Can anyone tell which one of these sentences are materially implies(implication) and logically implies(implication)?

Question

these are the contexts:

1):

If the number 0 has a certain property, and any natural number a has the property implies that a +1 also has the property, then every natural number has the property

2):

If a theory implies some phenomena and if observation shows that these phenomena do not happen, then we have no recourse but to conclude that the theory is false. Theory ⇒ false phenomena

3):

particles do not have free will ⇒ human beings do not have free will. From a scientific perspective this is not strange at all. After all, human beings are made out of particles. Abiding by the usual dictum of reductionism, scientists would have to say that the tendency of particles to follow the habitual laws of physics implies that humans must follow the habitual laws of physics.

4):

One of the main tools of logic is the law called modus ponens. This law says that if a statement P is true and the statement “P implies Q” is true, we can then derive that the statement Q is true. In symbols, we write this as

5):

Inside every computer there are literally billions of logical switches that perform the logical operations AND, OR, NOT, and IMPLY.

6):

It is not clear how an experimenter’s free will is impeded by the fact that a photon has knowledge of what freewill choice the experimenter will make. Even if the experimenter had knowledge of future choices, does that imply a lack of free will to choose?

7):

By Pythagoras’s famous theorem for a right triangle we have x2 + y2 = z2 or . both x and y have length 1. That implies that the diagonal, z, has length √2.

8):

Consider the predicate F(x) ≡ |x| → M. F(x) is true only if the logical sentence that corresponds to the number x implies M.

9):

A mathematical statement is a mathematical fact that can be put into symbols. We saw above that arithmetization is a correspondence between mathematical statements and the natural numbers. This implies that there are countably infinite mathematical statements.

I have to determine each one of these implies are materially implied or logically implied. As I know, material implication is when there are a conditional statement and means: if p is true, then q is also true. and there is not a causal relationship between them. but for logical implication, is a concept which describes the relationship between statements that hold true when one statement logically follows from one or more statements and A logically implies B if and only if any assignment that makes A true also will make B true. so, I am not an expert and still learning this stuff, so I will explain my guesses in a not formal way:

1) M. because I think this "implies" is written in this way:

∀P((P(0) ∧ ∀n(P(n) → P(s(n)))) → ∀nP(n))

2) L. because It has a causal relationship.

3) L. because It has a causal relationship.

4) M. because as I remember, this is the case in Modus ponens.

5) M. in this case I am really just guessing.

6) L. because It has a causal relationship.

7) L. because It has a causal relationship.

8) M. I am a little confused about this one. but since in the formalization it uses → and not a ⇒ , I'm going with Materially implies.

9) L. because It has a causal relationship

Can you say if my guesses are right or wrong?

Answer 1

Firstly, it would help to pin down what you mean by logical implication. It could mean syntactically that the antecedent proves the consequent, or semantically that all models of the antecedent are models of the consequent, or perhaps you have in mind some more informal notion that the conditional is necessarily true. Logical implication does not mean a causal relationship, so the edited version of your question rather confuses things.

Some of your examples are mathematical in nature, so they will depend on your preferred understanding of the philosophy of mathematics. If you consider mathematics to be reducible to logic, or at least that mathematical theorems are logical truths, or are necessarily true in the relevant sense, then I would say that 7 is a logical implication, provided we are assuming euclidean geometry. Number 9 also would come out as a logical implication provided we fill in the gaps about how mathematical formulas are defined recursively and how the arithmetization works. Number 1 is of course a statement of the rule of induction, and is usually taken to be an axiom schema, though it need not be assumed, and some arithmetics such as Robinson arithmetic do not include it, so it probably should not qualify as a logical implication. Number 8 is not entirely clear to me, though it appears to be saying something like the Gödel number of a sentence satisfies a predicate F iff the sentence whose Gödel number it is implies M. If this is the intended interpretation, then I don't see why the conditional need be stronger than a material implication: all one needs to claim about this sentence is that it holds true for all values of x.

The others all appear to be material implications. Number 3 is at best a claim about physical or natural necessity, and is likely just straightforwardly false. Particles do not fall in love, read books, make moral judgements, or die, but human beings do. All kinds of properties are emergent, or are properties of complex systems, or are supervenient upon fundamental physical properties. Number 6 is not clear: photons are not the kind of thing that can have knowledge, and it is debatable as to whether it is possible for a person to have knowledge of future choices, or whether knowledge precludes freedom of choice.

Answer 2

Implication ("If-then"), Material Implication, Material Equivalence, Logical Implication, Logical Equivalence!

P -> Q means: P 'implies' Q, where 'implies' can be rendered into the following conditional form: "If P, then Q".

The symbol (->) denotes "material implication", which sets up a sufficient condition between P and Q, such that P materially implies Q: that is, if P is the case (i.e., true), then Q follows from P. The material implication is the most logical sense of an implication; it is that implication is the "lowest common denominator of all sorts of implications (i.e., if-then statements). The material implication is that implication which is in common in all implication: the sufficiency of P for Q and equivalently to the necessity of Q for P.

The material conditional: P ->Q (If P then Q') logically entails the following:

• P => Q: P is a sufficient condition for Q.

• Q <= P: Q is a necessary condition for P.

                   **Material vs. Logical Implication**
  

Given the material conditional P -> Q, P is referred to as "antecedent" and Q is referred to as consequent in this form (forward implication from P to Q). Given the material conditional (if-then) statement (P -> Q), which can be stated equivalently as "Q if P", which in its turn is equivalent to stating "P only if Q", which implies that P is a sufficient condition for Q, which is represented as follows: P => Q.

• Original Implication: P -> Q; this means 'P is sufficient for Q'
• Converse of Original: Q -> P; this means 'Q is sufficient for P'

An implication and its converse taken together establish material equivalence between P and Q; that is, the conjunction ("and") of a material conditional [P -> Q] and its converse [Q -> P] yields a material biconditional [P <=> Q], which reads "P if and only if Q", and the bidirectional implication connective/truth-function/operator (<=>) is called "material equivalence". The operator (<=>) is also called 'iff' (which stands for 'if and only if').

The material equivalence (<=>) relation is a both necessary and a sufficient condition.

The "Original" Implication ("Forward Implication) (P -> Q) = ("If P, then Q") = ("Q if P") = (P only if Q), which sets up the sufficiency of P for Q: P => Q.

The Converse of the "Original" Implication ("Reverse Implication") (Q -> P) = ("If Q, then P") = ("P if Q") = ("Q only if P"), which sets up the sufficiency of Q for P: Q => P.

P is materially equivalent to Q 'if and only if' P is a sufficient condition for Q, and likewise, Q is a sufficient condition for P. P is materially equivalent to Q iff P and Q materially imply one another.

Therefore,

• (P <=> Q) = ('P if Q' AND 'P only if Q) = (P if and only if Q)

• (P <=> Q) = ('P is both a necessary and a sufficient condition for Q').

   **The Difference Between Logical and Material Equivalence:**
     (Logical Equivalence is a Subset of Material Equivalence!)
  

In the case of material equivalence (P <=> Q), P and Q must materially imply one-another; where the term "implies" is to be understood as setting up the sufficiency of the antecedent (P, "if-part" of conditional) for the consequent (Q, "then-part' conditional). Likewise, in the case of logical equivalence (P ≡ Q), P and Q must logically imply one-another; where "logically implies" (Log: P --> Q) means the antecedent (P) logically entails the consequent (Q), which can be restated as follows: P logically implies Q means that "The consequent Q is a logical consequence of the antecedent P".