Help with Logic: Deducing Argument Form in Formal Language

[Juval]

Hi there.

I need help deducing an argument form in formal language, but I am not sure that I can communicate the example very well, nor even that it can be expressed in a formal language for that matter.

The case at hand:

Rene is a cat.

Julie is not a cat.

Therefore, julie is a human.

Or:

People have died paragliding.

Bruce is going paragliding.

Therefore, bruce is going to die.

Of course, both of these examples are incorrect as they draw conclusions that are not necessary consequences of their premise. I guess what I am after is simply a counterexample to this sort of case where the conclusion is not necessarily true, even though there can be cases in which it is. How would you express such a counterexample in a formal language?



EDIT: How does one go about writing the language anyway? How do the guys here write down mathematical equations when none of the components exist on normal keyboards?

 

[Stephen Tashi]

How would you express such a counterexample in a formal language?

Look at the Wikipedia article on "First order logic". That would be one way. Expressing the difference between tenses of verbs ( for example: "go paragliding" vs "is going paragliding") isn't so cut and dried in such a formal language. If you want to do that right, you might need a formal language that implements "temporal logic".

EDIT: How does one go about writing the language anyway? How do the guys here write down mathematical equations when none of the components exist on normal keyboards?

On this forum and many other math forums, mathematical symbols may be written using the system called "LaTex" ( not the same as the rubbery substance "latex").

Pick a post that has such symbols in it and initiate the process "Reply with quote". You'll see how the equation is written in LaTex and you will see the "tags" "tex" and "/tex" or "itex" and "/itex" in square brackets. On this forum, these tags are used to begin and end a LaTex line. (On other forums, you might see the tags "math" and "/math" used.)

One attempt at the paragliding argument is to use the abbrevations:

$P(x): x$ is a person
$G(x) : x$ goes paragliding
$D(x): x$ dies
$b:$ bruce

One translation of the argument is

$$( (\exists x) \{P(x) \wedge G(x) \wedge D(x) \} \wedge ( P(b) \wedge G(b) ) )\Rightarrow D(b)$$

You can usually find out the appropriate Latex symbol for something by Googling for LaTex and some words that describe the type of symbol. For example, to do the above, I looked for words "LaTex logic".

 

[Juval]

One attempt at the paragliding argument is to use the abbrevations:

$P(x): x$ is a person
$G(x) : x$ goes paragliding
$D(x): x$ dies
$b:$ bruce

One translation of the argument is

$$( (\exists x) \{P(x) \wedge G(x) \wedge D(x) \} \wedge ( P(b) \wedge G(b) ) )\Rightarrow D(b)$$

Thanks for the input.

What do you mean when you say this is "one" attempt/translation of the argument? Is this because P, G, D, b are the symbols representing the given cases, which are not 'constant' or 'universal' in logic like the rules are, and thus the symbols representing them are also not? These non-established representative symbols are then structured and constructed into the argument forms by incorporating the established 'rules' around them. Is this correct? So P,G,D,b are the argument's 'semantics', and the established gibberish symbols are the 'syntax' or 'rules' that govern and structure the argument and give the argument 'form', literally, right?

And one more thing, why did you establish the person P separate to Bruce b, when in fact they are one and the same person? Why separate them?

Thanks again.

 

[Stephen Tashi]

why did you establish the person P separate to Bruce b, when in fact they are one and the same person? Why separate them?

"Bruce" could be a dog. We make many assumptions about normal speech and this is one reason that there is no unambiguous way to translate short verbal examples into a precise symbolic language. I chose to include the information that "Bruce" is a person. "P(b)" doesn't establish "a person P separate from b", it establishes that "b is a person". The thing represented by the variable "x" is not necessarily different than "b", although it might be.

You are correct that writing verbal expressions in symbolic logic requires that we define the semantics of our symbols, such as "P(x)". I would not call the symbolic expression of the argument, the "syntax" of the argument. The argument itself is not a language. The symbolic language in which the expresson is written has a syntax.


The symbols and syntax used in symbolic logic are not completely standardized. When you are reading a particular text, you must pay attention to how that text defines them.

Mathematicians who are not interested in symbolic logic still use the language of symbolic logic for the sake of clarity and brevity. I put myself in this category.

