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In Elliot Mendelson’s “Introduction to Mathematical Logic”, he states, “Sentences may be combined in various ways to form more complicated sentences. We shall consider only truth-functional combinations, in which the truth or falsity of the new sentence is determined to be the truth or falsity of its component sentences”. This is exercise 1.4 (g): exercise

Something that is confusing me about this problem is that “x is positive” and “x^2 is positive” are predicates. They don’t have a truth values. If I specified that x was 2, then “2 is positive” would be a sentence, and so would “2^2 is positive”. I think the answer they want is:

p: x is positive

q: x^2 is positive

p \implies q

I just don’t see how we can do this since we have predicates and also “x is positive implies x^2 is positive” is also a predicate.

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    The exercise is about prop logic. Thus the form will be: "if P, then Q".
    – Mauro ALLEGRANZA
    Commented Aug 4 at 9:19
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    Formally any complex sentence with one unbound variable (or any number of variables really) can also be interpreted as a complex predicate, so you're right about that, but then the logical connective would not be explicitly represented (but be part of the predicate). The exercise just wants you to either write it like you did "p -> q" or probably as "(x > 0) -> (x^2 > 0)" or "P(x) -> Q(x)".
    – mudskipper
    Commented Aug 4 at 13:41

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"is positive" is a predicate.

"x is positive" is a formula with an unbound variable and it can stand in sentence position.

So you can write,

x is positive → x↑2 is positive

That also is a formula with an unbound variable that is true for any real number x.

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  • Thank you! I see this makes sense. Would assigning a propositional variable like p to “x is positive” be incorrect, since “x is positive” is a formula and not a proposition?
    – Dr. J
    Commented Aug 4 at 17:35
  • If you write P for "x is positive" and Q for "x squared is positive" then your conditional is P → Q. But then you can't look inside the symbols P or Q to see what they are about. You have lost the information that these formulas relate to the same x. The significance of writing "x is positive → x squared is positive" is that the same variable x appears both in the antecedent and consequent. So we can generalise and understand it as saying, for any (real number) x, if it is positive then its square is positive.
    – Bumble
    Commented Aug 5 at 3:06
  • I see. What do you mean by “ you can't look inside the symbols P or Q to see what they are about”? Are you meaning that we’ll see that P represents “x is positive” and we’ll see that Q represents “x^2 is positive” but we won’t know if it’s the same x? Am I understanding this correctly?
    – Dr. J
    Commented Aug 6 at 0:22
  • Yes, that's pretty much it. If we use 'P' and 'Q' as symbols to represent propositional variables, then writing P → Q is opaque. It's useful if we wish to do propositional calculus. We could use propositional calculus to show that P → Q is logically equivalent to ¬(P & ¬Q) without knowing anything about the internal content of P and Q. But if we wish to express "for any x, x is positive → x squared is positive" we need to be able to exhibit the internals of P and Q to make it clear that they contain the same x. This is why quantifier logic is much more expressive than propositional logic.
    – Bumble
    Commented Aug 6 at 5:23
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Recall the definitions in first-order logic (also known as predicate logic):

  1. Signature: You begin with a signature consisting of three types of symbols:

    • Constant symbols
    • Function symbols
    • Relation symbols (synonymous with "predicate" in this context)

    You also inherit other symbols from first-order logic in general: quantifiers, logical connectives, and variables.

  2. Term-formation rules: These rules allow you to build "terms" from the symbols:

    • A 'term' is any variable symbol, any constant symbol, or any (recursive) application of a function symbol to any term.
    • "Application" can be thought of as concatenation.

    Examples:

    • In "Polish notation": fxy is a term (applying function f to variables x and y)
    • In standard notation: f(x, g(c, y)) is a term (applying f to x and g(c, y), where g(c, y) is itself a term)

    The infinite application of term formation rules to a starting set of terms is called a term algebra or free algebra.

  3. Formula-formation rules: These rules create "linguistic units" eligible for truth-values, called formulae. There are two types:

    a) Atomic formulae formation:

    • Asserting the equivalence of two terms: "The Queen of Albania is my mom"
    • Asserting that a relation applies to certain terms: "IsPositive(x)"

    b) Composite formulae formation:

    • Using quantifiers and logical connectives to combine atomic formulae
    • Example: "For all x, IsPositive(x)"

It's crucial to understand that formulae are "pre-alethic" - they can become true or false, but inherently, they don't possess a truth value. For instance, we can't determine the truth of "The Queen of Albania likes snowboarding" without knowing the referents of "The Queen of Albania" and "likes snowboarding".

This is why sentences with unbound ("open") variables are still formulae, even though they don't yield an immediate truth-value. They're candidates for truth-values upon interpretation.

To clarify this process, consider these stages:

  1. Form terms
  2. Form formulae
  3. Map symbols in the signature to elements in a structure
  4. Assign variable symbols to values (elements in the structure)

At each stage:

  • Stage 1: Entities without assertions; truth-valuation is inapplicable
  • Stage 2: Syntactic form of assertions, but no meaning; potential for truth-valuation exists
  • Stage 3: Fixed meaning for all symbols, but free variables prevent full evaluation
  • Stage 4: Every sentence is determined as "true" or "false"

At stage 3, sentences with free variables still determine sets of elements for which the sentence is true or false. For example, "x2 = 22" (where x is a real number) correlates with the set {√22, -√22}.

Sentences without free variables can be evaluated at stage 3, without waiting for variable assignment in stage 4.

To answer your specific question: "x is positive" is an (open) formula at stage 2. It requires mapping to a structure (stage 3) to determine the set of elements for which it's true, but the sentence itself isn't true or false until stage 4.

Here's a comparison between natural language, formal logic, and metaphysics:

Natural language Formal Logic Metaphysics
Determined nouns, proper nouns Constants Specific things
Undetermined nouns, pronouns Variables A class of things
Relative clauses Functions Relationships
Noun phrases Terms Things
Copula Equality Identity
Predicates Relations Properties
Clauses Atomic formulae Facts
Sentences Composite Formulae Facts
Pragmatically unresolved sentences Open formulae Proto-propositions; conditions?
Pragmatically resolved sentences Closed formulae Facts

Note: This comparison is a work in progress and open to refinement.

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