I'll state this caveat again:

When we are talking about classical logic, we need to be clear as to what classical logic is and what is the case about it in terms of its formulations and definitions. So, for example, when I say 'valid' in this context, I mean 'valid' in the sense of classical logic.

If one wishes to use terminology, such as 'valid', in some other sense, then that is fine, and we can discuss in that context too. But we need to be clear in any instance which context we're in.

Thus, since this discussion has focused on classical logic and a critique of classical logic, unless otherwise stated, my remarks pertain to what is the case with classical logic given its formulations and definitions. And, again, that is not a claim or attitude that only classical logic is admissible, but rather that when we are examining classical logic, we need to at least start by knowing what it is - what its formulations are definitions actually are.

Annotations do help to follow along in the proofs. But, depending on the formulation of the system, annotations may not be necessary. In the proofs I gave, annotations are not necessary. And that goes along with the fact that the statements of the rules do not require mentioning 'premise', 'assumption', 'supposition' or 'contradiction'. Unfortunately, the other poster understands none of this and knows jack about this subject.

The poster continues to substitute rhetoric for argument, utterly failing to engage in rational argumentation or inferential reasoning. — Leontiskos

I've given exact information, and clear explanations, demonstrations, reasoning. There is no lacuna in rationality there. The fact that I also mention that the poster is ignorant, confused and specious doesn't vitiate the on topic content I provide.

1. A -> (B & ~B) {1}
2. A {2}
3. B & ~B {1, 2}
4. ~A {1}
— TonesInDeepFreeze

The poster continues to assert his baseless arguments without answering the question and providing the rule of inference he purports to use in order to arrive at conclusion (4). — Leontiskos

The proof quoted is exactly correct. I've answered every central question. It's not my fault that you are unwilling to read the answers or are incapable of understanding them though they are clear and exact.

And for the 1000000th time, the rule is RAA.

How many times do people do have to tell you?:

The rule is RAA. The rule is RAA. The rule is RAA.

Do you not get it?

The rule is RAA.

The poster seems to suffer from psychological delusions and grandiosity. When faced with simple questions he retreats into himself, opting for 3rd-person rhetorical strategies and failing to engage in inferential reasoning. — Leontiskos

(1) I am hardly grandiose as I've said several times in this forum that I have only an intermediary knowledge of this subject and that I do make (corrected) mistakes.

(2) I have head on addressed the main contentions in this thread. Your replies often blatantly skip the key information, explanations and corrections given him.

(3) My reasoning has not been shown to be incorrect. On the other hand, your arguments are so often a jumble of ignorance, confusions and illogic.

(4) I've referred to you in the third person because you deserve to be referred to that way.

The poster continues to evidence a significant difficulty in using fairly basic forum features, such as quotes. — Leontiskos

Oh please! People often get snagged copying/pasting quote brackets. I almost always correct in edit though. That you try to make any kind of deal out that shows that you're quite the fundament.

So many of your claims have already been debunked in this thread. The truth-table approach to reductio was dispatched almost ten pages ago! — Leontiskos

For the half-dozenth plus one time:

The poster is doing it again! Trying to discredit interlocutors by painting them with a brush "truth-functional", even after I had at least a few times addressed that.

For about the half-dozenth time:

I am not a "truth functionalist". I study and enjoy classical logic, and appreciate its uses. But I am interested in other logics. I do not say that classical logic is the only logic that can be studied, enjoyed and used.

But when classical logic is being discussed, especially critiqued, it is crucial to say what actually is the case with classical logic. And in bringing clarity to what classical logic actually is, one needs to explain. Providing such explanations does not make one a "truth functionalist". — TonesInDeepFreeze

If you want to bring clarity you should explain what inference you used to draw (4). — Leontiskos

I explicitly said that they are examples of RAA. And the examples when given were earlier explicitly said as examples of RAA. And they were given this time in direct response to questions about RAA.

Again, as I said, for concision we may state RAA [emphasis added] without conjunction elimination:

If Gu{P} |- Q and if Gu{P} |- ~Q, then G |- ~P
is equivalent with
If Gu{P} |- Q & ~Q , then G |- ~P

If Gu{~P} |- Q and if Gu{~P} |- ~Q, then G |- P
is equivalent with
If Gu{~P} |- Q & ~Q, then G |- P

So, in this case:

(version 1)
1. A -> (B & ~B) {1}
2. A {2}
3. B & ~B {1, 2}
4. ~A {1}

is equivalent with

(version 2)
1. A -> (B & ~B) {1}
2. A {2}
3. B & ~B {1, 2}
4. B {1, 2}
5. ~B {1, 2}
4. ~A {1} — TonesInDeepFreeze

For the millionth time:

RAA. RAA. RAA.

Can you not read the words 'RAA' when they appear over and over?

