Exploring Logic: Finding Answers to Your Questions

[Nert]

Hey Guys,

I am currently a first year computer science student and I am taking discrete mathematics. After having done the logic portion of discrete, I wanted to learn more about logic so I went online and found Peter Smith's guide on learning logic. I followed his advice under "baby-logic" and picked up Paul Teller's book "A Modern Formal Logic Primer".

After having read the first few pages of Teller's book I got some questions that needs to be addressed, but I wasn't able to find the answers to my questions online. I try to consult my professor, but he told me to come back after learning discrete math!

So here are my questions:
-Can you guy recommend me a reference book on logic which will help to answer some of my questions.

Here is one of my question:

Teller defines an argument as such: Argument is a collection of declarative sentences one which is called conclusion and the rest of which are called the premises. I know that I am trying to get you to accept the conclusion as true by providing you with premises.

-Is the above definition of an argument solid ?
-What is the purpose of a premise in an argument? How does a premise relate to conclusion?

 

[jedishrfu]

Welcome to PF!

I would suggest you create a separate thread for each question. It makes it a lot easier for PF members to form a discusion with comingling multiple questions together. I would also suggest that you place your questions directly in the post and not in a pdf file as some people are less likely to look at it.

 

[Nert]

Sorry, this my first thread. I have made the some changes.

 

[Stephen Tashi]

An informal definition of the study of logic is:

Logic is the study of reliable methods of reasoning.

Logic isn't the study of "Truth". It doesn't deal whether statements about physics, politics or anything else are true or false. Instead it studies how one may deduce correct statements from other correct statements. So applying logic requires that you have some correct statements to begin with. These are the premises.

In common speech, saying a statement is "logical" means that it is plausible, believable, true etc. However, mathematical logic only deals with methods of proceeding from statements assumed to be true to other statements that must be true if those assumptions are correct.

Teller's definition is OK if he intends to use "argument" to mean a theorem of logic. I haven't read his book, but my guess is that he wants an "argument" to be something like:

Premises:
If A is true then B is true
A is true
----
Conclusion:
B is true

So he isn't defining an "argument" to be a "proof" since a proof would have intermediate "steps" between the Premises and the Conclusion.

 

[jedishrfu]

Have you done any geometric proofs? The proof is the argument, the premises are the step by step truths that you make to get to the last premise which is the conclusion.

Here's some simple examples and the associated proofs (click on the proof link to see it):

http://regentsprep.org/regents/math/geometry/gp4/praccongtri.htm

 

[Nert]

An informal definition of the study of logic is:
Instead it studies how one may deduce correct statements from other correct statements. So applying logic requires that you have some correct statements to begin with. These are the premises.

In common speech, saying a statement is "logical" means that it is plausible, believable, true etc. However, mathematical logic only deals with methods of proceeding from statements assumed to be true to other statements that must be true if those assumptions are correct.

Okay. I guess what you are trying to say is that:

-Logic is the process of how to deducing a new correct statement from a given correct statement. Also what do you mean by 'correct' are you talking about the truth?

But I am still not understanding the given definition of mathematical logic, because is seems very similar to the definition of logic

 

[jedishrfu]

Wikipedia has a write up on mathematical logic that may clear things up.

http://en.m.wikipedia.org/wiki/Mathematical_logic

 

[gopher_p]

Teller defines an argument as such: Argument is a collection of declarative sentences one which is called conclusion and the rest of which are called the premises. I know that I am trying to get you to accept the conclusion as true by providing you with premises.

-Is the above definition of an argument solid ?
-What is the purpose of a premise in an argument? How does a premise relate to conclusion?

I would say that Teller's definition of argument is fine in that it is (1) meaningful and (2) "approximately standard". Assuming everyone is on the same page - at least temporarily - with the use of the terms "collection" and "declarative sentences", this definition for argument is meaningful in that it is relatively unambiguous; you have a collection of declarative sentences, and there are two disjoint sub-collections, one of which has a single declarative sentence. It is "approximately standard" in that it basically matches up with the everyday meaning of "argument" (at least as it pertains to the act of trying to convince someone of something) as well as the meaning of the word as most mathematicians understand it.

