The main difference between sentences and statement is how they are quantified. For example:
Sentences are quantified existentially
o
Eg. The employee earns less than $55,000,
Statements are quantified universally
o
Eg. Every employee earns less than $55,000
Kinda different compared to English. Some things I've taken note of:
- A statement is always a sentence
- A sentence is not always a statement
- A statement can return true/false
Hmm.. this seems similar to the concept of Predicates! What's that? Am I going to use this exact same method to introduce more of what I've learned? Of course!
Predicates are kinda like giving properties
to objects we declare. I'll demonstrate with the example showed in class.
Assume L represents the employees who earn less than $55,000.
If we want to say
that employee "x" earns less than $55,000, we can write "x∈L,L(x)".
We can turn this into a statement by universally quantifying it:
eg. For each employee, the employee earns less than $55,000
We can turn this into a sentence by existentially quantifying it:
eg. There exists and employee who earns less than $55,000
Now, you might say, Hmm... can't we also use sets to say the same thing? Which one should we use? I have no clue. Personally I find it difficult to see the difference between a property and set
Now what does this imply? This implies that we should talk about implication. And yes, once again I forced the topic into this conversation. Anyways I consider implication like the if functions that we think of in computer science.
if(P=true)
{
Q=true
}
"if P, then Q". P is called the Antecedent (a.k.a. the assumption) and Q is the consequent (a.k.a. the conclusion). Generally can be summed up by "For P to be true, Q must be true." or "P implies Q", or "P requires Q" and probably many more ways to say this.
Anyways, this is sometimes different than its English counterpart. For example, "If I do my homework, then I can play video games.". This sentence in symbolic language would make more sense if it was "If I don't do my homework, I cannot play video games". Similarly, implication doesn't imply causation, for example "If it rains yesterday, then the sun rises tomorrow". This is an implication, as you cannot falsify it. However, to falsify an implication, you must find an example where the antecedent is true, but the conclusion is false. If you cannot falsify it, then the implication is true! Also, you can make something imply in both directions! P<=>Q, basically means P=>Q and P<=Q, as simple as that.
Now that I'm close to finishing this SLOG post, I just realized that I am very off-track from what I am supposed to write. More personal opinions and challenges, less course knowledge, got it. I'll keep this in mind for next SLOGS to come...
Now for the next post, be ready for more math symbols or/and maybe not? (<- thats the hint)
