“He claims he didn't do it, but I strongly suspect that he did.”
The disk tray of your DVD player is broken. You think that your son broke it, but he says that he didn't. Now you're telling your wife about it.
He claims he didn't do it, but I strongly suspect that he did.
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Use the word "claims" to describe what someone said, when you don't believe it or you're not ready to accept that it's true:
The suspect claims that he was at home at the time of the murder.
She claims that I agreed to sell it to her for $300, but that's bullshit. I never agreed to that.
In this example, the speaker used "claims" instead of "claimed". The son said it in the past, so why doesn't she say "claimed"? It's because the son still hasn't changed what he said. He said that he didn't do it, and if you ask him again he'll still say that he didn't do it. You can use "says" in the same way:
He says that he didn't do it.
He swears that he didn't do it.
("Swears" means "says very forcefully" or "promises".)
As with lots of other "that" clauses, people sometimes leave "that" out of the sentence in casual spoken English, like in the example at top.
"Suspect" means "think". But more specifically, it means "think that someone did something wrong" or "think that something bad has happened". You can use this word when you think someone did something bad, but you're not sure. For example:
I'm starting to suspect that she's been cheating on me.
In this example, you don't know for sure that she's cheated, but there is some evidence that makes you think it might be true.
However, there's another use of the word "suspect" when you're just making guesses about the future. This use isn't as common. It's used like this:
I suspect it might rain later this afternoon.
It's most common to use "suspect" for guessing about negative things like bad weather.
When your level of suspicion is high, you say that you "strongly suspect that ___"