People interested in symbolic logic for it's own sake can tell you more than I can. I can't resist noting that when such people discuss something, they often have fundamental disagreements about technicalities. So my impression of the field of symbolic logic is that it is not as standardized as other areas of mathematics.

You'll notice that I didn't answer your original question. You asked about how to express a counterexample symbolically. I think a counterexample must involve semantics, so I don't think expressing a counterexample (meaning a concrete, specific example like "Bruce Mermington went hang gliding, but he didn't die") in first order logic is purely a matter of using symbols. You can write the symbolic form of an argument and point out (in English) that it is not a valid form of reasoning. Elementary textbooks on symbolic logic do this. If you want a more advanced approach, get an expert in logic to tell you about "model theory".

 

[Juval]

"Bruce" could be a dog. We make many assumptions about normal speech and this is one reason that there is no unambiguous way to translate short verbal examples into a precise symbolic language. I chose to include the information that "Bruce" is a person. "P(b)" doesn't establish "a person P separate from b", it establishes that "b is a person". The thing represented by the variable "x" is not necessarily different than "b", although it might be.

You are correct that writing verbal expressions in symbolic logic requires that we define the semantics of our symbols, such as "P(x)". I would not call the symbolic expression of the argument, the "syntax" of the argument. The argument itself is not a language. The symbolic language in which the expresson is written has a syntax. The symbols and syntax used in symbolic logic are not completely standardized. When you are reading a particular text, you must pay attention to how that text defines them.

Mathematicians who are not interested in symbolic logic still use the language of symbolic logic for the sake of clarity and brevity. I put myself in this category.

People interested in symbolic logic for it's own sake can tell you more than I can. I can't resist noting that when such people discuss something, they often have fundamental disagreements about technicalities. So my impression of the field of symbolic logic is that it is not as standardized as other areas of mathematics.

I see now. I established that Bruce is going paragliding, but I did not establish the condition that Bruce is in fact a person.

What I meant when I said that the standardized symbols are the syntax, or more aptly, the "rules" that govern the structure of the given semantics is as follows: As I understand, like natural languages, formal languages also have syntax. For example, there are both atomic and molecular sentences, and of course, connectives such as conjunction, disjunction and negation. I know that in logic, negation is expressed by the symbol $\neg$. So what I mean is that I must personally provide the non-standardized semantics for my argument (whatever they may be), and upon having defined them, I may then structure them with the standardized rules such as $\wedge$ (and), or $\neg$ (not).

Concerning the math encoding, if I go to a philosophy forum for help with logic, it will have something such as Latex right? And also, is there any way that such language can be written out on non-math related forums?

You'll notice that I didn't answer your original question. You asked about how to express a counterexample symbolically. I think a counterexample must involve semantics, so I don't think expressing a counterexample (meaning a concrete, specific example like "Bruce Mermington went hang gliding, but he didn't die") in first order logic is purely a matter of using symbols. You can write the symbolic form of an argument and point out (in English) that it is not a valid form of reasoning. Elementary textbooks on symbolic logic do this. If you want a more advanced approach, get an expert in logic to tell you about "model theory".

I did notice that in the example you gave me, you did not express the condition making it necessary that anybody going paragliding dies... i.e the simple action of paragliding equals death.

Correct me is I'm wrong, but I think that you could well have written a counterexample for it. I imagine you could have done so by expressing a case in which bruce does go paragliding, but then simply negating the apparently necessary outcome of death... which is by all means possible within the laws of physics. What do you think?

 

[Stephen Tashi]

I see now. I established that Bruce is going paragliding, but I did not establish the condition that Bruce is in fact a person.

What I meant when I said that the standardized symbols are the syntax, or more aptly, the "rules" that govern the structure of the given semantics is as follows: As I understand, like natural languages, formal languages also have syntax. For example, there are both atomic and molecular sentences, and of course, connectives such as conjunction, disjunction and negation. I know that in logic, negation is expressed by the symbol $\neg$. So what I mean is that I must personally provide the non-standardized semantics for my argument (whatever they may be), and upon having defined them, I may then structure them with the standardized rules such as $\wedge$ (and), or $\neg$ (not).

