Stumped on Basic Sentential Logic!

[trickster]

I would avoid confusing the issue with what other things Tinker might possibly be. Female cat. Neutered cat. Self-impregnating alien space cat…

The only thing the argument subject hinges on is whether Tinker can or cannot be a “male cat”.

Yes, and that seems to be consistent with the fact that - at this stage of the game - sentential or propositional logic - (We’re not yet at predicate logic) those are five symbols. So that makes sense to me.

Thank you

Bruce

 

[trickster]

You have all missed the point.

No one has recognized the the relationship between Tinker (a TOM cat) and Clinton.

It should be obvious…

Well that makes the same sense that many other things in “logic” seem to make at this point in time… but I think Lion is correct.

Thanks

Bruce

 

[trickster]

If I am reading it correctly, the statement (b) Tinker will have kittens is indeterminate which means it might be true or it might be false, we don’t know. We can’t say that Tinker is a male cat, only that logically Tinker could be a male cat given the statements.

‘Could’ isn’t good enough In logic to suggest a universally TRUE statement. So we can’t say Tinker was a male cat, but likewise we can’t say Tinker was not a male cat.

Because Tinker will have kittens in indeterminate, then he might be male, or she might be female.

If i’m reading it right.

Sounds right to me! So I think that it is the indeterminant statement that prevents us from creating a logically valid arguements under the VERY elementary rules of sentential logic. The part I am confused about is how to read “Not both” “not” or “both” within the five logic operators I have to operate at this stage…in terms of symbolic formula. The operators I have to work with at this stage are “and”, “or”, if -then", If and only if" and the negation. - it is the negation in Not both…am I negating both Clinton and Dole, am I negating either Clinton or Dole… do you see my confusion here?

Bruce

Thanks!

 

[trickster]

This is the one that has me baffled. I understand the difference between an invalid argument and an unsound argument. And surely this argument is invalid because neither (a) nor (b), whether true or false, compel us to think that Clinton had to be president in 1999

But why is the first premise indicated as (F) False?

Re-worded
(a) Clinton and Dole were not both president in 1999. (It’s possible that two different people can be president in the same year.)
(b) Dole was not president in 1999
(c) Therefore: There can be no other possibilty other than Clinton was president in 1999 (Obviously there could have been other people who became president apart from Clinton or Dole.)

Yeah, its the negation that confuses me…

Ah, the struggle must continue !

Bruce

 

[trickster]

Understand the concepts of validity and soundness. Look for literature on conditional statements and understand what it is that we are trying to accomplish with respect to the various forms of conditional statements (more than one type of conditional relationship in terms of entailment…also research entailment). Anyway, I will try my best to help and I will only address the first argument (which is likely a material conditional):

P1 (premise 1): Is a conditional statement. It’s important that you understand the relationship between necessary and sufficient conditions so I will not reveal too much (you are more than capable of just reasoning through this one). But what I have said alone should motivate you to research the topic. Which if you do, I am positive will help you gain clarity.

P2 (premise 2): is an assertion. And as it pertains to this arguments form just so happens to be an assertion that negates (logical vocabulary also…should research) the consequent of the conditional established in P1.

And P3 (premise 3 or more properly the conclusion as indicated by: ‘therefore’) is the affirmation of the antecedent based SOLELY on the NEGATION of the consequent in P1.

Thus, the arguments form is invalid. It denies the consequent to affirm the antecedent. Usually, when you see an argument of this form you will know that it is invalid (proceed with caution though). As tautologies and necessary truths confuse the general framework for discerning which argument forms are logical or not. But that will come later if choose to continue training in logic.

Consider this argument:
If the grass is green then, the sky is blue
the sky is NOT blue
Thus, the grass is green

This argument takes the same shape as the one you have posted. If you see how this is invalid then you will see how the argument you posted is invalid.

Also just a personal note: begin by falsifying the conclusion and then looking to see if the premise could be true while the conclusion false. If the premises can all remain true while the conclusion false then the argument will always be invalid (except in rare occasion involving necessary truths).

Thank you Riz0! Wow. That’s great! My understanding of the conditional is that it is a “if and only if” statement… am I right? In any case, if you don’t mind, I want to share your advice with my professor whom I am meeting tommorrow to get some help…and with what you are saying and what he is teaching… I should get there… this is deep stuff for me

Bruce

 

[trickster]

The informal intuitive method: try to think of a counter example: Tinker is a spayed female cat for example. Then Tinker will not have kittens, and if Tinker were a male cat then (anything) because she’s not a male cat. This relies too much on the inner meaning of sentences to be a good answer in sentential logic, but this sort of thinking can help you figure out what direction you should be going when you move to the more formal method, at least until you get used to it.

An argument is valid if for all ways whatsoever of assigning True and False to the basic sentences (whether this way corresponds to reality or not) such that the premises are all true, the conclusion is also true.

Correspondingly, an argument is invalid if there is any way whatsoever of assigning True and False to the basic sentence letters so that the premises are all true and the conclusion is false - such a way being called a counterexample. Generally, the best way to show an argument is invalid is to find such a counterexample

The formal sentential method: Translate it into letters: Let A be Tinker is a male cat, B be Tinker will have kittens.

