(0,
122, 204); text-decoration: none;">Meaning Postulates Rudolf
Carnap (1952)</a><br>
Meaning postulates specify semantic logical entailment
syntactically.<br>
The best two examples from his paper: Bachelor(x) and Warmer(x,y):<br>
<br>
Bachelor(x) → ~Married(x)<br>
<br>
For example, let 'W' be a primitive predicate designating the
relation Warmer. Then 'W' is transitive, irreflexive, and hence
asymmetric in virtue of its meaning.<br>
<br>
In the previous example of the predicate 'W', we could lay down the
following postulates (a) for transitivity and (b) for irreflexivity;
then the statement (c) of asymmetry:<br>
(a)  ∀(x,y,z) Warmer(x,y) ∧ Warmer(y,z) → Warmer(x,z)<br>
(b)  ∀(x)      ~Warmer(x,x)<br>
(c)  ∀(X,Y)    Warmer(x,y) → ~( Warmer(y,x) )<br>
<br>
Copyright 2017 Pete Olcott
<div class="moz-signature">-- <br>
<meta charset="UTF-8">
<b>      Γ ⊢<sub><font style="font-size:8pt" size="1">FS</font></sub>
A   ↔   ∃Γ ⊆ FS Provable(Γ, A)</b>
    // MTT notational conventions <br>
<b>∀X True(X)   ≡   ∃Γ ⊆ MTT Provable(Γ, X)   </b> // MTT Truth
Predicate </div>
</body>
</html>
Pete Olcott
2017-12-20 18:57:47 UTC
Permalink
Raw Message
On 12/20/2017 12:24 PM, Pete Olcott wrote:

Syntactic versus Semantic Logical Consequence -
https://en.wikipedia.org/wiki/Logical_consequence#Syntactic_consequence

Formalizing the internal semantic meaning of propositional (sentential) variables using Meaning Postulates Rudolf Carnap (1952) bridges the gap between syntactic consequence (formal proof) and semantic consequence (logical entailment). -

Meaning Postulates Rudolf Carnap (1952) -
http://www.thatmarcusfamily.org/philosophy/Course_Websites/Readings/Carnap%20-%20Meaning%20Postulates.pdf

Meaning postulates specify semantic logical entailment syntactically. -
The best two examples from his paper: Bachelor(x) and Warmer(x,y): -

Bachelor(x) → ~Married(x) -

For example, let 'W' be a primitive predicate designating the relation Warmer. Then 'W' is transitive, irreflexive, and hence asymmetric in virtue of its meaning. -

In the previous example of the predicate 'W', we could lay down the following postulates (a) for transitivity and (b) for irreflexivity; then the statement (c) of asymmetry: -

(a) ∀(x,y,z) Warmer(x,y) ∧ Warmer(y,z) → Warmer(x,z) -
(b) ∀(x) ~Warmer(x,x) -
(c) ∀(x,y) Warmer(x,y) → ~( Warmer(y,x) ) -