In the elementary study of mathematical logic, examples like Bruce going paragliding, are presented as textbook exercises and to solve them, a student must specific a semantic interpretation. I have never seen any "serious" use of such verbal examples by mathematicians. So I'll say that it is true that one must personally provide the semantics in order to express verbal arguments in symbolic logic, but I don't want to give yo the impression that this is an important application of symbolic logic.

There may be people who have studied symbolic logic and attempt to apply it to problems stated in natural language. As far as I know, these are isolated cases and have not developed into any kind of intellectual discipline.

Some texts on logic use an informal syntax. The more advanced ones have rigorous syntax rules. If you are familiar with the syntax rules used by computer languages, that is an example of rigorous syntax. Natural languages don't have a rigorous syntax. Specifying the meaning of various symbols in terms of natural language isn't enough to define a rigorous syntax.

Concerning the math encoding, if I go to a philosophy forum for help with logic, it will have something such as Latex right? And also, is there any way that such language can be written out on non-math related forums?

I haven't joined any philosophy forums, so I don't know about them. I haven't tried writing equations on non-math related forums. Based on some directions for writing equations that I once saw on bautforum.com, you may be able to embed an html reference to a latex interpreter site in your message and have it display your LaTex. I don't recall the details of how to do this.

I did notice that in the example you gave me, you did not express the condition making it necessary that anybody going paragliding dies... i.e the simple action of paragliding equals death.

Correct me is I'm wrong, but I think that you could well have written a counterexample for it. I imagine you could have done so by expressing a case in which bruce does go paragliding, but then simply negating the apparently necessary outcome of death... which is by all means possible within the laws of physics. What do you think?[

First, let's make sure the goal of logic is clear. Logic is the study of reliable methods of reasoning, not the study of objective facts about the real world. To critique an argument from the "logical" point of view, we must ignore the semantics of the real world. A typical argument asserts that if some premise (which may be a complicated statement, consisting of several component statements) is true then some other statement is true. To test if an argument is valid or invalid, we may arbitrarily assigns the values "T" or "F" to any statement in he argument. If we discover an instance where the premise is 'T" and the conclusion is 'F' then the argument is invalid.

My remarks about a counterexample may be misleading since I used phrases such as "Bruce goes paragliding". That is merely an illustration of the fact that one may arbitrarily assign a "T" value to a statement in a argument while testing it's validity. So the "semantics" of evaluating an argument are no more than assigning a specific value ("T" or "F") to the statements in it.

For example, the argument:

Premise: (A implies B) and B is true
Conclusion: A is true

is invalid.

It can be presented verbally as:

Premise: (If a person has nonzero mass then that person has nonzero weight on the surface of the earth) and (The person Bruce had nonzero weight on the surface of the earth)

Conclusion: The person Bruce has nonzero mass

To show it is invalid, one may specify that case where "The person Bruce has nonzero mass" is "F". The real world semantics of such a specification contradict physical facts about people but it is permitted. (If the physical facts are to be taken into account, they would have to be added as premises to the argument.)

(Edit: In the above symbolism, I should have used quantifiers, and made something like
Premises: (For each x, (P(x) and M(x)) implies W(x)) and W(b)
Conclusion M(b)

but I hope you get the idea anyway.)
It's relatively pointless to appeal to symbolic logic in debates about real world issues such as politics or economics. A complete statement of the premises needed for an argument with a significant conclusion would be too vast. Although commentators enjoy criticizing each others "logic", it is a trivial game to play. No commentator can write a readable essay and state all the necessary premises. Most criticisms of "logic" in political debates are (correctly) criticisms about unstated assumptions being made. But there is always going to be debate about what assumptions to make and nobody can write an readable essay about a complicated real world problem that lists all the necessary assumptions. That's why all commentators need a somewhat "like minded" audience.

 

[Bacle]

Maybe to reinforce Stephen's points (and to show of the two things I remember about logic), the claims you make maybe true in some worlds (a world is a specification of the collection of subjects , predicates , and which predicates

apply to which subjects). Once your world is defined clearly-enough, your claims may or

not be satisfied , i.e., mapped into truths. If your claims are mapped into truths (by the precise rules of Tarski's definition), then your world (aka interpretation) is a model. If your claims are true for any/every model, then it is a tautology; if it is not satisfied by any model, I think it is defined to be a contradiction.

You can see more of this in the T.V show: Ultraproducts: America's new Supermodel.