The argument becomes:
A=>not B
B
Therefore A

This argument is invalid if there is any way to assign truth values to A and B such that the premises are true and the conclusion false. Assign F to A, T to B.

Then:

A=>B is F =>F, which is true according to truth table definitions.
B is True
And A is False.

So all the premises are true and the conclusion is false (under this truth assignment, or valuation), and so the argument is not valid.

Not ____ can be read “it is not the case that ___”

Do the letter thing again: let A be “Clinton was president in 1999” and B be “Dole was president in 1999”. Remember that whether an argument is valid has absolutely nothing to do with whether or not the premises ARE true, only if the conclusion must be true if they WERE true. (For example. the argument “All cats are purple”; “If all cats are purple, then cheese is made out of microwave radiation”; therefore “cheese is made out of microwave radiation” is perfectly valid, even if it is kind of stupid.)

This argument then becomes

NOT (A AND B)
NOT B
Therefore A

Note that NOT ___ is true whenever ___ is false, and vice versa. So NOT (A AND B) is true so long as (A AND B) is false, which is the case so long as at least one of A, B are false.

Again, intuitively, it is possible that someone other than Clinton or Dole was president in 1999. Now, Clinton actually was president in 1999, but that doesn’t actually matter - the question is, if all you knew was that Clinton and Dole were not both president in 1999 and also that Dole was not president in 1999, is that alone enough for you to be certain that Clinton was president?

So again, to do this legitimately, try to find a valuation that is a counter example - assign T and F to A and B in some way so that each premise is true and the conclusion is false.

Good luck on your exam.

Yes, counter arguments have proven problematic for me… all I know about them is their term (I’ve only been in class for three weeks), and I am following you, but I don’t pretend to understand everything you said. I will follow this through… and see how it works…thank you so much!

Bruce Ferguson

 

[rIz0]

Bruce of course for more than welcome to share this information with your professor.

An ‘if and only if’ conditional statement is unique and is what is known as a biconditional. An ‘if then’ statement is usually going to be either a counterfactual conditional claim or a material conditional claim. Each of these three types of conditional statements have different truth conditions. Also just a note that may help: premises or claims, assertions of/in an argument as being taken by themselves are never valid or invalid they are only ever true or false.

The material conditional is true in all cases except where the antecedent is true and the consequent false. You can express these conditions in a truth table if you wish but it’s much easier to just reason through it yourself.

Ex: if the sky is blue then, the sun is out

So to evaluate the truth of this claim with reference to the truth conditions set forth by the material condition all we have to do is look at the antecedent (if clause) and see if it is true or false. So here this may involve walking outside and looking at the sky. If the sky in fact is not blue then we know straightaway that the conditional claim is true (fine might be the better way to observe what we are talking about) logically speaking. This is where so many of us struggle because we want to say rather intuitively that the claim is false straightaway when we have disproven the epistemic truth of the first half of a conditional claim. But we must keep in mind what is being said with reference to the claim (being a conditional) and whether it is actually (the claim itself) false when taken as whole (and in reference to its truth conditions and the semantic rules of logic-- so important to understand necessary and sufficient conditions). Anyway, If yes the sky is blue then, we proceed to the antecedent: is the sun out? if yes the conditional claim is true and if false the conditional claim is as a whole false: because it’s antecedent was T while it’s consequent was F. What’s cool about this conditional claim and why I used it (not that’s it something reverent or unique haha) is that based on what we know about our material world it is going to hold true or false in a varying number of instances (weather, etc).

I won’t go into the semantic rules of the other conditional types but this should help a little. Most of the time what philosophers are debating about involves arguments whose premises contain counterfactual conditional. Which are simply conditional claims whose antecedent is from the get go contrary to fact (go figure) or false. This being said the semantic rules for establishing the truth of conditional claims of this type is much more interesting than are the others and much more controversial because they involve metaphysical and ontological considerations. Such as oh I don’t know the debate as to whether space is material or immaterial. Anyway, did this rambling in between work hope it helps a little and hope it isn’t as scatterbrained as it seemed to me while writing it. Check out a book called Language, Proof, and Logic. An introductory book which introduces you to first order predicate logic or symbolic logic. Which helps logicians as it is more equipped in it’s functions to handle claims that otherwise cannot be expressed with just the simple use of prepositional logic alone. BUT YOU MUST BUY: A RULEBOOK FOR ARGUMENTS: ANTHONY WESTON (short read and it will help you tremendously)

 

[Rhubarb]

A Modern Formal Logic Primer by Paul Teller is what we used for Intro to Sentential Logic. It was a very good textbook. I think you can find it for free if you look around, Professor Teller has made it available for free once the copywrite devolved back to him.

You can tell without a doubt that the Tinker argument is invalid through a truth table. That’s how we had to do it. We did a lot of truth tables >.>

Also don’t feel discouraged. Logic seems simple enough but it can get tricky very fast. Some people have no problem picking it up without even trying while some people struggle. I’m super-dyslexic myself and found the symbols VERY trying. Study hard, read and re-read your book and practice the proofs and you’ll do well.

