Is there something in logic that behaves more like "traditional" implication?

Question

In logic, implication refers to the following truth table. Initially, we can't know if an implication is true or false if all combinations of $P$ and $Q$ are possible.

Let's imagine we've ruled out the possibility that $Q$ is false when $P$ is true. From the truth table, we can see that the implication is true. This matches my real-world intuition.

But let's imagine a different scenario: we've ruled out the possibility for $Q$ to be true when $P$ is true. My intuition would say that the implication is now most definitely false. However, according to the truth table, we haven't learnt anything about the implication. It is still either true or false.

Clearly there is a discrepancy between my "intuitive" implication, and the version of implication defined in logic. I will call my implication "traditional implication".

Traditional implication would be defined as true if the logical implication is a tautology, otherwise false. (It can only be a tautology if we've ruled out the second row, i.e. $P \land \lnot Q$ is impossible)

From what I know, traditional implication is what's used to test the validity of an argument, as opposed to logical implication. Am I right in saying this? (I'm pretty sure this is correct).

Is there any such operation in logic that represents traditional implication? Is there any symbol for it? Is there a clean way of writing down this type of implication? Or do I have to resort to using the following, which is quite messy, and possibly incorrect?

$$\forall p, q (p \implies q)$$

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Edit: I just came across tautological implication and tautological consequence. Hopefully this is related to what I'm talking about, which I'm reading up on now.

Answer 1

(It can only be a tautology if we've ruled out the second row, i.e. $(P \land \lnot Q)$ is impossible)

No.

You're crucially forgetting that if $$P{\implies} Q$$ is tautological, then either $P$ and $Q$ are identical or at least one of them is a compound proposition. Thus, the truth-functional form $P\to Q$ is not a tautology.

If $(P{\implies}Q)$ is a tautology, then every row of its truth table is True, and its $2$nd row (out of $2$ or $4$ or $8$ or $16$ or etc.) is neither false nor ruled out.

Let's imagine we've ruled out the possibility that $Q$ is false when $P$ is true. From the truth table, we can see that the implication is true. This matches my real-world intuition.

But let's imagine a different scenario: we've ruled out the possibility for $Q$ to be true when $P$ is true. My intuition would say that the implication is now most definitely false. However, according to the truth table, we haven't learnt anything about the implication. It is still either true or false.

In the two intuition-matching exercises above, even though you claim to be considering the truth of the implication $(P{\implies} Q),$ you've almost certainly actually carelessly reverted to considering instead, the truth of its conclusion $Q$ true given that its premise $P$ is true.

Clearly there is a discrepancy between my "intuitive" implication, and the version of implication defined in logic. I will call my implication "traditional implication".

Your ‘traditional/intuitive implication’ is really just regular implication ignoring the case of a false premise.

‘Traditional implication’ would be defined as true if the ‘logical implication’ is a tautology.$\tag{*}$

(BTW, a technical point: in propositional logic, ‘logical implication’ and ‘tautological implication’ are synonyms, by definition. What you call ‘logical implication’ above is least ambiguously called ‘material conditional’, but for this entire Answer, let's continue calling it just ‘implication’, without the adjective ‘logical’.)

As I pointed out above, your ‘traditional implication’ is nothing more than the usual implication restricted to the first two rows. In fact, paraphrasing $(*)$ in light of this results in a self-evident lemma:

   “The first two rows of $(P{\implies}Q)$ are true if all rows of $(P{\implies} Q)$ are true.”$\tag{*}$

From what I know, ‘traditional implication’ is what's used to test the validity of an argument, as opposed to ‘logical implication’. Am I right in saying this? (I'm pretty sure this is correct).

An argument ​is said to be valid when its conclusion is a logical consequence of its premises; in other words, a valid argument is precisely a logical (i.e., tautological or valid) implication!

On the other hand, your ‘traditional implication’ says nothing about the validity of an argument with a false premise.

is the following what I'm looking for to test the validity of my earlier comment's argument? $$\forall P,Q\: \big((A \land B) {\implies} Q\big)\tag{#}$$

Every argument's corresponding conditional is of the form $$P\to C,$$ where any quantification occurs within its premises $P$ and conclusion $C.$ So, $(\#)$ does not represent an argument.

More: False implies anything.

Answer 2

In daily discourse, if we know that a proposition is false, we don't usually give much thought to its implications. In mathematical proofs, however, if a proposition $A$ is assumed or proven to be false, we can infer that $A$ implies $B$, no matter what the truth value of $B$ may be. Symbolically, we can state this so-called principle of vacuous truth as

$$\neg A \implies [A \implies B]$$

As a kind logical "firewall," we would not be able to infer anything about the truth value of $B$ from this statement since $A$ is assumed to be false. In this case, more information would be required to infer anything about the truth value of $B$.

This principle can be derived by other more basic rules of logic as follows using a form of natural deduction (screenshot from my proof checker):

If the principle of vacuous truth is somehow be disallowed, one or more these basic rules of logic must be eliminated or otherwise restricted in their application. The "cure" would probably be worse than "disease" IMHO. There would be little to gain.

Answer 3

The problems of interpreting the propositions in order to give them a truth value and that of defining logical consequence, implication, satisfiability, etc go well further than propositional logic into model theory, proof theory and math foundations.

Propositional calculus is about understanding complex connections of propositions and calculating their truth value. Just take it as it is: there are 2^4 possible truth tables for binary "combinations" of propositions with a specific truth value. For various reasons only few of these "combinations" have become standard and have received a symbol.

If (y)our desired/intuitive table doesn't correspond to a dedicated symbol (you)we can still use it by combining various unary/binary symbols or by defining a new symbol which represents (y)our desired truth combination.

Answer 4

General comment :

It seems to me you are looking for a truth table of the logical operator " logical implication". But there is no such truth table. Logical implication is not a logical operator ( belonging to the object language) ; it is a higher level relation between 2 sentences, so to say a " meta" relation, that is asserted in the " metalanguage" ( by someone who is observing the logical language " from the outside").

Note : in the same way, it is only in the metalanguage that one can say that a reasoning is valid; from the outside, the observer says that the conjunction of the premises logically imply the conclusion, or, equivalently, from the outside, the observer asserts that the material implication $P_1\land P_2 \land P_3 ... \rightarrow C$ is a tautology.

The first order relation " P materially implies Q" holds iff factually, in the actual world, it is not the case that P is true and Q is false. ( But the material imùplication relation still holds when it could have been the case P to have been true and Q to have been false. At the material implication level, only actual truth values count.)

The second order relation " P logically implies Q" holds iff , not only P materially implies Q , but also, there is no possible case / situation / world in which P is true and Q is false.

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let's imagine a different scenario: we've ruled out the possibility for Q to be true when P is true. My intuition would say that the implication is now most definitely false.

Consider this sentence " IF you've found a whole number between 3 and 4 THEN I was born before my father".

Both sentences are false in the actual world, meaning that the material implication is true.

Not only are they false, but they could not be true, meaning that it is impossible to be on the first row of the " if ... then " truth table.

This does not prevent the first sentence to imply logically the second.

The reason is that: since the first sentence cannot be true, not only we are not factually in the true / false case, but such a case is impossible.

And such an impossibility is all we need to have a logical implication.

Reference : Seymour Lipschutz, Outline Of Set Theory And Related Topics, chapter on the algebra of logic.