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Can someone help me derive the truth table of the logical conditional from the material conditional?

Here are three statements about logical implication from my logic professor:

  1. Given the argument: S:= {A1, A2, ..., An |= C}, where A1 through An are premises and C is the conclusion and the symbol |= is called double turnstile and it stands for "entails"/is a logical consequence of, i.e., C is entailed by the premises.

  2. Argument S may also be represented as S:= {A1, A2, ..., An} =log=> B, where =log=> stands for logically implies (as opposed to materially implies).

  3. Given that S may also be expressed as follows: S:= {A1, A2, ..., An} --> B is a tautology.

How do I derive the truth table for logical implication given --> the material implication? Can somebody help me understand how logical implication depends on the material implication? The truth table of logical implication would be highly appreciated.

Here is what I understand about the difference between material and logical implication:

  • P -> Q: P materially implies Q means P is a sufficient condition for Q.

  • P--> Q: P logically implies Q means P is necessarily sufficient
    for Q; it is a logical truth that P is sufficient for Q.

P -> Q goes to the structure of a formula (ex., a 'line' in an argument, i.e., a proposition as premise/conclusion in an argument); whereas, P --> Q goes to the structure of an argument (made up formulas: i.e., premises and conclusion(s)); that is, the logical implication has to do with logical form and thereby goes to the relationship between lines: how the premises relate to the conclusion of an argument, and whether the (final) conclusion of an argument is a logical consequence of the premises stated: i.e., whether the premises entail the conclusion: whether the premises necessitate the conclusion (i.e., is it a 'necessary truth' (ex., tautology) that the conclusion follows from the premises, as opposed to merely being a material consequence (i.e., a consequent of the material implication/conditional (statement)).

Where I need help is in putting these two concepts together and synthesize how they are truth-functionally related, illustrated with truth tables. How would one demonstrate that P -> Q is a necessary (i.e., logical) truth, for it to suffice to be a logical implication?

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    See What's the difference between material implication and logical implication? on Math SE. => does not have a truth table because it is not a connective, it is a meta-symbol. – Conifold Nov 28 '20 at 12:56
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    Logical implication can't be represented by a truth table, but as you noted (were you quoting from somewhere?), to say that premises {A1, A2, ..., An} logically imply a conclusion B is equivalent to saying that the proposition {A1, A2, ..., An} --> B is a tautology (where --> is the material conditional). – Hypnosifl Nov 28 '20 at 21:46
  • I consulted the notes I had taken in class. The source is our professor of the three different statements of logical implication. The part that I don't get is how to demonstrate that A ->C is a tautology. The main connective of A -> C is the arrow (->), so how do you make use of it to evaluate whether the conditional comes out as a tautology or not. I don't see how that material implication can give a truth value of T (true) for every valuation. – Karlen Karapetyan Nov 30 '20 at 8:22
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Conditional is a Logical connective used to form expressions (formulas) of the language.

Logical implication is a relation holding between set of formulas and a formula.

Thus, they are two different concepts.

But they are strongly related: in propositional logic, for example, we have that:

the formula A → B is a tautology iff B is a logical consequence of A [i.e. A logically implies B].

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We need to clear up a couple of points to begin with. In the standard way in which logic is treated today, we distinguish between the object language and the metalanguage. This amounts to distinguishing between saying something in a language and saying something about a language. Material implication is a connective within the object language of classical logic. What you are calling logical implication, which is also sometimes called logical consequence, is a statement about the propositions in a language. Material implication is sometimes written as P ⊃ Q, or as P → Q, or as P ⇒ Q.

Another important distinction is between syntax and semantics. Syntactically, we say that some set of propositions Γ proves a proposition β, or β is derivable from Γ, or β is a theorem on Γ, if there is some way to prove β from Γ using the axioms or rules of our logic. This relation is written as Γ ⊢ β. Semantically, we say that β is the logical consequence of Γ, or the semantic consequence of Γ, or that Γ entails β, if the truth of Γ guarantees the truth of β according to some semantic framework. This relation is written as Γ ⊨ β. The usual framework is called model theory and we say that an argument is semantically valid if every model of Γ is a model of β. First order classical logic is sound and complete, which means that Γ ⊢ β and Γ ⊨ β will always both be true or both be false.

