need help understanding truth functional entailment

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ProfAlexHartdegen
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Joined: Mon Jul 31, 2017 5:43 pm

need help understanding truth functional entailment

Post by ProfAlexHartdegen » Mon Jul 31, 2017 5:48 pm

I am reading a book called The Logic Book. There is a section about Truth Functional Entailment and Truth Functional Validity.

It states:

A set Γ of sentences of symbolic logic truth functionally entails a sentence P if and only if there is no truth value assignment on which every member of Γ is true and P is false.

I am able to follow the truth table examples that they present however it is not clear to me what the actual meaning and purpose of the statement is - what is its' usefulness? Can someone help me to understand and clarify this? thanks

Averroes
Posts: 55
Joined: Thu Jul 20, 2017 8:48 pm

Re: need help understanding truth functional entailment

Post by Averroes » Tue Aug 08, 2017 8:27 am

ProfAlexHartdegen wrote: Can someone help me to understand and clarify this?
I do not know but I can try share some thoughts with you.
ProfAlexHartdegen wrote:It states:

A set Γ of sentences of symbolic logic truth functionally entails a sentence P if and only if there is no truth value assignment on which every member of Γ is true and P is false.

I am able to follow the truth table examples that they present however it is not clear to me what the actual meaning and purpose of the statement is - what is its' usefulness?
The definition you provided can be concisely rewritten as:

Γ ⊨ p, where '⊨' means semantically(or truth functionally) entail.

Validity is a semantic concept, i.e. semantic entailment and validity are essentially the same notion.

As you are able to follow the truth table examples that are presented, you can understand that the following entailments are instances of 'Γ ⊨ P':

1. A → B, A ⊨ B

A....B....A→B
T....T.......T...←
T....ꓕ.......ꓕ
ꓕ....T.......T
ꓕ....ꓕ.......T

2. A v B, ~A ⊨ B

A....B...A v B....~A
T....T......T........ꓕ
T....ꓕ......T........ꓕ
ꓕ....T......T........T...←
ꓕ....ꓕ......ꓕ.......T

(I use the '....' for formatting purposes. The computer language in which the forum was written (PHP) ignores multiple white spaces and considers them as one.)

In these two examples, one observes that each time when the premises are true, the conclusion as well is true.
There are many other semantic sentential sequents which will exhibit this property. What if we want to discuss all such type of sequents generally? In order to do that we use a general notation of the form 'Γ ⊨ p'.
In 'Γ ⊨ p', 'Γ' and 'p' are called metalanguage variables. In the metalanguage, they stand for sentences (well-formed formulas) in the object language, such as 'A', 'A→B', 'A v B', '~A' etc…

As an aside, note here that we have two levels of generalization. Firstly from natural language (English here) arguments to sentential logic representation (first level of abstraction); secondly from sentential logic representation to metalogical representation (second level of generalization).

E.g.
English argument:
If ProfAlex studies hard, he will succeed.
ProfAlex studies hard.
Therefore, ProfAlex will succeed.

Object language (symbolic) representation:
A→ B, A ⊨ B

Metalogical representation:
Γ ⊨ p

For now, you main aim is the study of the first level of generalization. In the course of your study you get the opportunity to delve into some metalogic as well, and this can cause some confusion if you do not have a map of the various levels of generalizations involved. As may have been the case here! A convention often used which can help you recognize to distinguish metalogical from logical symbols is that for logical symbolism, the symbols used are upper case Latin letters (A, B, C etc) and for metalogic, the symbols used are upper case Greek letters (Γ ,Ψ, Σ etc…).

Anyway, a particular and important use of this general notation Γ ⊨ p is when we discuss soundness and completeness of a set of rules of inference.
To be able to understand that, first you must make a distinction between deductive entailment and semantic entailment.

May be later in your book you will learn about rules of inference. To get a general idea of what it is about see this thread (read the OP and my posts):viewtopic.php?f=26&t=19022

If you do not get it for now, it is alright. It is easier than it appears at first sight. So relax, just get the feel of how it looks like. Take one step at a time.

When the conclusion of a sequent can be deduced from its premises by the application of some inference rules in some system, the premises are said to deductively entail its conclusion. Here, we do not use truth values at all. When a set of premises Γ deductively entail its conclusion we write:
Γ ⊢ p, where '⊢' means deductively entail.

Provability is a syntactic concept.

Pay particular attention to the notation, there is a subtle difference:

Γ p, is called a syntactic sequent, and it is one in which the conclusion p is deduced by the use of rules of inference from the premises Γ.
Γ p, is called a semantic sequent, and it is one for which there is no truth value assignment on which every member of Γ is true and P is false.
In semantic sequents, we deal with truth value assignments for sentential logic and in deductive sequents we deal with rules of inferences.

Now, we will want our deductive sequents to be semantic sequents as well. This is what the concept soundness is about.
  • A system of rules of inference is sound if and only if whenever Γ ⊢ p then Γ ⊨ p.
Note: A potential source of confusion might be for you to confuse soundness of an argument with soundness of a set of rules of inference. They are different concepts. An argument is sound if it has a valid argument form and has true premises.

We may also want our semantic sequents to be deductive sequents as well. The latter is captured in the concept of completeness.
  • A system of rules of inference is complete if and only if whenever Γ ⊨ p then Γ ⊢ p.
It turns out (by proof) that sentential logic (the one you are currently studying) and more broadly First Order Logic (of which sentential logic is a part) is both sound and complete. But higher order logic does not have such a fortunate fate! Anyway, you should not be concerned about this for now. As you asked a question about the use of the notation, this is just to give you a general idea of how it is used. As you progress in your studies, all this might become clearer to you, and you might learn other uses as well. For example in metatheory, such notation is widely used. But the latter is a bit more advanced mathematical subjects which if you just know by name only for now is already an achievement in itself!

A useful paradigm to take when learning a new subject is that first one must learn the alphabet by heart. When one has fully grasped the alphabet, then one can combine each of them to form syllables. Later the syllables are combined to form words. When words are able to be written, we start learning some grammar so that we can join these words to form grammatical sentences. Sentences can subsequently be grouped meaningfully to form paragraphs. Paragraphs to form chapters, chapters to form books, and books to form volumes! So relax, keep moving and enjoy your studies. Remember, one step at a time.

ProfAlexHartdegen
Posts: 20
Joined: Mon Jul 31, 2017 5:43 pm

Re: need help understanding truth functional entailment

Post by ProfAlexHartdegen » Thu Aug 10, 2017 10:27 pm

thank you for your detailed answer - this helped to clarify the issue for me. I do have many other questions as I progress through the book; Actually, one right now. But I will post it as a separate thread because it is a different topic!

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