Forming a Bridge between Proof Theory and Model Theory

Discussion:

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Pete Olcott

2017-12-20 18:24:19 UTC

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<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> 
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). 
 
<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> 
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
<div class="moz-signature">-- 
<meta charset="UTF-8">
 Γ ⊢FS
A ↔ ∃Γ ⊆ FS Provable(Γ, A)
// MTT notational conventions 
∀X True(X) ≡ ∃Γ ⊆ MTT Provable(Γ, X) // MTT Truth
Predicate </div>
</body>
</html>

Pete Olcott

2017-12-20 18:57:47 UTC

Permalink

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

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<div class="moz-cite-prefix">On 12/20/2017 12:57 PM, Pete Olcott
wrote: 
</div>
<blockquote type="cite"
cite="mid:x4WdnVVSTOI2L6fHnZ2dnUU7-***@giganews.com">On
12/20/2017 12:24 PM, Pete Olcott wrote:
 
 
Syntactic versus Semantic Logical Consequence -
 
<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>
 
 
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) -
 
<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>
 
 
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.
 
 
 
</blockquote>
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. 
 
The Provable(Γ, X) predicate is shown below in terms of
Propositional Logic: 
 
X = "It is raining outside." 
Y = "You go outside." 
Z = "You get wet." 
 
These three all say the same thing: 
(1) if (X and Y) then Z 
(2) X ∧ Y → Z 
(3) Provable( {X, Y}, Z ) 
 
 

<div class="moz-signature">-- 
<meta charset="UTF-8">
 Γ ⊢FS
A ≡ ∃Γ ⊆ FS Provable(Γ, A) // MTT notational conventions 
∀X True(X) ≡ ∃Γ ⊆ MTT ∧ Axioms(Γ) Provable(Γ, X) // MTT
Truth Predicate </div>
</body>
</html>

Pete Olcott

2017-12-21 00:47:49 UTC

Permalink

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<div class="moz-cite-prefix">On 12/20/2017 6:20 PM, Pete Olcott
wrote: 
</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: 
</div>
<blockquote type="cite"
cite="mid:x4WdnVVSTOI2L6fHnZ2dnUU7-***@giganews.com">On
12/20/2017 12:24 PM, Pete Olcott wrote: 
 
Syntactic versus Semantic Logical Consequence - 
<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>
 
 
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) - 
<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>
 
 
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. 
 
 
</blockquote>
</blockquote>
 
I was trying to be concise yet someone on the [ Mathematics of
Semantics ] Facebook group pointed out the gap that I left open: 
 
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. 
 
The Provable(Γ, X) predicate is shown below in terms of
Propositional Logic: 
 
W = "It is raining outside." 
X = "You go outside." 
Y = "You are unprotected from the rain." 
Z = "You get wet." 
 
These four all say the same thing: 
(1) "It is raining outside and you go outside and you are
unprotected from the rain then you get wet." 
(2) if (W and X and Y) then Z 
(3) W ∧ X ∧ Y → Z 
(4) Provable( {W, X, Y}, Z ) 
 
 

<div class="moz-signature">-- 
<meta charset="UTF-8">
 Γ ⊢FS
A ≡ ∃Γ ⊆ FS Provable(Γ, A) // MTT notational conventions 
∀X True(X) ≡ ∃Γ ⊆ MTT ∧ Axioms(Γ) Provable(Γ, X) // MTT
Truth Predicate </div>
</body>
</html>