Not actually Philosophy, but Logic.

Sojourner<>< said:

That being so what is the formal syntax for "A if B and B if A"?

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Assuming this statement is bracketed as "(A if B) and (B if A)", you are expressing the conjunction of two conditionals. "A if B" becomes "B=>A" and "B if A" becomes "A=>B". Their conjunction is then

"(A=>B) & (B=>A)"

This statement is equivalent to "A <=> B". In fact, in most formalisations of the propositional calculus, "A <=> B" is just shorthand for "(A => B) & (B => A)"

Statements in the form of A only if B are statements of implication. (usually denoted by something like A->B or A=>B) "=>" is an binary operation on propositions with the truth value:

A B A=>B
T T T
T F F
F T T
F F T

In common English. A statement in the form of: If A then B is ONLY false if A is true and B is false. A=>B means if A is true you are absolutely guarantied that B is also true. A=>B isn&#8217;t a statement about A or B, it&#8217;s a statement about a relationship between A and B.

For example:
If My name is Jon then I own a cat &#8211; is true since my name is Jon and I do own a cat

If I wear glasses then all fish have ten legs &#8211; is false since I do wear glasses, but fish don&#8217;t have legs

If I own 50 million dollars then i am posting on Christianforums.com &#8211; is true since I don&#8217;t own 50 million dollars, but I am posting on CF

If all cows are pink then all turtles are purple &#8211; this is also true since cows aren&#8217;t pink and turtles aren&#8217;t purple.


A if and only if B means that A and B have both of the following relations: A => B and B=>A.

JonF said:

In common English. A statement in the form of: If A then B is ONLY false if A is true and B is false. A=>B means if A is true you are absolutely guarantied that B is also true. A=>B isn&#8217;t a statement about A or B, it&#8217;s a statement about a relationship between A and B.[/SIZE][/FONT]

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And yet it seems perverse to agree to the following statements:

"If the world is approximately 1000 miles in diameter, then London is the capital of France."

"If 1+1=2 then George Washington was the first president of the United States."

Or consider asking the question: "Who won the last American presidential election?" A correct answer is apparently:

"Well, if the wind blew in a south-easterly direction in Taiwan in 1956, then George Bush did."

These seem strange because the above truth table does not generally capture the meaning of "if...then..." statements, and to stress this, the connective "=>" is often referred to as the material conditional. Conditionals in everyday English often express causal connections or relations of logical entailment, neither of which factor into the semantics of the material conditional. Furthermore, many "if...then..." constructions, namely counterfactual claims, require the subjunctive mood:

"Had America not entered World War 2, Germany would have won."

Consider the components of this conditional:

"Had America not entered World War 2"
"Germany would have won."

Both of these are grammatically incomplete, making it clear that the whole statement (a subjunctive conditional) cannot be analysed into independent components as required by the material conditional.

The reason the material conditional is still important is not because it is an accurate reflection of English usage, but because it suffices to express a variety of propositions in certain domains, such as mathematics. In my experience, this point is not adequately explained in the textbooks, leaving students to believe the truth-table above is a thorough and correct analysis of "if...then..." constructions. Instead, the meaning of those constructions has been weakened, in order to make "=>" truth-functional and hence give it a simple semantics (a truth-functional connective is one whose truth-value is uniquely determined by its component propositions). The drawback is that we must be careful in interpreting it, lest we think there is some genuine causal or logical connection between "4=5" and "The moon is green cheese". To that end, I suggest reading "A => B" as its equivalent disjunction

"Either A is false, or B is true, or both."

For example:
If My name is Jon then I own a cat &#8211; is true since my name is Jon and I do own a cat

If I wear glasses then all fish have ten legs &#8211; is false since I do wear glasses, but fish don&#8217;t have legs

If I own 50 million dollars then i am posting on Christianforums.com &#8211; is true since I don&#8217;t own 50 million dollars, but I am posting on CF

If all cows are pink then all turtles are purple &#8211; this is also true since cows aren&#8217;t pink and turtles aren&#8217;t purple.

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While your analyses here are correct according to the semantics of "=>," they read quite perversely, and you might see how these kinds of assertion could be a source of humour.

The lesson then is not to take the truth-table for the material conditional too seriously. While great for mathematics and other limited contexts, it is woefully deficient at capturing the meaning of English conditionals, which are better analysed in a more expressive modal logic.

lawtonfogle said:

This is a logic question, and knowing not where it went, I have placed it here.

What does "A only if B" mean?

Does it mean "A if and only if B", or "If B, then A"?

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It would mean:

-(A&-B)

lawtonfogle said:

This is a logic question, and knowing not where it went, I have placed it here.

What does "A only if B" mean?

Does it mean "A if and only if B", or "If B, then A"?

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The first.

sparklecat said:

I disagree.

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Those were the lines along I was thinking, too, so at first I wondered why you objected. I think I see my mistake: I didn´t pay attention. I responded to "only if", and not to "if and only if".

quatona said:

Those were the lines along I was thinking, too, so at first I wondered why you objected. I think I see my mistake: I didn´t pay attention. I responded to "only if", and not to "if and only if".

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I don't disagree, then.

lawtonfogle said:

This is a logic question, and knowing not where it went, I have placed it here.

What does "A only if B" mean?

Does it mean "A if and only if B", or "If B, then A"?

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"A only if B" means that B is a necessary condition of A and that A is a sufficient condition of B.

In other words, it is the equivalent of "if A, then B".

It is easy to solve these problems by substituting concrete events for the letters.

"I will go out only if it is sunny."
This is equivalent to: "If I go out, then it is sunny."

A is a sufficient condition for B to happen. This means that if A is true, then B also is.
B is a necessary condition for A to happen. If B is false, so is A.

If I'm right, then both of the options you gave are incorrect.
"If A, then B" is different from "If B, then A"- "If I go out, then it is sunny" is different from "If it is sunn, I will go out"

It is also different from "A if and only if B" - in this case A is both a necessary and sufficient condition of B, and vice-versa, which is not what the original proposition states.

JonF said:

The truth table accurately captures the meaning of a conditional in everyday English. The truth value of a conditional is not a clam about either the antecedent's truth value or the consequence's, but merely the relation between the two.

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It is spelled "consequent", and your last sentence is utterly confused.

A truth value is not a claim of anything. A proposition (a conditional in this case) is a claim. Material conditionals are truth functional, so they can be interpreted entirely as restrictions on the possible combination of truth values taken by the antecedent and consequent. However, strict conditionals often express a non-truth functional relationship between the content of the antecedent and consequent (such as a causal relationship, or logical entailment) which cannot be captured by any truth-functional connective.

"If the water in the pot is at 50 degrees celsius, then it is boiling."

If the majority of English speakers understand that this sentence is always false, then it cannot be a material conditional, and therefore the material conditional does not always capture the meaning of everyday "if...then..." constructions.