I don't get to say:

P→Q
P
~Q
∴ Q {See truth table for 1, 2} — Leontiskos

First, you don't need '~Q' there.

And I didn't say "see truth table" in the proof.

There are two separate things: the deduction system, (such as natural deduction) which is syntactical, and truth evaluation (such as truth tables), which is semantical.

But we have the soundness and completeness theorem that states that a formula P is provable from a set of formulas G in the deduction system if and only if there is no row in the truth table such that all the members of G are true and P is false.

The proof would be this:

1. P -> Q {1}
2. P {2)
3. Q {1, 2}

The rule applied there is modus ponens:

From P and P -> Q, infer Q and charge it with all lines charged to P and to P -> Q.

The truth table would be:

P true, Q true ... P -> Q true

P true, Q false ... P -> Q false

P false, Q true ... P -> Q true

P false, Q false ... P -> true

There are two rows in which both P and P -> Q are true, and Q is true in both of those rows. There is no row in which both P and P -> Q are true but Q is false, so modus ponens is valid.

But if you want to include ~Q:

P→Q
P
~Q
∴ Q {See truth table for 1, 2; avert eyes from 3 at all costs. I repeat: do not allow 3 a seat at the truth table!} — Leontiskos

There's no "avert eyes", "don't allow 3 a seat"

1. P -> Q {1}
2. P {2}
3. ~Q {3}
4. Q {1, 2}

or if we are required to use (3):

1. P -> Q {1}
2. P {2}
3. ~Q {3}
4. Q {1, 2}
5. Q & ~Q {1, 2, 3}
3. Q {1, 2, 3}

Truth table:

P true, Q true ... P -> Q true ... ~Q false

P true, Q false ... P -> Q false ... ~Q true

P false, Q true ... P -> Q true ... ~Q false

P false, Q false ... P -> Q true ... ~Q true

There are no rows in which P -> Q, P, and ~Q are all true and Q is false. The argument from {P -> Q, P, ~Q} to Q is valid. ~Q is in the truth table.

As it happens, truth tables don't adjudicate contradictions. — Leontiskos

An odd thing to say, since a contradiction will have "F" all the way down it's main operator

(The fact that you think this sort of thing can be adjudicated by a truth table is proof that non-truth-functionality is in your blind spot.) — Leontiskos

The rule and truth tables agree. They agree and are independent. They are independent in the sense that are formulated separately without reference to each other. But there is the theorem that connects them with an equivalency:

Definition: An inference from G to P is valid if and only if there are no rows in the which all the members of G are true and P is false.

Theorem: An inference is allowed by the rules if and only if the inference is valid.

Moreover, the truth table method is algorithmic, thus sentential logic is decidable. So, actually the truth table method, if appropriately formulated, can itself be used as a deduction system.

Which is what Wittgenstein was working towards when he created truth tables.

4.31 We can represent truth-possibilities by schemata of the following kind (‘T’ means ‘true’, ‘F’ means ‘false’; the rows of ‘T’s’ and ‘F’s’ under the row of elementary propositions symbolize their truth-possibilities in a way that can easily be understood):

4.4 A proposition is an expression of agreement and disagreement with truth-possibilities of elementary propositions.
— Tractatus

One source (I don't know whether reliable) says Peirce invented truth tables, then later Wittgenstein and Post independently. Of course, Boole invented Boolean algebra (though maybe there were precursors?).

Ludwig Wittgenstein is generally credited with inventing and popularizing the truth table in his Tractatus Logico-Philosophicus, which was completed in 1918 and published in 1921.[2] Such a system was also independently proposed in 1921 by Emil Leon Post.[3] — Wiki.

Yeah.

Another source (from search of 'history of sentential logic') says, "The truth table system for Sentential Logic was invented in 1902 by the American logician Charles Peirce to display how the truth of some sentences will affect the truth of others. Truth tables were rediscovered independently by Ludwig Wittgenstein and Emil Post."

Oh, yeah - I was just editing the post to acknowledge that. My understanding is that Pierce tabulated some bits of binary logic, but it appears that the use of truth tables to demonstrate tautology and contradiction is down to Wittgenstein or Wittgenstein and Russell in about 1912. It is an issue of some disagreement.

So after outlining the process, he says

4.46 Among the possible groups of truth-conditions there are two extreme cases. In one of these cases the proposition is true for all the truth-possibilities of the elementary propositions. We say that the truth-conditions are tautological. In the second case the proposition is false for all the truth-possibilities: the truth-conditions are contradictory . In the first case we call the proposition a tautology; in the second, a contradiction.

This is original.

In any case, "truth tables don't adjudicate contradictions" is yet another error on the part of @Leontiskos.

Why do you reject the claim that Peirce came up with truth tables? Why do you omit Post?