 

[trickster]

Bruce of course for more than welcome to share this information with your professor.

An ‘if and only if’ conditional statement is unique and is what is known as a biconditional. An ‘if then’ statement is usually going to be either a counterfactual conditional claim or a material conditional claim. Each of these three types of conditional statements have different truth conditions. Also just a note that may help: premises or claims, assertions of/in an argument as being taken by themselves are never valid or invalid they are only ever true or false.

The material conditional is true in all cases except where the antecedent is true and the consequent false. You can express these conditions in a truth table if you wish but it’s much easier to just reason through it yourself.

Ex: if the sky is blue then, the sun is out

So to evaluate the truth of this claim with reference to the truth conditions set forth by the material condition all we have to do is look at the antecedent (if clause) and see if it is true or false. So here this may involve walking outside and looking at the sky. If the sky in fact is not blue then we know straightaway that the conditional claim is true (fine might be the better way to observe what we are talking about) logically speaking. This is where so many of us struggle because we want to say rather intuitively that the claim is false straightaway when we have disproven the epistemic truth of the first half of a conditional claim. But we must keep in mind what is being said with reference to the claim (being a conditional) and whether it is actually (the claim itself) false when taken as whole (and in reference to its truth conditions and the semantic rules of logic-- so important to understand necessary and sufficient conditions). Anyway, If yes the sky is blue then, we proceed to the antecedent: is the sun out? if yes the conditional claim is true and if false the conditional claim is as a whole false: because it’s antecedent was T while it’s consequent was F. What’s cool about this conditional claim and why I used it (not that’s it something reverent or unique haha) is that based on what we know about our material world it is going to hold true or false in a varying number of instances (weather, etc).

I won’t go into the semantic rules of the other conditional types but this should help a little. Most of the time what philosophers are debating about involves arguments whose premises contain counterfactual conditional. Which are simply conditional claims whose antecedent is from the get go contrary to fact (go figure) or false. This being said the semantic rules for establishing the truth of conditional claims of this type is much more interesting than are the others and much more controversial because they involve metaphysical and ontological considerations. Such as oh I don’t know the debate as to whether space is material or immaterial. Anyway, did this rambling in between work hope it helps a little and hope it isn’t as scatterbrained as it seemed to me while writing it. Check out a book called Language, Proof, and Logic. An introductory book which introduces you to first order predicate logic or symbolic logic. Which helps logicians as it is more equipped in it’s functions to handle claims that otherwise cannot be expressed with just the simple use of prepositional logic alone. BUT YOU MUST BUY: A RULEBOOK FOR ARGUMENTS: ANTHONY WESTON (short read and it will help you tremendously)

Riz0. OMG thank you so much! I don’t pretend to understand everything you say, but slowly I am understanding more… and I take that as a good thing. It seems that I need a dictionary when getting the explanations to my questions and by the time I get through the definitions I have already forgotten my question Anyways, I look forward to many more talks with you on this stuff… predicate logic is next …can’t wait (Well yeah I can

What is your relationship to philosophy? Is it a natural interest. I can see how it will be influencial in helping me think through what other people argue and already i can see how it will help me form better arguements… how has this logic helped or hindered you in life?

Bruce

 

[trickster]

A Modern Formal Logic Primer by Paul Teller is what we used for Intro to Sentential Logic. It was a very good textbook. I think you can find it for free if you look around, Professor Teller has made it available for free once the copywrite devolved back to him.

You can tell without a doubt that the Tinker argument is invalid through a truth table. That’s how we had to do it. We did a lot of truth tables >.>

Also don’t feel discouraged. Logic seems simple enough but it can get tricky very fast. Some people have no problem picking it up without even trying while some people struggle. I’m super-dyslexic myself and found the symbols VERY trying. Study hard, read and re-read your book and practice the proofs and you’ll do well.

Rhubarb, thank you for the reference and the encouragement… I am a right brain creative kind of guy, so this left brain stuff is throwing me for a loop But as you can see in this thread, lots of great support from fellow Catholics. We do have a great tradition of philosophy in our church…I am even thinking that I need to start praying to the saints who were philosophers to get them to help me

Anyways Rhubarb, look forward to your thoughts and comments as I move forward in this stuff…predicate logic is next…oh goodie!

Bruce

 

[Rhubarb]

I’m starting predicate logic winter quarter this year and wondering if I’m really masochistic enough to give meta and modal a try. Whenever I got stuck, I always found it was helpful to go back to the truth table. It’s a pain filling out tables, especially with a lot of connectives and especially moreso with 4 or more atomic sentences in the argument but it really clarified what I was looking at.

 

[Iron_Donkey]

I’m starting predicate logic winter quarter this year and wondering if I’m really masochistic enough to give meta and modal a try. Whenever I got stuck, I always found it was helpful to go back to the truth table. It’s a pain filling out tables, especially with a lot of connectives and especially moreso with 4 or more atomic sentences in the argument but it really clarified what I was looking at.

I particularly find modal logic interesting, but it definitely builds off predicate/propositional logic. If those come naturally to you, then modal likely will as well, but if any issues with them will also likely carry over.