Thoughts on symbolic logic to assess the truth of arguments

Main article: Logic

Assessing validity using propositional logic

Symbolic logic can be used to assess the validity of arguments without distracting, or even deceiving, ourselves with English verbiage. Here we will confine ourselves to propositional logic in which propositions (or statements) have a truth value and can be combined together to form conclusions.

Let’s look at an example:

P1) John is at the Library or he is Studying (L or S, ie ${\textstyle L\lor S}$ )
P2) John is not Studying ( ${\textstyle \neg S}$ )
C) John is at the Library ( ${\textstyle L}$ )

The truth table for this situation is as follows and simply involves writing all the possible True/False combinations of L and S followed by the calculated values of the propositions (or premises) and conclusion. We check for invalidity by seeing if there are cases where all the premises are true and the conclusion false. If this is not the case, the conclusion is valid.

${\textstyle L}$	${\textstyle S}$	${\textstyle \neg S}$	${\textstyle L\lor S}$	${\textstyle L}$	Validity
T	T	F	T	T	ok
T	F	T	T	T	ok
F	T	F	T	F	ok
F	F	T	F	F	ok

The conclusion is valid because there is no case in which the premises are all true and the conclusion false. If we had concluded that John is not at the library, our truth table would look like:

${\textstyle L}$	${\textstyle S}$	${\textstyle \neg S}$	${\textstyle L\lor S}$	${\textstyle \neg L}$	Validity
T	T	F	T	F	ok
T	F	T	T	F	invalid
F	T	F	T	T	ok
F	F	T	F	T	ok

The argument is invalid because it allows for a plainly self-contradictory situation. This is the standard method by which we assess propositional logic arguments. Note that one big advantage of this is that once the translation from English statements to symbolic statements is done, the methodology can be automated.

Generally speaking we will be using the following logical symbols:

Negation, not A: ${\textstyle \neg A}$

A And B: ${\textstyle A\land B}$

A Or B: ${\textstyle A\lor B}$

If A then B: ${\textstyle A\implies B}$

A if and only if B: ${\textstyle A\iff B}$

There are more but let’s start here.

For reference the truth table for these operations is as follows:

${\textstyle A}$	${\textstyle B}$	${\textstyle A\land B}$	${\textstyle A\lor B}$	${\textstyle A\implies B}$	${\textstyle A\iff B}$
T	T	T	T	T	T
T	F	F	T	F	F
F	T	F	T	T	F
F	F	F	F	T	T

We will be using these in the material that follows.

Let’s apply this to a climate change argument. Suppose we state that climate change is real (C) if and only if temperature of the earth is rising (T) and humans have caused the temperature to rise (H):

${\textstyle C\iff (T\land H)}$

This is our premise. Now let’s construct a truth table to verify the following conclusions:

${\textstyle (T\land H)\implies C}$
${\textstyle (H\implies T)\implies C}$
${\textstyle \neg H}$

${\textstyle T}$	${\textstyle H}$	${\textstyle C}$	${\textstyle C\iff (T\land H)}$	${\textstyle (T\land H)\implies C}$	${\textstyle (H\implies T)\implies C}$	${\textstyle \neg H}$
T	T	T	T	T	T	F (INV)
T	T	F	F	F	F	F
T	F	T	F	T	T	T
T	F	F	T	T	F (INV)	T
F	T	T	F	T	T	F
F	T	F	T	T	T	F (INV)
F	F	T	F	T	T	T
F	F	F	T	T	F (INV)	T

Here we see that the first conclusion is valid, which is expected. The second conclusion is not valid which may seem a little strange. This would be an attempt to argue that if humans caused the temperature to rise then the temperature would rise and cause climate change. This may sound reasonable and might even be true but it is not logically supported by the premise. In fact we cannot even draw the simpler conclusion ${\textstyle H\implies T}$ , which also sounds quite reasonable. The premise does not delve into the relationship between ${\textstyle H}$ and ${\textstyle T}$ . It only provides the conditions by which we can state that climate change is real.

Note however that a skeptic of climate change might try to negate H by using the invalidity of ${\textstyle (H\implies T)\implies C}$ . They may say that because the statement is invalid it must mean that humans are not the agents of climate change. This, while it might in fact be true, is really just another logical error, built on top of the first one. It can be countered by simply adding ${\textstyle \neg H}$ to the truth table and filling it in.