Note that he also defines the words premise and conclusion here as well; the conclusion is the declarative sentence in the sole member of the singleton sub-collection, and the premises are members of the other sub-collection. And while we can presume what the intended meanings of those words are based on our current understanding of those words, we need to be careful assuming that we know how he intends to use the premises and conclusion of the argument. Now he does give us that little hint there - the part in italics after the definition - to let us know that he does eventually intend for premise and conclusion to mean something approximating their ordinary meanings. But we'll need to wait to see how he does that.

Also note that he has not yet defined what it means for an argument to be correct, nor has he stipulated that an argument be correct or even coherent. He also has not stipulated that the premises of the argument be "true".

At the end of the day, they're his definitions, and you'll need to accept them if you want to continue reading his book. If his definitions contradict something that you think is "true" about the words argument, premise, and conclusion, then you might have to temporarily discard your biases about the meaning of the words in order to progress without confusion. One of the more difficult parts of studying mathematical logic in my opinion is the necessity for relieving oneself of one's preconceived notions of what all of the words and symbols and ideas are "supposed" to mean in order to understand what the author is actually saying. It's often the case that the author is saying something much simpler than the reader is thinking. In this case, the author is saying,

Hey, there's this thing that I'm going to call an argument, and that thing is a collection (whatever that means) of declarative sentences (whatever those are). Furthermore, I can split this collection into two smaller collections, the premises and the conclusion. The conclusion is just one of the declarative sentences of the argument. Parenthetically, I intend for the premises and the conclusion to be used in the way that you think they should be used, but we'll get there when we get there.

 

[Stephen Tashi]

Okay. I guess what you are trying to say is that:

-Logic is the process of how to deducing a new correct statement from a given correct statement.

Yes.

Also what do you mean by 'correct' are you talking about the truth?

One can write down strings of words and symbols that have no meaning. Very formal mathematical logic needs rules about what strings of symbols form a "statement - just as computer languages need rules about legal syntax.. So I picked the word "correct" to indicated "meaningful and true".

But I am still not understanding the given definition of mathematical logic, because is seems very similar to the definition of logic

What do you count as the usual definition of logic? There is a study of the logic people use in debating that includes psychological things like "ad hominem arguments". (I think that's called the study of "rhetoric", or at least that term was used in the old days). Since the book refers to "declarative sentences" it sounds as if it's not yet dealing with mathematical logic. "Declarative sentences" sounds like a term from "rhetoric".

 

[WWGD]

I think another way of seeing it is that Sentence Logic, aka, Truth-Functional Logic, which is what I think you are using, is the study of valid argument, which comes down to separating valid arguments from non-valid ones. Here, each atomic sentence is assigned , basically somewhat-arbitrarily (meaning without any concern for the inner-structure of the sentence), a truth value in {T,F} , and the connectives, which are binary functions : {T,F}x{ T,F} --> {T,F} , provide the rules whereby a sentence put together from atomic sentences and connectives is true or not.
A(n) argument in TFL is valid if whenever the premises are true, the conclusion must be true. A counterexample is an invalidation of an argument, meaning an assignment of truth values to the atomic sentences in the set of premises that makes the set of premises true and the conclusion false.

For a version of logic where the intrinsic content of atomic sentences matter, you go into predicate logic, where truth-values can be assigned more in relation to the intrinsic meaning of the atoms. This is part of Model Theory. Unfortunately, while TFL is decidable, Sentence Logic is not.

 

[verty]

There is a rule called Modus Ponens, it relates a premise to a conclusion. Specifically it says, if A implies B, and A is true, then B must be true. Essentially it encodes half of the meaning of "A implies B". There is an opposite statement, Modus Tollens. If A implies B, and B is false, then A must be false. You will have a number of statements of the sort "A,B,C imply D,E,F", you will have a number of premises "A, ~B, C" and by applying those two rules, you see what you can derive. That's the best answer I can give in a short time.

 

[atyy]

I liked the explanation in the introduction (p2-6) of Cori and Lascar's book https://www.amazon.com/dp/0198500483/?tag=pfamazon01-20.

Robert Wolf's https://www.amazon.com/dp/0883850362/?tag=pfamazon01-20 is also quite friendly.