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I'm having trouble understanding how to determine the logical form of a statement.

Here's my current understanding (I'm not sure if all of them are correct though):

  • p is a propositional variable means that p can be any proposition, so the Truth Value of p has 2 possibilities: T or F(but we don't know which one yet until we assign p to be a certain statement). I think it's kind of analogous to numerical variables, like let x be a natural number, then the "evenness" of x has 2 possibilities: even or odd (but we don't know which one yet until we assign a value to x, like 4 or 123...)

  • A logical form is an expression consist of the component statement variables and the logical connectives, and the truth table of logical form shows all the possible truth values of the form for all combinations of Truth Values of the statement variables. I like to think of the truth table for the logical form to be what all statements of that logical form mean logically (replacing the propositional variables with the respective component statements, of course).

If my understanding of propositional variable and logical form is correct, then here's what I don't understand: when determining the logical form of a statement, the general procedure is to first identify the component statements and give them names as variables, then translate the whole statement into an expression consist of the propositional variables and appropriate logical connectives. For example, say "the car is black and the bike is white". When identifying the component statement, the solution always says that denote p:= the car is black and q:= the bike is white. How can that work?

I thought what I should do is to completely remove the component statements and replace them with 2 propositional variables p and q, which results in "p and q", and then set up the truth table for that logical form and get to the logical form which is p∧q. If we denote p and q to be the component statements, then p and q are not propositional variables anymore, right? therefore, there will be no truth table for the initial statement (because "p and q" now is not a logical form, it's a statement so it can only have a single truth value). Then how can we determine what is the logical connective for the initial statement's logical form?

I guess I also want to ask is that can there be a truth table for a statement? I thought truth tables only exist for logical forms.*

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  • $\begingroup$ i put the keywords form, statement, variable to italic and hope this makes more readable. If you dont like this you can rollback the changes. $\endgroup$ – miracle173 Feb 20 at 4:31
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  • truth tables are for logical forms.

  • statements can only be true or false at one time

  • if the structure of a statement can be expressed in a logical form containing one or more logical connectives, then there will be more than 2 rows to the truth table of that logical form. However the statement will only correspond to one row of the truth table for which the component statements have the correct truth value

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The notions of logical form and truth table really only make sense when speaking about formulas where the component statements are undetermined and independent, which means each component can be true and false without influencing the truth value of other components. In particular, the truths value of the components are not fixed a priori.

This reading of (atomic) components as undetermined and independent is made explicit by the use of variables: If I use different variables $p$ and $q$ in a formula, I signal to the reader that all $4$ combinations of truth values for $p$ and $q$ can occur.

Sometimes people use descriptive statements such as "the car is black" or "the bike is white" as components. This is indeed a source of confusion, as it might suggests that the truth values are fixed -- we only have to look at the world to find out whether the car is black, or the bike is white and so on. In this fixed-value interpretation, it may make sense to say that the statement "the car is black and the bike is white" has the linguistic form of a conjunction, but from a logical perspective it is actually trivial: Either the statemenent is true, in which case it is equivalent to $\top$ (or any other tautology) or it is false and therefore equivalent to $\bot$ (or any other contradiction).

So long story short, if you are asked to determine the logical form of "The car is black and the bike is white" it is implicitly assumed that the components "The car is black" and "the bike is white" do not have a fixed truth value (and are independent), i.e. they behave like variables. Keeping this in mind, it is still fine to use "the car is black" as a variable name instead of something less descriptive like "p".

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  • $\begingroup$ So if "the car is black" behaves like a variable now(and can be a variable name), then the whole p:=" the car is black" is just to make the variable name more compact right? I also have another question: Since a component statement can be denoted by any variable name, does that means that a statement can have multiple logical forms? $\endgroup$ – ncc291203 Feb 23 at 4:22
  • $\begingroup$ Yes to your first question. Concerning the second, I think when speaking about logical forms one is usually willing to identify logical formulas which differ only in an inessential way. The naming of variables would be such an inessential difference (the bracketing of long conjunctions might be another one: Is "the car is black and the bike is yellow and snow is white" of the form $(p\land q)\land r$ or of the form $p\land(q\land r)$?). These matters depend on convention, and consequently there is no completely rigorours definition of "the logical form of a natural language statement". $\endgroup$ – Timo Feb 23 at 8:07
  • $\begingroup$ I just have one more question, somewhat related to my second question above: When p and q are the same statement, like "the Sun is white", then p∨(¬q) is the statement "the Sun is white or the Sun is not white". But the logical form for that statement, using the above procedure, will be p∨(¬p). So which one is the right form for that statement? I think p∨(¬p) is the right form, but I don't know if p∨(¬q) is a also the right form or not (I mean from the definition of logical form I don't think there's anything wrong with it) $\endgroup$ – ncc291203 Feb 23 at 15:15
  • $\begingroup$ The same statements must be replaced by the same variable - this is important (otherwise you get formulas which are not even equivalent). So $p\lor\lnot p$ is the right choice here. $\endgroup$ – Timo Feb 23 at 16:57

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