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I have been following Brexit on the BBC website, for example, this article. They have consistently said it's unlikely that Brexit will be cancelled (at times I think they said "very unlikely"), but given the failure to get the required legislation passed, isn't it the logical outcome, unless there is significant political movement?

Personally, I see little chance of the House of Commons agreeing to an exit deal or to a no-deal exit.

So why does the BBC say it is unlikely? What arguments, if any, have they provided for saying that? Since the first vote on the PM's deal was defeated, I have seen it as the most likely outcome, while admitting I could turn out to be wrong.

Edit:

On checking the latest BBC guide to Brexit, it no longer says revoking article 50 is unlikely. Instead it now says:

The European Court of Justice has ruled that it would be legal for the UK to unilaterally revoke Article 50 to cancel Brexit (without the need for agreement from the other 27 EU countries).

With the government still committed to Brexit, it's very likely that a major event such as a further referendum or change of government would have to happen before such a move.

However, any further delay to Brexit would certainly lead to questions about whether the ultimate destination was going to be a reversal of the 2016 referendum.

It's not totally clear what the process would be. But an act of Parliament calling for Article 50 to be revoked would probably be sufficient.

So there seems to have been a change of tune there.

Absent action from the House of Commons, the UK will leave the EU at 23:00 BST on 12 April, regardless of whether or not the Withdrawal Agreement has been ratified. That, as per both Article 50 of the Treaty of the European Union and the EU Withdrawal Act (2018), are the legal default positions as of this writing.

The House has voted on numerous occasions that it doesn't want a "No-Deal Brexit", but it hasn't voted on anything that would stop that from being the default legal position. They would need to, for example, pass a bill to authorize a second referendum, or to repeal the EU Withdrawal Act (2018). It does not seem that there is a majority in the House to do either of these things.

Yes, on the 5th of April, the BBC published an article: Brexit: What happens now? in which it explains the no Brexit option (as well as many others). In particular relating to your question, they wrote the following on the 'no Brexit' option:

With the government still committed to Brexit, it's very likely that a major event such as a further referendum or change of government would have to happen before such a move.

However, any further delay to Brexit would certainly lead to questions about whether the ultimate destination was going to be a reversal of the 2016 referendum.

It's not totally clear what the process would be. But an act of Parliament calling for Article 50 to be revoked would probably be sufficient.

Indeed, given how the situation has taken so much time and how it's such a big part of the agenda now, it would certainly be anticlimactic to see that was all for nothing.

Imagine you made a mistake and someone mentioned it. Then you have a choice, you either admit the mistake or you double down on your original action. In this case, it's not easy to admit the mistake (or reverse your course of action, or whatever you want to call it) because those arguing for and acting towards a Brexit have been very passionately about it. To make a U-turn now would damage their credibility very much. Indeed there's almost no new information, if they wanted to back out of their position because they realised they're at an impasse they could've done so months ago.

In sociology (but also in some cultures), this problem the politicians find themselves in is called losing face. It's a bit too broad to explain here, but the linked Wikipedia article has a lot of information on it looking at it from different cultures and academic fields.

The first paragraph of this question seems to be predicated on the false assumption that Brexit will not occur without further action ("but given the failure to get the required legislation passed..."), while the article linked to says the opposite: "The current default position - ie, if nothing changes - is for the UK to leave without a deal on 12 April" - which, at least at the time of publication, appears to be correct (and I am not aware of any development that has changed that.)

That article sets out the options available at the time, but is generally free of speculation about which is more likely. One place it does is here: "The European Court of Justice has said the UK could cancel Brexit altogether without the agreement of other nations, but politically, that's not likely to happen." [my emphasis]. If, however, you follow the link from that sentence, you will find that the British government opposed the question even being raised at the ECJ. That is an objective basis for an argument that the British government is unlikely to act on the ruling, whether you like it or not.

More generally, the existence of an option does not, by itself, invalidate arguments that it is not likely to be taken. Similarly, subsequent events do not retroactively invalidate arguments that were reasonable at the time they were made.