Copyright 2017 Pete Olcott -

Posted again for text only news readers. Dashes to prevent Google hiding.
--
*      Γ ⊢_FS A   ↔   ∃Γ ⊆ FS Provable(Γ, A)*     // MTT notational conventions
*∀X True(X)   ≡   ∃Γ ⊆ MTT Provable(Γ, X)   * // MTT Truth Predicate
Pete Olcott
2017-12-21 00:20:52 UTC
Permalink
Raw Message
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
</head>
<body text="#000000" bgcolor="#FFFFFF">
<div class="moz-cite-prefix">On 12/20/2017 12:57 PM, Pete Olcott
wrote:<br>
</div>
<blockquote type="cite"
cite="mid:x4WdnVVSTOI2L6fHnZ2dnUU7-***@giganews.com">On
12/20/2017 12:24 PM, Pete Olcott wrote:
<br>
<br>
Syntactic versus Semantic Logical Consequence -
<br>
<a class="moz-txt-link-freetext" href="https://en.wikipedia.org/wiki/Logical_consequence#Syntactic_consequence">https://en.wikipedia.org/wiki/Logical_consequence#Syntactic_consequence</a>
<br>
<br>
Formalizing the internal semantic meaning of propositional
(sentential) variables using Meaning Postulates Rudolf Carnap
(1952) bridges the gap between syntactic consequence (formal
proof) and semantic consequence (logical entailment).  -
<br>
<br>
Meaning Postulates Rudolf Carnap (1952) -
<br>
<a class="moz-txt-link-freetext" href="http://www.thatmarcusfamily.org/philosophy/Course_Websites/Readings/Carnap%20-%20Meaning%20Postulates.pdf">http://www.thatmarcusfamily.org/philosophy/Course_Websites/Readings/Carnap%20-%20Meaning%20Postulates.pdf</a>
<br>
<br>
Meaning postulates specify semantic logical entailment
syntactically. -
<br>
The best two examples from his paper: Bachelor(x) and Warmer(x,y):
-
<br>
<br>
Bachelor(x) → ~Married(x) -
<br>
<br>
For example, let 'W' be a primitive predicate designating the
relation Warmer. Then 'W' is transitive, irreflexive, and hence
asymmetric in virtue of its meaning. -
<br>
<br>
In the previous example of the predicate 'W', we could lay down
the following postulates (a) for transitivity and (b) for
irreflexivity; then the statement (c) of asymmetry: -
<br>
<br>
(a) ∀(x,y,z) Warmer(x,y) ∧ Warmer(y,z) → Warmer(x,z) -
<br>
(b) ∀(x)      ~Warmer(x,x) -
<br>
(c) ∀(x,y)    Warmer(x,y) → ~( Warmer(y,x) ) -
<br>
<br>
Copyright 2017 Pete Olcott -
<br>
<br>
Posted again for text only news readers. Dashes to prevent Google
hiding.
<br>
<br>
<br>
</blockquote>
<p><font face="Segoe UI Symbol">The key prerequisite to
understanding provability theory is a knowledge of propositional
logic. Propositional Logic is very easy to understand because
all of its symbols stand for simple English words. <br>
<br>
The Provable(Γ, X) predicate is shown below in terms of
Propositional Logic: <br>
<br>
X = "It is raining outside."<br>
Y = "You go outside."<br>
Z = "You get wet."<br>
<br>
These three all say the same thing:<br>
(1) if (X and Y) then Z<br>
(2) X ∧ Y  → Z <br>
(3) Provable( {X, Y}, Z )<br>
<br>
</font><br>
</p>
<div class="moz-signature">-- <br>
<meta charset="UTF-8">
        <b>Γ ⊢<sub><font style="font-size: 8pt" size="1">FS</font></sub>
A ≡ ∃Γ ⊆ FS Provable(Γ, A)</b> // MTT notational conventions<br>
<b>∀X True(X) ≡ ∃Γ ⊆ MTT ∧ Axioms(Γ) Provable(Γ, X) </b> // MTT
Truth Predicate </div>
</body>
</html>
Pete Olcott
2017-12-21 00:47:49 UTC
Permalink
Raw Message
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
</head>
<body text="#000000" bgcolor="#FFFFFF">
<div class="moz-cite-prefix">On 12/20/2017 6:20 PM, Pete Olcott
wrote:<br>
</div>
<blockquote type="cite"
cite="mid:4O6dna83vZ_4Y6fHnZ2dnUU7-***@giganews.com">
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<div class="moz-cite-prefix">On 12/20/2017 12:57 PM, Pete Olcott
wrote:<br>
</div>
<blockquote type="cite"
cite="mid:x4WdnVVSTOI2L6fHnZ2dnUU7-***@giganews.com">On
12/20/2017 12:24 PM, Pete Olcott wrote: <br>
<br>
Syntactic versus Semantic Logical Consequence - <br>
<a class="moz-txt-link-freetext"
href="https://en.wikipedia.org/wiki/Logical_consequence#Syntactic_consequence"
moz-do-not-send="true">https://en.wikipedia.org/wiki/Logical_consequence#Syntactic_consequence</a>
<br>
<br>
Formalizing the internal semantic meaning of propositional
(sentential) variables using Meaning Postulates Rudolf Carnap
(1952) bridges the gap between syntactic consequence (formal
proof) and semantic consequence (logical entailment).  - <br>
<br>
Meaning Postulates Rudolf Carnap (1952) - <br>
<a class="moz-txt-link-freetext"
href="http://www.thatmarcusfamily.org/philosophy/Course_Websites/Readings/Carnap%20-%20Meaning%20Postulates.pdf"
moz-do-not-send="true">http://www.thatmarcusfamily.org/philosophy/Course_Websites/Readings/Carnap%20-%20Meaning%20Postulates.pdf</a>
<br>
<br>
Meaning postulates specify semantic logical entailment
syntactically. - <br>
The best two examples from his paper: Bachelor(x) and
Warmer(x,y): - <br>
<br>
Bachelor(x) → ~Married(x) - <br>
<br>
For example, let 'W' be a primitive predicate designating the
relation Warmer. Then 'W' is transitive, irreflexive, and hence
asymmetric in virtue of its meaning. - <br>
<br>
In the previous example of the predicate 'W', we could lay down
the following postulates (a) for transitivity and (b) for
irreflexivity; then the statement (c) of asymmetry: - <br>
<br>
(a) ∀(x,y,z) Warmer(x,y) ∧ Warmer(y,z) → Warmer(x,z) - <br>
(b) ∀(x)      ~Warmer(x,x) - <br>
(c) ∀(x,y)    Warmer(x,y) → ~( Warmer(y,x) ) - <br>
<br>
Copyright 2017 Pete Olcott - <br>
<br>
Posted again for text only news readers. Dashes to prevent
Google hiding. <br>
<br>
<br>
</blockquote>
</blockquote>
<br>
<b>I was trying to be concise yet someone on the [ Mathematics of
Semantics ] Facebook group pointed out the gap that I left open:</b><b><br>
</b><br>
The key prerequisite to understanding provability theory is a
knowledge of propositional logic. Propositional Logic is very easy
to understand because all of its symbols stand for simple English
words. <br>
<br>
The Provable(Γ, X) predicate is shown below in terms of
Propositional Logic: <br>
<br>
W = "It is raining outside."<br>
X = "You go outside."<br>
Y = "You are unprotected from the rain."<br>
Z = "You get wet."<br>
<br>
These four all say the same thing:<br>
(1) "It is raining outside and you go outside and you are
unprotected from the rain then you get wet."<br>
(2) if (W and X and Y) then Z<br>
(3) W ∧ X ∧ Y  → Z <br>
(4) Provable( {W, X, Y}, Z )<br>
<br>
<br>
<font face="Segoe UI Symbol"></font>
<div class="moz-signature">-- <br>
<meta charset="UTF-8">
        <b>Γ ⊢<sub><font style="font-size: 8pt" size="1">FS</font></sub>
A ≡ ∃Γ ⊆ FS Provable(Γ, A)</b> // MTT notational conventions<br>
<b>∀X True(X) ≡ ∃Γ ⊆ MTT ∧ Axioms(Γ) Provable(Γ, X) </b> // MTT
Truth Predicate </div>
</body>
</html>

Loading...