Why do you claim that the notions of logically true and logically false were original from Wittgenstein?

He's hopeless.

Several times he was told that the rule used was RAA, and the proofs were stated in situ as being RAA, and yet he keeps demanding that it hasn't been said what rule was used.

That's just for starters.

I'm not familiar with Peirce, so I'm not rejecting the notion that he used truth tables. I don't see anything that indicates Peirce used them such that "the truth table method, if appropriately formulated, can itself be used as a deduction system". In 4.46, Wittgenstein sets out how to use a truth table to adjudicate tautology and contradiction. In the absence of an alternative, I believe that to be original, but let me know if Pierce did anything similar.


What is original is that Witti points out how to use a truth table to determine tautology or contradiction.

One source says Peirce came up with truth tables in 1902. If that is correct, then why rule out that he didn't also see that we can use them to make inferences and infer that a sentence is a logical truth or logical falsehood or neither? He had to have been a pretty smart guy, so it is unlikely that he would look at truth tables and not notice that we can use them to make inferences and check whether a sentence is logically true or logically false or neither.

See for example Irvine Anellis. I don't see that Anellis carries the case that Pierce's approach was complete. From the little I've seen Pierce used them to set out the permutations of three variables and so on, but I can't see anywhere that he made the connection with tautology and contradiction. I might be mistaken. But on a quick look around Anellis seems to be alone in his claim.

So unless a stronger case can be made, I'll credit Wittgenstein. In any case, it was the Tractatus that brought truth tables to the attention of the greater philosophical community. You are of course welcome to take a different opinion.

Number of letters: "he noted there that, for a proposition having n-many terms, there would be 2^n-many sets of truth values."

Tautologies: "For many years, commentators have recognized that Peirce anticipated the truth-table method for deciding whether a wff is a tautology.”

/

Aside from the question of invention, some salient and useful points about sentential logic:

(1) Sentential logic is, in a certain exact sense, isomorphic with the Tarski-Lindenbaum algebra.

(2) There are 16 binary truth functions. And, for all n, any n-ary truth function can be reduced to a binary truth function.

(3) All 16 binary functions can be derived from just one binary function (either Sheffer stroke or Nicod dagger).

(4) The truth table method can be formulated algorithmically. Sentential logic is decidable. And since sentential logic is decidable, it suffices to have just one inference rule: If G tautologically implies P then G proves P (e.g. 'A Mathematical Introduction To Logic' by Enderton, though he does it by saying that all tautologies are axioms).

(5) Soundness and completeness.

...anticipated... — TonesInDeepFreeze

Yep.

I don't know exactly what the author meant by "anticipated"

Meanwhile, may I take it that the point is made about the number of letters? He wouldn't have to display a truth table with n number of letters for every natural number n for us to grasp that there is no finite bound on the number of letters in a truth table.

The author says, "But the discovery by Zellweger of Peirce’s manuscript of 1902 does permit us to unequivocally declare with certitude that the earliest, the first recorded, verifiable, cogent, attributable and complete truth-table device in modern logic attaches to Peirce, rather than to Wittgenstein’s 1912 jottings and Eliot’s notes on Russell’s 1914 Harvard lectures." Whether the author properly makes the case for that would deserve more scrutiny of the paper.

But again,

...complete... — TonesInDeepFreeze

I'll maintain that the cardinal step, to using truth tables as a device for determining tautology and contradiction, was taken by Witti.

Meanwhile, may I take it that the point is made about tautologies? — TonesInDeepFreeze

Yep.

I edited. Not 'tautology' there. I meant 'the number of letters'.

I'll maintain that the cardinal step, to using truth tables as a device for determining tautology and contradiction, was taken by Witti. — Banno

That might be the case; but hardly clear that it is.

It seems amazing that it wasn't invented a lot earlier. Such a simple idea by now. It shows how much we take for granted in intellectual products.

"And Richard Zach reminds us that "Peirce, Wittgenstein, and Post are commonly credited with the truth-table method of determining propositional validity.""

It seems doubtful to me that anyone who sees that a truth table tests validity would not remark that some sentences are true in all rows and some sentences are false in all rows, whether such sentences are given the names 'logical truth' or 'logical falsehood' respectively. Indeed, to say that we test for validity is to say that we test whether the sentence is true in all rows (i.e. whether the sentence is a logical truth (aka 'tautology')).

Yeah, I can see that last, but what is actually quoted in the literature from Pierce seems to be about listing permutations of Boolean operators rather than showing truth. I dunno. Just unconvinced. Whereas the quote from Wittgenstein is pretty unequivocal. My prejudice is showing, of course.

I wonder if there is a "deductive system" using truth tables - say a proof of the completeness and consistency of prop logic using only truth tables... Someone must have done it. Might have a look around.