Can logic be used to prove any belief?
It depends what you mean by "prove."
Two different concepts in logical argumentation are soundness and validity.
To better explain soundness and validity, I will use an analogy:
In January of 2004, mars rovers “spirit” and “opportunity” landed on mars. 8 years later, another rover, “Curiosity,” landed. Curiosity was equipped various instruments for collecting data:
- Video cameras for capturing photographs
A Rock Abrasion Tool (RAT). The outside of a boulder, exposed to martian air, is chemically different from the inside of the rock; like a candy wrapper surrounding a candy. The RAT was able to create holes 45 mm (about 2") in diameter and 5 mm (1/5") deep in the rock so that the inside could be studied.
Logic can be thought of as "what can a mars rover do if you turn off its video-cameras, spectrometers, and other data collectors." Logic is closely related to the study of what a computer can compute. Logic is also the study of how to take some information as input, and output new information.
For example, Bob is allowed a 30 minute break at the middle of his work shift. He wants to know what the clock will say when it's time to go back to work.
- INPUT 1.... break begins at 8:20PM
- INPUT 2 .... break is 30 minutes long
- OUTPUT ...... break will end at 8:50PM
If humans turn off their metaphorical sensors (ears, eyes, sense of touch, etc...), then humans can still make new inferences. This is because old data can be recombined to give us new data. Bob did not initially know that break time ended at 8:50PM. Although the calculation is easy to make, it still takes computational effort. to add 30 minutes to 8:20PM. Although it may happen in a fraction of a fraction of a second, even an electronic man-made computer still requires a modicum of time to compute 2+3.
- loosely speaking an argument is sound if the data collected by the video-cameras, microphones, tactile sensors, etc... was correct.
- an argument valid if we pretend the input data is correct, and use high quality reasoning to infer new facts from old facts.
Let us look at an example of a valid, unsound argument. In the year 2018, Emily Levine gave a TED talk titled, "How I made friends with reality." Ms. Levine made the following argument:
- For any person, if that person is afraid of death, then that person is anti-woman (axiom)
- I am a person (axiom)
- If I am afraid of death, then I am anti-woman (from lines 1 & 2)
- If I am anti-woman, then I am not a feminist (axiom)
- I am a feminist (axiom)
- Therefore, I am not afraid of death (output/result)
Although Emily Levine's argument is ridiculous, it is 100% valid. The argument is valid, but unsound. If the inputs were correct, then the output would be correct. However, the inputs are wrong. Notably, the input that afraid of death does not make a person "anti-woman."
Part of the original Emily Levine is as follows,
"my real problem with the mindset that is so out to defeat death is if you're anti-death, which to me translates as anti-life, which to me translates as anti-nature, it also translates to me as anti-woman, because women have long been identified with nature."
It is true that women have long been identified with nature. However, the rest of that stuff is nonsense. In the past, there have existed women, some of whom were ardent feminists (very pro-woman), who did not want to die, but got old, got sick, and died anyway.
The following are two definitions of "proof:"
- proof(1)...... an argument is a proof if it is logically valid and logically sound.
- proof(2)...... an argument is a proof if that argument is logically valid. The argument may or may not be logically sound.
Can logic be used to prove any belief?
logic can be used to type-2 prove any belief. That is, for any belief
B there exists a 100% valid logical proof that
B is true.
Without loss of generality, take belief
B be "all horses have wings, eat rainbows, and live in Australia."
- If at least one person on planet earth ate a hot dog before the year 2020 then all horses have wings, eat rainbows, and live in Australia
- least one person on planet earth ate a hot dog before the year 2020
- From line 1, line 2, and modus ponens, all horses have wings, eat rainbows, and live in Australia.
The above was a completely valid proof that all horses have wings, eat rainbows, and live in Australia. The proof was unsound.
However, logic cannot be used to type-1 prove any belief. That is, there exist beliefs
B such that every valid proof of
B is unsound.
To correctly answer a question like, "Can logic be used to prove any belief?" you must first decide what it means to "prove" something.
The statement "logic can be used to prove any belief" contains a grain of truth if you are looking for logical validity only, and you ignore soundness.
Logic is like a coffee grinder. You put coffee beans into the machine and coffee grounds come out. For the logic machine, you put feed data in, and new data comes out. Logic does not care where the input facts came from, or whether those input facts are correct or not. You could put corn in the coffee grinder, and maybe the output would be ground corn. Garbage-in, garbage out.