Symbolic Logic

[Tyr]

Has anyone taken an introductory course in symbolic logic? Not the really tough stuff. I'm taking it at the moment, it's amazingly fun and absurdly mind-breaking at the same time. What sucks is, we get no checks whatsoever on our assignments, and my TA is not very helpful (to put it politely)...

Has anyone taken a course on it, or is familiar with "Language, Proof, and Logic" text by Barwise? I could use some help, I foresee myself tanking hard, soon.

[Denchi]

I took intro and I'm currently taking Advanced Symbolic Logic. If we are using the same language then I can probably help you. What kind of stuff are you doing at the moment?

[Fistandantilus]

I took it last year, was one of my favorite classes. That stuff is pretty fun honestly, I don't think I missed more than like two days the whole quarter.

[Korietsu]

Propositional Calculus! Basically the same as most boolean/digital logic. Gotta love it as an EE student.

[Tyr]

I took it last year, was one of my favorite classes. That stuff is pretty fun honestly, I don't think I missed more than like two days the whole quarter.

help a brutha out =/

And no, this is I think very basic stuff.

I have to prove the following without a premise:
A v ~(A ^ B)

And I'm getting stuck =/

[Blubbartron]

There's two cases for that problem. Either A is true, or it is false. If A is true, then the entire statement is true via the left side. If A is false, then the interior (A^B) is false, but ~ makes it true, so the statement is yet again true. I don't know if a plain old symbolic logic course requires a proof any more rigorous than that, but it's effectively what you have to show.

[Korietsu]

help a brutha out =/

And no, this is I think very basic stuff.

I have to prove the following without a premise:
A v ~(A ^ B)

And I'm getting stuck =/

What exactly are you trying to prove? That it is equivalent to another proposition?

A or not (a and b)?

A | B | A and B     | ¬ (A and B)     | A or !(A and B)  |
0 | 0 |      0      |        1        |         1        |
0 | 1 |      0      |        1        |         1        |
1 | 0 |      0      |        1        |         1        |
1 | 1 |      1      |        0        |         1        |

Easiest way to do the truth table is piece by piece.

Going to say that http://en.wikipedia.org/wiki/Propositional_calculus + http://www.utdallas.edu/~cpb021000/T...0Functions.pdf or seeing if you can get Discrete Math by Kenneth and Rosin might help.

[Tyr]

We're currently learning introduction & elimination rules and formal/informal proofs. In the problem I'm currently on, it's mainly a proof by contradiction, but the catch is there's no premise. So here's a near-identical one that I did correctly ("law of the excluded middle")

Connectives:
~ = negation (not)
^ = conjunction (and)
v = disjunction (or)
| = contradiction

Goal: P v ~P
1. No premise given (intentional)
2. (Subproof) ~(P v ~P)
3. (Sub-Subproof) P
4. (Sub-Subproof) P v ~P (disjunction intro; 3)
5. (Sub-Subproof) | (contra. intro; 2, 4)
Close Sub-Subproof (back to subproof)
6. (Subproof) ~P (disjunction intro; 3-5)
7. (Sub-Subproof) ~P
8. (Sub-Subproof) P v ~P (disj. intro; 7)
9. (Sub-Subproof) | (contra. intro; 2, 8)
Close sub-subproof (back to subproof)
10. (Subproof) ~~P (negation intro; 7-9)
11. (Subproof) | (contra. intro 6, 10)
Close subproof
12. ~~(P v ~P) (negation intro; 2-11)
13. P v ~P (negation elimination; 12)


So now, my question I'm given is:
1. No premise (intentional)
...
...
prove: A v ~(A ^ B)

I know it's generally the same thing, but I keep getting tripped up.

[Korietsu]

We're currently learning introduction & elimination rules and formal/informal proofs. In the problem I'm currently on, it's mainly a proof by contradiction, but the catch is there's no premise. So here's a near-identical one that I did correctly ("law of the excluded middle")

Connectives:
~ = negation (not)
^ = conjunction (and)
v = disjunction (or)
| = contradiction

Goal: P v ~P
1. No premise given (intentional)
2. (Subproof) ~(P v ~P)
3. (Sub-Subproof) P
4. (Sub-Subproof) P v ~P (disjunction intro; 3)
5. (Sub-Subproof) | (contra. intro; 2, 4)
Close Sub-Subproof (back to subproof)
6. (Subproof) ~P (disjunction intro; 3-5)
7. (Sub-Subproof) ~P
8. (Sub-Subproof) P v ~P (disj. intro; 7)
9. (Sub-Subproof) | (contra. intro; 2, 8)
Close sub-subproof (back to subproof)
10. (Subproof) ~~P (negation intro; 7-9)
11. (Subproof) | (contra. intro 6, 10)
Close subproof
12. ~~(P v ~P) (negation intro; 2-11)
13. P v ~P (negation elimination; 12)


So now, my question I'm given is:
1. No premise (intentional)
...
...
prove: A v ~(A ^ B)

I know it's generally the same thing, but I keep getting tripped up.

Multiply the not through using de morgan's law. Should simplify it.

[Suiram]

With this being a first-year course in symbolic logic, he's probably being taught whatever formal system is preferred by the author of his textbook or his instructor, and is being introduced one-by-one to the various rules of inference in that system and given, at each step, a few problems that utilize those new rules. For that reason, it won't be that helpful to use a truth-table (as it seems they're looking for a syntactic proof), and he may not be able to use De Morgan's laws if they haven't yet been presented in his class--his homework next week could very possibly be to derive them, for that matter.

