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

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?

• Use the definitions : "material implication" must refer to the conditional connective : it must be used in the formal language to build compelx formulas with simpler ones. – Mauro ALLEGRANZA Jun 20 at 5:59
• "Logical implication" is a relation between a set of sentences/formulas (premises) and a single sentence (conclusion) and is the formalization of the concept of valid deductive argument. – Mauro ALLEGRANZA Jun 20 at 6:00
• In both cases, no "causal relationship ". – Mauro ALLEGRANZA Jun 20 at 6:00
• So, in a nutshell, if you are "modelling" a single sentence, you need the connective, while if you are modelling an argument, you need the relation of consequence. – Mauro ALLEGRANZA Jun 20 at 6:04
• But there is a "trick" : if Γ is a finite set of formulas, say { a,b,c }, we prove that : Γ ⊨ φ iff (a ∧ b ∧ c) → φ. – Mauro ALLEGRANZA Jun 20 at 6:14