In spoken Spanish both "no lo sé" and "no sé" (I don't know) are used very often, but it is often difficult for learners to know when it's possible to omit the article "lo".
In direct responses to questions
When using the phrase as a direct response to a question that someone has formulated, we can use both:
-Ese actor se llama Richard West, ¿verdad? -No sé.-That actor is called Ricard West, isn't he? -I don't know.
-Ese actor se llama Richard West, ¿verdad? -No lo sé.-That actor is called Richard West, isn't he? -I don't know.
-¿Tienes ganas de salir a tapear? -No sé. No tengo mucha hambre ahora.Do you feel like going out to have some tapas? -I don't know. I am not very hungry right now.
-Tienes ganas de salir a tapear? -No lo sé. No tengo mucha hambre ahora.Do you feel like going out to have some tapas? -I don't know. I am not hungry right now.
When we say "no lo sé", "lo" is referring to the antecedent in the question:
¿Tienes ganas de salir a tapear? -No lo sé. [lo = "si tengo ganas de tapear"]
Ese actor se llama Richard West, ¿verdad? -No lo sé. [lo = "si se llama Richard West"]
Not a response or not immediately after a question
If it is not part of a response or we use it as a response but not immediately after, we need "lo":
-¿Cómo se escribe esa palabra? -Pues voy a mirarla en el diccionario porque no lo sé.
[lo = "cómo se escribe "]-How do you spell that word? -I 'll look it up in the dictionary because I don't know.
Si supiera su teléfono te lo diría, pero es que no lo sé. [lo = "su teléfono"]If I knew his number I'd tell you, but the thing is I don't know.
Yo debería saber dónde trabaja mi hija, pero no lo sé. [lo = "dónde trabaja"]I should know where my daughter works, but I don't know.
When the direct object is explicitly stated
When the direct object is explicitly stated, very often in the form of a subordinate clause, we can only use "No sé":
No sé de qué manera voy a convencer a Luis.I don't know how I am going to convince Luis.
No sé montar en bicicleta.I don't know how to ride a bike.
No sé cuánto tiempo tardaré en llegar.I don't know how long it will take me to get there.
In these cases it would be incorrect to use "no lo sé" as there is no antecedent.
"No lo sé montar en bicicleta."
With no clear antecedent and expressing "doubt"
In a conversation where there is no clear antecedent, and it is not a response to a direct question, we use "no sé". Using "no lo sé" would sound strange because of the lack of antecedent. In this case what the speaker is conveying is "doubt", as if saying "I am not sure [about that]," , "I have my doubts [about that], "I am not sure [that's the best idea] . For example:
-Vamos a tomarnos algo. -No sé, yo creo que ya hemos bebido demasiado.-Let's have a drink -I'm not sure, I think we've drunk too much already.
-Vamos a esperar un poco más a Laura, ¿vale? -No sé, ya debería haber llegado.-Let's wait for Laura a bit longer, OK? -I am not sure, she should have arrived by now.
Remember that the verb sé always has a written accent to differentiate it from the pronoun se.
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