This premise, however, does not go very far in allowing us to see how misinformation can be inserted into climate change arguments from both sides (pro and con). Let’s use a different argument where the con side (the skeptic) claims that climate change is the result of natural causes rather than man-made ones. The pro side, of course, claims that climate change is the result of man-made causes. The pro side concedes that natural climate change can occur and let’s assume that the con side concedes the logic that if there were a human behavior that caused climate change, that it would in fact occur:

N = naturally occurring agent of warming temperatures (eg earth tilts a little toward the sun)
M = man made agent of warming temperatures
C = Climate change

Premise: ${\textstyle N\implies C}$ (if N then C)
Premise: ${\textstyle M\implies C}$ (if M then C)
We can combine this as: ( ${\textstyle M\lor N)\implies C}$ (if (M or N) then C)

The skeptic then concludes:

${\textstyle C\implies N}$ (if C then N)

The truth table:

${\textstyle N}$	${\textstyle M}$	${\textstyle C}$	${\textstyle M\lor N)\implies C}$	${\textstyle C\implies N}$
T	T	T	T	T
T	T	F	F	T
T	F	T	T	T
T	F	F	F	T
F	T	T	T	F (INV)
F	T	F	F	T
F	F	T	T	F (INV)
F	F	F	T	T

Similarly the pro side concludes:

${\textstyle C\implies M}$ (if C then M)

The truth table:

${\textstyle N}$	${\textstyle M}$	${\textstyle C}$	${\textstyle M\lor N)\implies C}$	${\textstyle C\implies M}$
T	T	T	T	T
T	T	F	F	T
T	F	T	T	T
T	F	F	F	T
F	T	T	T	F (INV)
F	T	F	F	T
F	F	T	T	F (INV)
F	F	F	T	T

Since there is at least one case where a true premise leads to a false conclusion, both the pro and con arguments are invalid.

The argument above requires both sides to concede the obvious proposition that if humans are doing something to cause climate change then climate change would be the result. Likewise it requires both sides to concede that if nature is doing something to cause climate change then climate change would be the result. Many folks will readily do this but some will not. Instead, they will construct an argument, logical or not, based on false premises. This strategy, of course, occurs for any argument, not just climate change, regardless of which side is taken (pro or con). These would need to be refuted at the factual level, by showing evidence that contradicts the erroneous belief.

Some propositions used at this level might include:

Claiming the existence of “experts” even though they are not trained in the subject at hand.
Inventing conspiracy theories to show that the opposing side is faking data, etc. to advance some ideological goal.
Incorrectly using data, ie passing off local data as global, etc.

and so on. In these cases, we could only try to identify the incorrect fact or method and provide the correct one.

Software for doing propositional logic

Some results from searching the web for logic software:

Proplogic: https://karinhowe.com/home/courses/kenny/software
Aris: https://www.gnu.org/software/aris/
Carnap: https://carnap.io/
The Logic Machine: https://logic.tamu.edu/

These are capable of evaluating logical expressions and doing/checking proofs. They have built-in a large number of standard logic rules (eg Modus Ponens). However, none of them seem capable of translating English propositions into logical syntax.

ChatGPT, however, can mostly do this to the extent of the examples above. It was able to reduce the English statements to logical statements (with variables and operators), produce a truth table for the premise/conclusion, and decide if the conclusion was valid. It would, however, sometimes make mistakes in calculating the values of logical statements. This may be similar to the known difficulty it has doing normal math computations.

Here is an example session:

Question:

Ok, that’s great. Now let’s try a new problem. The premise is: if there is a man-made behavior that causes warming temperatures or a natural phenomenon that causes warming temperatures then climate change is real. The conclusion is that if climate change is real then there is a natural phenomenon that causes warming temperatures. Can you do the truth table for this and show if it is a valid conclusion?

ChatGPT Answer:

Note that the 2nd and 3rd row from the bottom are incorrect for the last column ( ${\textstyle C\implies N}$ ):

The only case for which ${\textstyle C\implies N}$ is false is when ${\textstyle C}$ is true and ${\textstyle N}$ is false. For all other cases it is true. ChatGPT clearly erred here.

It would seem that ChatGPT gets the hard part right and but has a strange tendency to get the easy part wrong. We could probably use the software mentioned above for the easy part (computation) and ChatGPT for the hard part (translating the English wording to logic).

Another point to note is that questions posed to ChatGPT were worded carefully so as to maximize its chance of correctly translating to logic. Earlier I had tried a few loosely worded premises and ChatGPT would get the wrong answer or get confused and say it needed more information.