Anyway, to the OP, you shouldn't feel intimidated by the fact that there are no premises in the assignment. Looking at the problem you went through above, you already have a handful of derivation rules that use sub-derivations to generate formulas without needing any premises. It might be instructive to try to imagine what the sentence you're attempting to prove "means," more-or-less? Something like, "a sentence is either true or it's false that both that sentence and another are true," or however you best understand it. From there, think about why that has be true for all propositions A and B (or in particular, for your exercise, why it's impossible that this could be false) and then see if you can't translate that thinking into a formal proof using the rules you've been taught in the class.

[Zalius]

fuck logic in all of its forms, must math taint all of this world

[Woozie]

Symbolic logic was one of the most fun courses I've ever taken. This is the only class I've ever taken where I looked forward to doing homework. It's kinda like how some people like to do puzzles or sudoku or crosswords and stuff, if they had a symbolic logic section in the newspaper I'd buy one every day.

Edit: The first two thoughts to come to my mind when looking at this is truth table and De Morgan's laws, both of which have already been mentioned. In my class, when we were to prove something was a tautology, we usually used truth tables. If we were to prove a conclusion given certain premises, that's when we'd do it analytically.

[Darus Grey]

Has anyone taken an introductory course in symbolic logic? Not the really tough stuff. I'm taking it at the moment, it's amazingly fun and absurdly mind-breaking at the same time. What sucks is, we get no checks whatsoever on our assignments, and my TA is not very helpful (to put it politely)...

Has anyone taken a course on it, or is familiar with "Language, Proof, and Logic" text by Barwise? I could use some help, I foresee myself tanking hard, soon.

I took it, and several advanced versions when I was an under-grad.

Honestly I was pissed to find out later that it's all bullshit. The premise of philosophical logic is flawed at the base. Because it assumes we manipulate symbols at the level of base-thought in our heads. Which these days we know to be false.

Unless you need it as a degree requirement I wouldn't recommend it, it's useless. I guess it was fun, but it was fun to me in that sense that I thought it was applicable to the world at the time, instead of just an incredibly out-dated set of hypothesis that's taught just out of tradition at this point.

[Sirrus_Asura]

The premise of philosophical logic is flawed at the base. Because it assumes we manipulate symbols at the level of base-thought in our heads. Which these days we know to be false.

What logic did you use to come to this conclusion? (har har)

[Suiram]

That's an...interesting assessment lol Suffice it to say that a lot of people wouldn't agree with much of what you just wrote.

[Darus Grey]

The Neurosciences and Conceptual Metaphor have went a long way in explaining human logic in a way that fills the gaps that classical logic never could, and simply formulated tons of bizarre rules to work around or outright ignore.

As Lakoff points out, the entire basis of classical logic is circular, in that all studies concluded it to be true due to the false premise of using classical logic to prove for itself. Oh and I'm fully aware that most logicians don't agree, but whether they agree or not is irrelevant. Most linguists are still Chomskian linguists despite there being 0 evidence for his hypothesizes, hell I'm sure there's still physicists that believe in Steady State. Academics is like that, when the majority are proven to be wrong, it takes almost a century for universities to catch up, because you will never convince them they're wrong, you simply have to teach the next generation the new theories and hope they return the favor.

[Suiram]

You seem to be approaching the question of its utility from a fairly narrow perspective lol I'm not able to address the particular points you brought up regarding the relation of logic to actual human thought (simply because it's beyond the scope of my own knowledge), but it's somewhat beside the point. Let me just point out a few things in your first post that I'd personally take issue with.

The premise of philosophical logic is flawed at the base. Because it assumes we manipulate symbols at the level of base-thought in our heads. Which these days we know to be false.

What premise would that be? If the suggestion here is that symbolic logic is a model for how actual thought takes place, I'd disagree that this is the premise at all. As I would regard it, the premise is more about the analysis of true propositions that derive their truth from their logical form, rather than by facts about the world. Now whether or not that premise is flawed is, I suppose, also a question that's open to debate, but recognize that the question itself is at least one step removed from how the human mind perceives and actually analyzes those propositions. Either way, this other question is an interesting and perhaps unresolvable one, not the kind you could just dismiss as flawed and move on.

Unless you need it as a degree requirement I wouldn't recommend it, it's useless. I guess it was fun, but it was fun to me in that sense that I thought it was applicable to the world at the time, instead of just an incredibly out-dated set of hypothesis that's taught just out of tradition at this point.

I would also dispute that it's useless. For one thing, in mathematics, there are several fields with a lot of work going on in them for which this class would be an early introduction. But even if he's not interested in mathematics or computer science (or philosophy although you'll probably object if I lump that in here), it's still useful in that it works a person's mind in a way that will make it more natural to analyze arguments and rhetoric down the line, just as a side-effect. And we agree it's fun, although I'm more with woozie in that the problems were fun because of their puzzle-type nature, and not because--like you, apparently--one enters into it with a misconception of its value and utility in certain respects. Sure, if you think that studying logic is going to unlock the human mind (or allow you to prove any sentence in first-order arithmetic) you might be setting yourself up for disappointment, but that doesn't mean that the field or even a cursory study of it is without redeeming value.

[Jodwahh]

I think my brain just melted via this thread.

[Kwijiboe]

Had to master this for LSAT but the notation you use is so foreign to me, lol

[isladar]

Oh man, I'm doing this shit now too; I thought I'd hate it but truth tables are just lovely and elegant. They seem to be giving you a different set up that what I've been exposed to (keep in mind this is a course designed for filthy humanities students), so:

Multiply the not through using de morgan's law. Should simplify it.

^

Oh, and biconditionals are awesome too.