Help understanding deductive arguments

Question

I am currently finding it hard to understand deductive arguments. I am taking a module called 'Effective Reasoning'. I've been finding it so hard to understand some things (as simple as this).

I decided to read the provided textbook to understand better (Salmon Merrillee - Introduction to logic and critical thinking) but still have some questions.

The definition of a deductive argument according to the book: "In a deductive argument if the premisses are true the conclusion cannot be false".

I am a little confused with the part where it says that the conclusion cannot be false if the premises are true. Consider a deductive argument with a true premise, then what they are saying is that whatever the conclusion is, it will be true (?) then wouldn't any conclusion be true - even one that is obviously false? I'm sure they do not mean it this way, but can someone define it in simpler terms?

Given some examples:

How exactly do we check if a premise guarantees the truth? E.g.

Jack is a bachelor
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(Conclusion) Jack has no wife

Is this how I should approach the argument?: Look at the premise "Jack is a bachelor" and derive the meaning: bachelor means "a man who is not and has never been married." (from dictionary)

And then I look at the conclusion "Jack has no wife". This matches the premise, hence it is a deductive argument?

And what about this example?

Whatever is done as an expression of love is morally acceptable.

Mrs. X, who believed her child's soul was possessed by demons that
could be driven out only by beating the child, beat her child severely
because she loved him.
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(Concl.) Mrs. X did something morally acceptable when she beat the child.

According to the original definition of a deductive argument, I would think that the above is a sound argument.

Why is the sentence false? Considering the first premise says that:

1)Whatever is done as an expression of love is morally acceptable.

and the second premise says that Mrs X. beat her child .... because she loved him.

However, the textbook shows this, which I do not quite understand.

Answer 1

I think what's confusing you is the distinction between syntax and semantics. Logic is entirely concerned with syntax: with the rules governing the structure and transformation of symbols. Semantics - the meaning of symbols, and the relationship of symbols to the external world — is a separate matter, one that lies outside of logic properly put. 'Truth' is a semantic property, not a logical one; we can assert that a premise is true or false, but we can't prove it. Logic allows us to transform the truth value of our premises in order to demonstrate that other statements must be true if the premises are true.

A deductive argument — a chain of connected propositions — that follows all of the syntactic rules of logic is called valid, meaning that we can properly demonstrate that conclusions are true or false with respect to the truth or falsehood of the premises. So when your textbook says:

In a deductive argument if the premisses are true the conclusion cannot be false.

It means that a properly constructed deductive argument, if the premises are (semantically) true, then the conclusions must be (semantically) true, because the argument is (syntactically) valid.

To use your example:

• "Jack is a bachelor": This is an assertion about the real world (a matter of semantics). It may be true that jack is a bachelor, or it may be false, but we are asserting it as true.
• "'Bachelor' means 'has no wife'": This is a definition, which is a valid syntactic structure.
• "Jack has no wife": This is an extension of the definition across an identity (equivalent to 'a=b and b=c so a=c'), which is also a valid syntactic structure.

Because each step in the process is syntactically valid, then the conclusion must be true if the premise is true: e.g., if Jack is a bachelor, then Jack is unmarried.

Answer 2

I take it your "not quite understand" refers to the method of proving something false by starting with a statement of that something along with some known truths and deducing a false conclusion.

A classic example of this method is the mathematical proof that the square root of two is an irrational number:

Premises:

• A: Definition: Any rational number can be written as a/b, where a and b are relatively prime integers.
• B: Definition: 2X is even for any integer X.
• C: Known: if a is odd, then a² must be odd (implying that if a² is even, then a must be even).
• D: Assumed: √2 is rational.

Deductions:

• 1. We have (#A and #D): √2 = a/b
• 2. Square: 2 = a²/b²
• 3. Rearrange: 2b² = a²
• 4. (#C): since 2X is even, a² must be even, so a must be even, say 2k
• 5. This means: 2b² = (2k)² = 4k²
• 6. Or: b² = 2k²
• 7. (#B): 2k² is even, so b² must be even
• 8. (#C): b must be even

Conclusion:

• #4: a is even
• #8: b is even

Contradiction:

• #A: a and b are relatively prime, so they can't both be even.

Result:

• We deductively proved something that is false, so at least one of our original premises must be false.
• All premises but D were known to be true.
• D must be false.

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Answer 3

What you seem not to understand is the use one can make of the concept of deductively valid argument.

(1) First case. You have premises and do not really care whether they are true or not. What you want to know is which conclusion can be derived from these premises, using logic ( that is, which conclusion would follow in case the premises were true). The goal here is simply to learn logic.

Example: (1) if pigs can fly, then the moon is made of cheese (2) pigs can fly (3) therefore ????

(2) You have premises and believe ( or even know) that they are true; you use logic to find which conclusion ( which " new truth") can be derived using logic . The goal here is to discover " new truth", to extend our knowledge ( using logic).

(1) If this man is guilty, he has left town after the crime (2) this man has not left town after the crime (3) therefore ???

(3) You are presented with an argument the conclusion of which you know is false, or cannot believe is true. Here , you can use logic

• to show that the argument is not valid , saying " ok, the premises are true, but the conclusion does not follow, for the reasoning is not valid, does not obey the rules of valid reasoning; so I am not " compelled" to accept the conclusion "

Example : (1) if God exists, natural events are governed by laws (2) natural events are governed by laws (3) therefore God exists

• to show that , since the argument is valid, there must be someting wrong in at least one of the premises ( for, the argument being valid, in case the premises were true, the conclusion would also be true") --> this is the case which corresponds to the example given in the handbook.