Concludia Instructions

Have you ever heard someone make an assertion about the way the world works, or about something that should be done, and doubted whether they were correct? You've wanted to ask them "Why?" but were not able to get a satisfying response. Or perhaps you simply believed they were wrong, either from bad reasoning, or because they might be basing their assertion on something they believe is true, but is actually false.

Or perhaps you yourself have come to a conclusion - one that is counterintuitive. It has a deep argument behind it that is difficult to communicate. You believe it to be convincing, but your audience might disagree with it on a number of different grounds that might vary from person to person. You have answers for each person, but have difficulty reaching them.

Concludia is a way for people to communicate these arguments to each other, fully and completely. It displays conclusions, reasoning, the premises behind them, and the logical relationships between each. It also shows the truth of the premises that the conclusions rest upon, and it propagates truth throughout the arguments to display the provability of each conclusion.

Concludia uses three-valued truth. A premise can be True, False, or Unknown.

If you create a box that says "All humans live on Jupiter", it is completely allowable to set it to True. This is because Concludia has no semantic-level understanding of sentences. Concludia does, however, have syntax-level understanding. If you create two premises and set them to True, and then join them with an AND junctor, the output will by definition be True. If either or both premises are set to False, the output will by definition change to False. And since each output can in turn be an input to another conclusion, the truth values spread throughout the argument. We call this "truth propagation".

Because of this machinery, two things are true:

• Users can only adjust truth values on premises; all other truth values propagate from premises.
• All arguments are guaranteed valid on a syntax level.

Let's say you have joined A and B together, via an AND junctor, to point to a conclusion C. As described above, T ^ T = T, while T ^ F = F.

With Concludia, these Truth values correspond to provability. If one or more of the premises are False, it does not mean that the conclusion is false. Instead, it means that it is false that the conclusion is proven.

This should match an intuition about long proofs or arguments. If you have a conclusion backed by a lengthy argument, and you believe the conclusion to be true, but then discover one of its underlying premises is false, it does not necessarily mean your conclusion is false. It may still be true; you would just have to find a different way to prove it.

Note, however, that in common usage of the word, if you do successfully prove a conclusion, it is allowable to also say it is True.

Just because a sentence is proven on a syntax level does not mean it is actually proven on a semantics level. As discussed above, Concludia's machinery does not understand the semantic meaning of sentences.

We rely on each other as humans to check each other's meanings. So if you come across a premise that is marked True, and believe it is not actually True, Concludia allows this to be contested. This is called an 'undermine': a counterpoint that contests a premise.

Similarly, if you come across a conclusion that is marked proven but you disagree that its sources justify the conclusion, Concludia allows a way for you to contest the proof. This is called an 'undercut': a counterpoint that contests an inference.

Currently, Concludia supports argument navigation, counterpoints, and adjustment of truth values in premises. Truth value changes are currently shared across all viewers. In the future, users will be able to set their own truth for Values. If a graph includes counterpoints, you can view them via the “Show counterpoints” toggle. (Future plans include the ability to challenge reasoning, and the ability to create and add to existing arguments.)

Any conclusion or lemma - that is, any box that is not a premise - is prepended with a symbol that indicates its provability state:

• ⊢: This sentence is proven. The necessary premises it depends on are True.
• ◇: This sentence is possibly proven. Of its necessary premises, at least one is Unknown. All others are True.
• ⊬: This sentence is not proven. (This does not mean the sentence is proven false.) It is contested: a required premise is False, or a counterpoint blocks the inference.

A sentence displays its immediate sources and targets. You can navigate up and down the argument by clicking on them.

Any premise is appended with a symbol that indicates its truth state: T, F, or U - indicating True, False, or Unknown respectively.

Here are two common patterns of navigation:

• If you disagree with a proven sentence, but agree that its sources, if true, would hypothetically prove the sentence, click on the source that you disagree with. Continue in this manner until you reach an inference or premise that you disagree with.
• If you come across a contested conclusion or lemma, one or more of its sources will be contested, by definition. Click it, and continue in this manner until you find the relevant counterpoint.

In either case, changing the truth value of a premise will immediately propagate the truth upward through all relevant lemmas and conclusions.

To set a premise's truth value, navigate to a premise. An inline truth toggle will appear allowing you to set the truth value for that premise.

While all Concludia arguments are guaranteed valid on a syntax level, they are not guaranteed valid on a semantics level. This means that even if premises are marked true, they may not actually - in meaning - prove the conclusion they immediately point to. In the future, registering counterpoints will be possible, where Concludia will ask you to highlight the problematic junction, and state the missing assumption that:

• if true, would prove the conclusion
• is currently false, and why

If people use Concludia to fully communicate their arguments, then other people can use Concludia to fully understand these arguments. Additionally, through community engagement, the arguments themselves can be strengthened. The hope is that as usage increases, people will better be able to understand each other and respect each other's points of view.

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