Material implication, like other connectives, can be defined either syntactically or semantically, and the two are provably equivalent. For comparison, conjunction can be defined syntactically using an introduction rule: P; Q therefore P ^ Q and an elimination rule: P ^ Q therefore P; Q. These rules are sufficient to characterise the classical conjunction connective exactly. Conjunction can be defined semantically by a truth table: T/F/F/F. The two can be proved to be equivalent using techniques that you will typically find in an intermediate level textbook. In the case of material implication, it can be defined syntactically by the introduction rule Γ ⋃ {P} ⊢ Q therefore Γ ⊢ P → Q, and the elimination rule P; P → Q therefore Q. Semantically, its truth table is T/F/T/T. The introduction rule looks a little odd, but it is provable using the deduction theorem.

Now that we have this in place, it is straightforward to answer your questions. Material implication has a truth table because it is a truth functional connective in the object language. The derivability and logical consequence relations do not themselves have truth tables, because they are relations between sets of propositions. To understand the relationship between them, take a look at what happens to the introduction rule for material implication when Γ is empty. It simplifies down to P ⊢ Q therefore ⊢ P → Q. In other words, material implication is provable in just those circumstances in which Q is provable from P. And since by soundness and completeness we can substitute ⊨ for ⊢, it also follows that P ⊨ Q therefore ⊨ P → Q. In other words, P → Q is a tautology in just those circumstances in which Q is the logical consequence of P.

So, as an example, suppose we wish to test whether this argument is valid:

P1:  a → b
P2:  c → d
Therefore 
C:  (b → c) → (a → d)

First we form the corresponding conditional of the argument. This is done by conjoining the premises and having these materially imply the conclusion, thus:

((a → b) ^ (c → d)) → ((b → c) → (a → d))

Then we plug this into a truth table generator, of which there are many on the Web. This yields the following. As you can see, the corresponding conditional has T in all rows, so it is a tautology, and the argument is valid.

+---+---+---+---+---------------------------------------------+
| a | b | c | d | (((a → b) ∧ (c → d)) → ((b → c) → (a → d))) |
+---+---+---+---+---------------------------------------------+
| F | F | F | F |  T                                          |
+---+---+---+---+---------------------------------------------+
| F | F | F | T |  T                                          |
+---+---+---+---+---------------------------------------------+
| F | F | T | F |  T                                          |
+---+---+---+---+---------------------------------------------+
| F | F | T | T |  T                                          |
+---+---+---+---+---------------------------------------------+
| F | T | F | F |  T                                          |
+---+---+---+---+---------------------------------------------+
| F | T | F | T |  T                                          |
+---+---+---+---+---------------------------------------------+
| F | T | T | F |  T                                          |
+---+---+---+---+---------------------------------------------+
| F | T | T | T |  T                                          |
+---+---+---+---+---------------------------------------------+
| T | F | F | F |  T                                          |
+---+---+---+---+---------------------------------------------+
| T | F | F | T |  T                                          |
+---+---+---+---+---------------------------------------------+
| T | F | T | F |  T                                          |
+---+---+---+---+---------------------------------------------+
| T | F | T | T |  T                                          |
+---+---+---+---+---------------------------------------------+
| T | T | F | F |  T                                          |
+---+---+---+---+---------------------------------------------+
| T | T | F | T |  T                                          |
+---+---+---+---+---------------------------------------------+
| T | T | T | F |  T                                          |
+---+---+---+---+---------------------------------------------+
| T | T | T | T |  T                                          |
+---+---+---+---+---------------------------------------------+
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  • Material conditonal belongs to the language of propositional logic; you use it to build sentences.

  • Logical implication belongs to the meta-language ; you use this notion in order to talk about the logical relations between a set of sentences ( the premises) and a sentence called the conclusion.

  • Logical implication has no truth table. But when you use it, you say something about the truth table of a given sentence, namely the truth table of the corresponding ( material) conditional of the reasoning you evaluate.

  • Example.

Premise (1) (P&Q) --> R

Premise (2) ~R

Premise (3) Q

Conclusion : ~P

Let's build the corresponding material condtional [ ((P&Q) --> R) & ~R & Q ] --> ~P.

This material conditional only has truth value T in its truth table.

So the whole antecedent [ ((P&Q) --> R) & ~R & Q ] logically implies the consequent ~P.

Or, equivalently the conclusion ~P is a logical consequence of the set of premises { P1, P2, P3}.

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